Recent zbMATH articles in MSC 13https://zbmath.org/atom/cc/132024-09-13T18:40:28.020319ZUnknown authorWerkzeugFrieze matrices and friezes with coefficientshttps://zbmath.org/1540.050232024-09-13T18:40:28.020319Z"Maldonado, J. P."https://zbmath.org/authors/?q=ai:maldonado.juan-pabloSummary: Frieze patterns are combinatorial objects that are deeply related to cluster theory. Determinants of frieze patterns arise from triangular regions of the frieze, and they have been considered in previous works by \textit{D. Broline} et al. [Geom. Dedicata 3, 171--176 (1974; Zbl 0292.05009)], and by \textit{K. Baur} and \textit{R. J. Marsh} [J. Comb. Theory, Ser. A 119, No. 5, 1110--1122 (2012; Zbl 1239.05031)]. In this article, we introduce a new type of matrix for any infinite frieze pattern. This approach allows us to give a new proof of the frieze determinant result given by Baur-Marsh [loc. cit.].Kazhdan-Lusztig polynomials of braid matroidshttps://zbmath.org/1540.050292024-09-13T18:40:28.020319Z"Ferroni, Luis"https://zbmath.org/authors/?q=ai:ferroni.luis"Larson, Matt"https://zbmath.org/authors/?q=ai:larson.mattSummary: We provide a combinatorial interpretation of the Kazhdan-Lusztig polynomial of the matroid arising from the braid arrangement of type \(\mathrm{A}_{n-1}\), which gives an interpretation of the intersection cohomology Betti numbers of the reciprocal plane of the braid arrangement. Moreover, we prove an equivariant version of this result. The key combinatorial object is a class of matroids arising from series-parallel networks. As a consequence, we prove a conjecture of \textit{B. Elias} et al. [Adv. Math. 299, 36--70 (2016; Zbl 1341.05250)] on the top coefficient of Kazhdan-Lusztig polynomials of braid matroids, and we provide explicit generating functions for their Kazhdan-Lusztig and \(Z\)-polynomials.Move-reduced graphs on a torushttps://zbmath.org/1540.051212024-09-13T18:40:28.020319Z"Galashin, Pavel"https://zbmath.org/authors/?q=ai:galashin.pavel"George, Terrence"https://zbmath.org/authors/?q=ai:george.terrenceSummary: We determine which bipartite graphs embedded in a torus are move-reduced. In addition, we classify equivalence classes of such move-reduced graphs under square/spider moves. This extends the class of minimal graphs on a torus studied by \textit{A. B. Goncharov} and \textit{R. Kenyon} [Ann. Sci. Éc. Norm. Supér. (4) 46, No. 5, 747--813 (2013; Zbl 1288.37025)], and gives a toric analog of \textit{A. Postnikov}'s [``Total positivity, Grassmannians, and networks'', Preprint, \url{arXiv:math/0609764}] and \textit{D. P. Thurston}'s [Proc. Cent. Math. Appl. Aust. Natl. Univ. 46, 399--414 (2017; Zbl 1407.52021)] results on a disk.Waring numbers over finite commutative local ringshttps://zbmath.org/1540.111332024-09-13T18:40:28.020319Z"Podestá, Ricardo A."https://zbmath.org/authors/?q=ai:podesta.ricardo-a"Videla, Denis E."https://zbmath.org/authors/?q=ai:videla.denis-eSummary: In this paper we study Waring numbers \(g_R(k)\) for \((R, \mathfrak{m})\) a finite commutative local ring with identity and \(k \in \mathbb{N}\) with \((k, | R |) = 1\). We first relate the Waring number \(g_R(k)\) with the diameter of the Cayley graphs \(G_R(k) = C a y(R, U_R(k))\) and \(W_R(k) = C a y(R, S_R(k))\) with \(U_R(k) = \{x^k : x \in R^\ast \}\) and \(S_R(k) = \{x^k : x \in R^\times \}\), distinguishing the cases where the graphs are directed or undirected. We show that in both cases (directed or undirected), the graph \(G_R(k)\) can be obtained by blowing-up the vertices of \(G_{\mathbb{F}_q}(k)\) a number \(| \mathfrak{m} |\) of times, with independence sets the cosets of \(\mathfrak{m}\), where \(q\) is the size of the residue field \(R / \mathfrak{m}\). Then, by using the above blowing-up, we reduce the study of the Waring number \(g_R(k)\) over the local ring \(R\) to the computation of the Waring number \(g(k, q)\) over the finite residue field \(R / \mathfrak{m} \simeq \mathbb{F}_q\). In this way, using known results for Waring numbers over finite fields, we obtain several explicit results for Waring numbers over finite commutative local rings with identity.Iwasawa-Cohen-Lenstra heuristicshttps://zbmath.org/1540.111482024-09-13T18:40:28.020319Z"Greither, Cornelius"https://zbmath.org/authors/?q=ai:greither.corneliusSummary: In this note we propose an analog of the well-known Cohen-Lenstra heuristics for modules over the Iwasawa algebra \(\Lambda\). It turns out that only the analog of the real-quadratic situation leads to a convergent series and hence to potential predictions. We determine the sum of this series, which runs over all isomorphism classes of finite \(\Lambda\)-modules, and we discuss the partial sum that arises by restricting to cyclic \(\Lambda\)-modules. We demonstrate that this subsum is almost as large as the total sum. No attempt is made to test the heuristics numerically.
For the entire collection see [Zbl 1462.11006].Irreducibility properties of Carlitz' binomial coefficients for algebraic function fieldshttps://zbmath.org/1540.111722024-09-13T18:40:28.020319Z"Tichy, Robert"https://zbmath.org/authors/?q=ai:tichy.robert-franz"Windisch, Daniel"https://zbmath.org/authors/?q=ai:windisch.danielSummary: We study the class of univariate polynomials \(\beta_k (X)\), introduced by Carlitz, with coefficients in the algebraic function field \(\mathbb{F}_q (t)\) over the finite field \(\mathbb{F}_q\) with \(q\) elements. It is implicit in the work of Carlitz that these polynomials form an \(\mathbb{F}_q [t]\)-module basis of the ring \(\operatorname{Int}(\mathbb{F}_q [t]) = \{f \in \mathbb{F}_q (t) [X] | f(\mathbb{F}_q [t]) \subseteq \mathbb{F}_q [t]\}\) of integer-valued polynomials on the polynomial ring \(\mathbb{F}_q [t]\). This stands in close analogy to the famous fact that a \(\mathbb{Z}\)-module basis of the ring \(\operatorname{Int}(\mathbb{Z})\) is given by the binomial polynomials \(\left(\begin{smallmatrix} X \\ k \end{smallmatrix}\right)\).
We prove, for \(k = q^s\), where \(s\) is a non-negative integer, that \(\beta_k\) is irreducible in \(\operatorname{Int}(\mathbb{F}_q [t])\) and that it is even absolutely irreducible, that is, all of its powers \(\beta_k^m\) with \(m > 0\) factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that \(\beta_k\) is not even irreducible if \(k\) is not a power of \(q\).FastMinors package for Macaulay2https://zbmath.org/1540.130012024-09-13T18:40:28.020319Z"Martinova, Boyana"https://zbmath.org/authors/?q=ai:martinova.boyana"Robinson, Marcus"https://zbmath.org/authors/?q=ai:robinson.marcus"Schwede, Karl"https://zbmath.org/authors/?q=ai:schwede.karl-e"Yao, Yuhui"https://zbmath.org/authors/?q=ai:yao.yuhuiSummary: In this article, we present \texttt{FastMinors.m2}, a package in \textit{Macaulay2} designed to introduce new methods focused on computations in function field linear algebra. Some key functionality that our package offers includes: finding a submatrix of a given rank in a provided matrix (when present), verifying that a ring is regular in codimension n, recursively computing the ideals of minors in a matrix, and finding an upper bound of the projective dimension of a module.Generalized transforms of graded strong Mori domainshttps://zbmath.org/1540.130022024-09-13T18:40:28.020319Z"Kim, Dong Kyu"https://zbmath.org/authors/?q=ai:kim.dongkyuLet \(D\) be a commutative integral domain with quotient field \(K\) and \(F(D)\) be the set of nonzero fractional ideals of \(D\). For \(I \in F(D)\), let \(I^{-1} = \{x \in K \mid xI \subseteq D\}\), \(I_v = (I^{-1})^{-1}\), and \(I_t = \bigcup\{J_v \mid J \in F(D)\) is finitely generated and \(J \subseteq I\}\). Now let GV\((D) = \{J \in F(D) \mid J\) is finitely generated and \(J_v = D\}\). The \(w\)-operation on \(D\) is a mapping \(I \mapsto I_w\) of \(F(D)\) into \(F(D)\) defined by \(I_w = \{x \in K \mid xJ \subseteq I\) for some \(J \in\) GV\((D)\}\). Let \(* = v\), \(t\), or \(w\). Then (i) \((xD)_* = xD\) and \((xI)_* = xI_*\), (ii) \(I \subseteq I_*\); \(I \subseteq J\) implies \(I_* \subseteq J_*\), and \((I_*)_* = I_*\) for all \(0 \neq x \in K\) and \(I, J \in F(D)\). An \(I \in F(D)\) is called a \(*\)-ideal if \(I_*= I\) and \(*\)-Max\((D)\) denotes the set of all ideals which are maximal among proper integral \(*\)-ideals of \(D\). Then \(t\)-Max\((D) = w\)-Max\((D)\), and \(t\)-Max\((D) \neq \emptyset\) if and only if \(D\) is not a field. A \(*\)-ideal \(I\) is of \(*\)-finite type if \(I = J_*\) for some finitely generated ideal \(J\) of \(D\). A \(*\)-Noetherian domain is an integral domain in which every \(*\)-ideal is of \(*\)-finite type, so a \(v\)-Noetherian or \(t\)-Noetherian domain is a Mori domain and a \(w\)-Noetherian domain is a strong Mori domain (SM domain).
Let \(\mathcal{S}\) be a multiplicative set of ideals of \(D\). Then \(D_{\mathcal{S}} = \{x \in K \mid xI \subseteq D\) for some \(I \in \mathcal{S}\}\) is an overring of \(D\). Matijevic's result says that if \(\mathcal{S}\) is generated by the set of maximal ideals of a Noetherian domain \(D\), then \(D_{\mathcal{S}} = D^g\) is called the global transform of \(D\) and every ring between \(D\) and \(D^g\) is a Noetherian domain. Let \(D\) be an SM domain. As the SM domain analog, in [J. Pure Appl. Algebra 172, No. 1, 79--85 (2002; Zbl 1036.13015)], \textit{M. H. Park} introduced the notion of \(w\)-global transform \(D^{wg}\), which is defined by \(D_{\mathcal{S}}\) for \(\mathcal{S}\) generated by \(t\)-Max\((D)\), and showed that every \(t\)-linked overring of \(D\) contained in \(D^{wg}\) is an SM domain. Park also showed that \(D_{\mathcal{S}}\) is an SM domain for any \(\mathcal{S}\).
Let \(\Gamma\) be a torsion-free cancellative commutative monoid and \(R = \bigoplus_{\alpha \in \Gamma}R_{\alpha}\) be a graded integral domain. In this paper, among other things, the author studies the graded integral domain analogs of the aforementioned results on SM domains. For example, the author introduces the notion of homogeneous \(w\)-global transform \(R^{hwg}\) of a graded SM domain \(R\) and shows that every homogeneously \(t\)-linked overring of \(R\) contained in \(R^{hwg}\) is a graded SM domain; in particular, if h-\(w\)-dim\((R)=1\), then every homogeneously \(t\)-linked overring of \(R\) is a graded SM domain with h-\(w\)-dim\((R) \leq 1\). Finally, the author also shows that if \(\mathcal{S}\) is a multiplicative set of ideals of a graded (strong) Mori domain \(R\) generated by homogeneous ideals of \(R\), then \(R_{\mathcal{S}}\) is a graded (strong) Mori domain.
Reviewer: Gyu Whan Chang (Incheon)On the ideal avoidance propertyhttps://zbmath.org/1540.130032024-09-13T18:40:28.020319Z"Chen, Justin"https://zbmath.org/authors/?q=ai:chen.justin"Tarizadeh, Abolfazl"https://zbmath.org/authors/?q=ai:tarizadeh.abolfazlSummary: In this article, we investigate the avoidance property of ideals and rings. Among the main results, a general version of the avoidance lemma is formulated. It is shown that every idempotent ideal (and hence every pure ideal) has avoidance. The avoidance property of arbitrary direct products of avoidance rings is characterized. It is shown that every overing of an avoidance domain is an avoidance domain. Next, we show that every avoidance \(\mathbb{N} \)-graded ring whose base subring is a finite field is a PIR. It is also proved that the avoidance property is preserved under flat ring epimorphisms. Dually, we formulate a notion of strong avoidance, and show that it is reflected by pure morphisms.On the complement of the total zero-divisor graph of a commutative ringhttps://zbmath.org/1540.130042024-09-13T18:40:28.020319Z"Visweswaran, S."https://zbmath.org/authors/?q=ai:visweswaran.subramanian|visweswaran.shyamIn this paper, author considered commutative rings with identity and which are not integral domains. Let \(R\) be a ring, let \(Z(R)\) be the set of all zero-divisors of \(R\) and \(Z(R)\setminus \{0\}=Z(R)^*.\) Let \(R\) be a ring such that \(Z(R)^* \neq \phi.\) With such a ring \(R,\) \textit{A. Ðurić} et al. [J. Algebra Appl. 18, No. 10, Article ID 1950190, 16 p. (2019; Zbl 1423.13055)] introduced and investigated an undirected graph denoted by \(ZT(R)\) whose vertex set is \(Z(R)^*\) and distinct vertices \(x\) and \(y\) are adjacent in \(ZT(R)\) if and only if \(xy = 0\) and \(x + y\in Z(R).\) Let \((ZT(R))^c\) be the complement of the graph \(ZT(R).\) In this paper, author determined when \((ZT(R))^c\) is connected and in the case \((ZT(R))^c\) is connected, author has obtained the diameter and radius of \((ZT(R))^c.\) Having obtained diameter and radius, author has provided suitable examples to substantiate the results obtained in this paper.
Reviewer: T. Tamizh Chelvam (Tirunelveli)Finite \(F\)-representation type for homogeneous coordinate rings of non-Fano varietieshttps://zbmath.org/1540.130052024-09-13T18:40:28.020319Z"Mallory, Devlin"https://zbmath.org/authors/?q=ai:mallory.devlinSummary: Finite \(F\)-representation type is an important notion in characteristic \(p\) commutative algebra, but explicit examples of varieties with or without this property are few. We prove that a large class of homogeneous coordinate rings in positive characteristic will fail to have finite \(F\)-representation type. To do so, we prove a connection between differential operators on the homogeneous coordinate ring of \(X\) and the existence of global sections of a twist of \((\text{Sym}^m\Omega_X)^\vee\). By results of \textit{S. Takagi} and \textit{R. Takahashi} [Math. Res. Lett. 15, No. 2--3, 563--581 (2008; Zbl 1147.13001)], this allows us to rule out finite \(F\)-representation type for coordinate rings of varieties with \((\text{Sym}^m\Omega_X)^\vee\) not ``positive.'' By using positivity and semistability conditions for the (co)tangent sheaves, we show that several classes of varieties fail to have finite \(F\)-representation type, including many Calabi-Yau varieties and complete intersections of general type. Our work also provides examples of the structure of the ring of differential operators for non-\(F\)-pure varieties, which to this point have largely been unexplored.Values of the \(F\)-pure threshold for homogeneous polynomialshttps://zbmath.org/1540.130062024-09-13T18:40:28.020319Z"Smith, Karen E."https://zbmath.org/authors/?q=ai:smith.karen-e"Vraciu, Adela"https://zbmath.org/authors/?q=ai:vraciu.adela-nSummary: We find a formula, in terms of \(n, d\), and \(p\), for the value of the \(F\)-pure threshold for the generic homogeneous polynomial of degree \(d\) in \(n\) variables over an algebraically closed field of characteristic \(p\). We also show that in every characteristic \(p\) and for all \(d\geqslant 4\) not divisible by \(p\), there \textit{always} exist reduced polynomials of degree \(d\) in \(k[x, y]\) whose \(F\)-pure threshold is a truncation of the base \(p\) expansion of \(\frac{2}{d}\) at some place; in particular, there always exist reduced polynomials \(f\) whose \(F\)-pure threshold is \textit{strictly} less than \(\frac{2}{\deg (f)} \). We provide an example to resolve, negatively, a question proposed by \textit{D. J. Hernández} et al. [Mich. Math. J. 65, No. 1, 57--87 (2016; Zbl 1342.13007)], as to whether a list of necessary restrictions they prove on the \(F\)-pure threshold of reduced forms are ``minimal'' for \(p\gg 0\). On the other hand, we also provide evidence supporting and refining their ideas, including identifying specific truncations of the base \(p\) expansion of \(\frac{2}{d}\) that are always \(F\)-pure thresholds for reduced forms of degree \(d\), and computations that show their conditions suffice (in \textit{every} characteristic) for degrees up to eight and several other situations.
{{\copyright} 2023 The Authors. \textit{Journal of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.}The separating variety for \(2 \times 2\) matrix invariantshttps://zbmath.org/1540.130072024-09-13T18:40:28.020319Z"Elmer, Jonathan"https://zbmath.org/authors/?q=ai:elmer.jonathanLet \(G = \mathrm{GL}_2(\mathbb{C})\) act on the \(\mathbb{C}\)-vector space \(\mathcal{M}_2^n\) of \(n\)-tuples of \(2 \times 2\) matrices by simultaneous conjugation. Recently it was proven that a well known minimal generating system \(\mathrm{S_n}\) of the invariant subalgebra \(\mathbb{C}[\mathcal{M}_2^n]^G\) is also a minimal separating system with respect to inclusion. The cardinality of this system is \(|\mathrm{S_n}| = \frac{1}{6} (n^3 + 11n)\).
The main theorem of this paper (Theorem 1.4) shows that any separating set for \(\mathbb{C}[\mathcal{M}_2^n]^G\) has cardinality at least \(5n-5\) (hence no separating set of size \(\mathrm{dim}(\mathbb{C}[\mathcal{M}_2^n]^G) = 4n-3\) exists for \(n \geq 3\)). This result is obtained by a detailed study of the geometric structure of the separating variety \(\mathcal{S}_{G,\mathcal{M}_2^n}\) consisting of pairs of points of \(\mathcal{M}_2^n\) which are not separated by any invariant. The main result follows after using several dimension counting arguments.
Moreover, for \(n \geq 3\) a separating set \(\mathrm{S'_n}\) of \(\mathbb{C}[\mathcal{M}_2^n]^G\) consisting of \(|\mathrm{S'_n}| = \frac{1}{2} (n^2 + 9n-16)\) homogeneous polynomials is constructed (Theorem 1.6). One can deduce the following censequences of Theorem 1.4 and Theorem 1.6: \(\mathrm{S_3}\) has minimum cardinality as a separating set. For \(n=4\) it may happen, that there exists a separating set smaller than \(|\mathrm{S_4}|\); for \(n \geq 5\), \(|\mathrm{S'_n}| < |\mathrm{S_n}|\).
Similar results for the left-right action of \(\mathrm{SL}_2(\mathbb{C})\times\mathrm{SL}_2(\mathbb{C})\) on \(\mathcal{M}_2^n\) are discussed in the last Section of the paper.
Reviewer: Barna Schefler (Budapest)Automorphisms of the Koszul homology of a local ringhttps://zbmath.org/1540.130082024-09-13T18:40:28.020319Z"Iyengar, Srikanth B."https://zbmath.org/authors/?q=ai:iyengar.srikanth-b"Rüping, Henrik"https://zbmath.org/authors/?q=ai:ruping.henrik"Stephan, Marc"https://zbmath.org/authors/?q=ai:stephan.marcSummary: This work concerns the Koszul complex \(K\) of a commutative noetherian local ring \(R\), with its natural structure as a differential graded \(R\)-algebra. It is proved that under diverse conditions, involving the multiplicative structure of \(H(K)\), any \(\text{dg}\,R\)-algebra automorphism of \(K\) induces the identity map on \(H(K)\). In such cases, it is possible to define an action of the automorphism group of \(R\) on \(H(K)\). On the other hand, numerous rings are described for which \(K\) has automorphisms that do not induce the identity on \(H(K)\). For any \(R\), it is shown that the group of automorphisms of \(H(K)\) induced by automorphisms of \(K\) is abelian.Homogeneous coordinate rings as direct summands of regular ringshttps://zbmath.org/1540.130092024-09-13T18:40:28.020319Z"Mallory, Devlin"https://zbmath.org/authors/?q=ai:mallory.devlinThe aim of the paper is to give conditions under which a given ring can be realized as a direct summand of a regular ring. The question is studied through the case of homogeneous coordinate rings. Several distinct classes of examples (quadric hypersurfaces, del Pezzo surfaces, Fano threefolds) are analized. In many of the cases it is given a complete classification of such coordinate rings that can occur as direct summands.
The paper is organized as follows: Section 1 is dedicated to list the new results. In Section 2 one can find a detailed revision of the most important knowledge of this topic.
In Section 3 it is given a restrictive criteria for the homogeneous coordinate ring of a smooth variety to be a finite graded direct summand of a regular ring. (Theorem 1.1). In Section 4 the coordinate rings of quadric hypersurfaces are analyzed.
Section 5 is about del Pezzo surfaces. It turns out, that the homogeneous coordinate ring of a complex (smooth) del Pezzo surface \(X_d\) of degree \(d\) is a direct summand of a regular ring exactly when \(d\geq 5\) (Theorem 1.2). As a consequence, it is given an answer to a question of Hara about the finite \(F\)-representation type (FFRT) property of the quintic del Pezzo. In Section 6 the case of smooth hypersurfaces of degree \(d\geq 3\) and prime Fano threefolds are discussed, while in Section 7 the author develops techniques for when \(\mathrm{Proj} R\) is singular.
Reviewer: Barna Schefler (Budapest)Complemented zero-divisor graphs associated with finite commutative semigroupshttps://zbmath.org/1540.130102024-09-13T18:40:28.020319Z"Bender, Chase"https://zbmath.org/authors/?q=ai:bender.chase"Cappaert, Paul"https://zbmath.org/authors/?q=ai:cappaert.paul"DeCoste, Rachelle"https://zbmath.org/authors/?q=ai:decoste.rachelle-c"DeMeyer, Lisa"https://zbmath.org/authors/?q=ai:demeyer.lisaIn this interesting paper, the authors study complemented zero-divisor graphs of finite commutative semigroups. Recall that a simple graph is called complemented if every vertex has an incident edge which is not the edge of any triangle cycle. The complemented zero-divisor graphs, as well as the algebraic properties of the corresponding finite semigroups, are described in terms of the clique numbers. Also, infinite families of the complemented zero-divisor graphs are provided.
Reviewer: Tongsuo Wu (Shanghai)Bi-amalgamations subject to J-clean and weakly J-clean propertieshttps://zbmath.org/1540.130112024-09-13T18:40:28.020319Z"Khalid Adarbeh, Mohammad Adarbeh"https://zbmath.org/authors/?q=ai:khalid-adarbeh.mohammad-adarbehLet \(A\) denote a commutative ring with unity, \(U(A) \) the set of all units of \(A\), Id\((A) \) the set of all idempotents of \(A\), and \(J(A)\) the Jacobson radical of \(A\). Recall that an element \(x\) in a ring \(A\) is called clean (after [\textit{W. K. Nicholson}, Trans. Am. Math. Soc. 229, 269--278 (1977; Zbl 0352.16006)]) if there is \(u \in U(A)\) and \(e \in \) Id\((A)\) such that \(x = u + e.\) An element \(a \in A\) is called \(J\)-clean element , if there is \( j \in J(A)\) and \(e \in \) Id\((A)\) such that \(a = j+e\). A ring \(A\) is called a \(J\)-clean ring (or, semi-boolean ring after [\textit{W. K. Nicholson} and \textit{Y. Zhou}, J. Algebra 291, No. 1, 297--311 (2005; Zbl 1084.16023)]) if every element \(a \in A\) is \(J\)-clean.
In 2017, \textit{P. V. Danchev} [JP J. Algebra Number Theory Appl. 39, No. 3, 261--276 (2017; Zbl 1373.16066)] generalized the notion of \(J\)-clean ring into the notion weakly \(J\)-clean ring: a weakly \(J\)-clean ring is a ring \(A\) in which each element is either the sum or the difference of an element from \(J(A)\) and an element from Id\((A)\).
This paper investigates the transfer of commutative \(J\)-clean and weakly J-clean properties in several cases of bi-amalgamations (after [\textit{S. Kabbaj} et al., J. Commut. Algebra 9, No. 1, 65--87 (2017; Zbl 1390.13008)]). The main results recover previous results that concern with amalgamations (after [\textit{M. D'Anna} et al., in: Commutative algebra and its applications. Proceedings of the fifth international Fez conference on commutative algebra and applications, Fez, Morocco, June 23--28, 2009. Berlin: Walter de Gruyter. 155--172 (2009; Zbl 1177.13043)]) and trivial extensions (after Nagata, 1962) of these rings. Moreover, bi-amalgamations are used to provide new examples as well as counterexamples of \(J\)-clean and weakly \(J\)-clean rings.
Reviewer: Marco Fontana (Roma)Almost \(\phi\)-integrally closed ringshttps://zbmath.org/1540.130122024-09-13T18:40:28.020319Z"Gaur, Atul"https://zbmath.org/authors/?q=ai:gaur.atul"Kumar, Rahul"https://zbmath.org/authors/?q=ai:kumar.rahul"Singh, Anant"https://zbmath.org/authors/?q=ai:singh.anant-pratapThe authors introduce and study almost \(\phi\)-integrally closed rings. Let \(R\) be a commutative ring in the class \(\mathcal H\) introduced by Badawi, which means that \(\mathrm{Nil}(R)\) is a divided prime ideal of \(R\). Let \(\phi\) be the canonical homomorphism \(T(R)\to R_{\mathrm{Nil}(R)}\) satisfying \(\phi(x)=x\) for all \(x\in R\), where \(T(R)\) is the total quotient ring of \(R\). The ring \(R\) is an \textit{almost \(\phi\)-integrally closed ring} if \(\phi(R)\) is integrally closed in \(\phi(R)_{\phi(\mathfrak p)}\) for each nonnil prime ideal \(\mathfrak p\) of \(R\). The ring \(R\) is an almost \(\phi\)-integrally closed ring, if and only if \(R/\mathrm{Nil}()\) is an almost integrally closed domain, as defined by Dobbs and Shapiro. In particular, if \(R\) is an integral domain, then \(R\) is almost \(\phi\)-integrally closed if and only if \(R\) is almost integrally closed, A \(\phi\)-integrally closed ring is almost \(\phi\)-integrally closed, but the converse is false in general. The authors characterize almost \(\phi\)-integrally closed rings of dimension \(\le 1\). They define \(\phi\)-pseudo-valuation rings, and characterize the \(\phi\)-pseudo-valuation rings that are \(\phi\)-almost integrally closed. Further results in this context concern divided rings, \(\phi\)-treed rings, etc.
Reviewer: Moshe Roitman (Haifa)Minimal lattice points in the Newton polyhedron and application to normal idealshttps://zbmath.org/1540.130132024-09-13T18:40:28.020319Z"Al-Ayyoub, Ibrahim"https://zbmath.org/authors/?q=ai:al-ayyoub.ibrahimLet \(S\) be a ring and \(I\) be an ideal in \(S\). An element \(f\in S\) is \textit{integral} over \(I\), if there exists an equation
\[
f^k+c_1f^{k-1}+\cdots +c_{k-1}f+c_k=0\text{ with }c_i\in I^i.
\]
The set of elements \(\overline{I}\) in \(S\) which are integral over \(I\) is the \textit{integral closure} of \(I\). The ideal \(I\) is called \textit{integrally closed}, if \(I=\overline{I}\), and \(I\) is said to be \textit{normal} if all powers of \(I\) are integrally closed. Now, consider a monomial ideal \(I=(x_1^{a_1}, \ldots, x_n^{a_n}) \subset R=K[x_1, \ldots, x_n]\) with each \(a_i\) is a positive integer and \(K\) is a field. Let \(\mathbf{I}(a_1, \ldots, a_n)\) denote the integral closure of the ideal \(I\). The main aim of this paper is to present an elementary and simpler proof of Theorem 5.1 in [\textit{L. Reid} et al., Commun. Algebra 31, No. 9, 4485--4506 (2003; Zbl 1021.13008)]. To do this, the author uses the elementary definition of convex sets, and in particular, a simple characterization of the exponents of the minimal generators of \(\mathbf{I}(a_1, \ldots, a_n)\). In fact, let \(L= \mathbf{I}(a_1, \ldots, a_n, a_{n+1} +l)\) and \(J=\mathbf{I}(a_1, \ldots, a_n, a_{n+1})\), where \(l=\mathrm{lcm}(a_1, \ldots, a_n)\). Then the author proves that:
{Corollary 1.} If \(L\) is normal, then \(J\) is so.
{Corollary 2.} If \(J\) is normal and \(a_{n+1} \geq l\), then \(L\) is so.
Reviewer: Mehrdad Nasernejad (Lens)The finitely presented torsion-free SG-projective modules are not necessarily projectivehttps://zbmath.org/1540.130142024-09-13T18:40:28.020319Z"Xing, Shiqi"https://zbmath.org/authors/?q=ai:xing.shiqiThe paper elaborates on the concept of $w$-invertibility over infra-Krull domains through the $w$-conductor, examining its implications in the context of SG-projectivity. As a consequence of this study, the article disproves the belief that ``a finitely presented torsion-free module is SG-projective if and only if it is projective.'' Specifically, Example 5.3 demonstrates that the $w$-conductor $\mathfrak{m}[Y]$ of $D = R[Y]$, where $R = L + \mathfrak{m}$ and $\mathfrak{m} = XF [X]$ with a field extension $L \subseteq F$ of degree 2, is finitely presented torsion-free SG-projective but not projective. The article is well-discussed and well-written, effectively highlighting the intricate relationships between different types of modules and ideals within the domain.
Reviewer: Nitin Bisht (Katra)The multivariate Serre conjecture ringhttps://zbmath.org/1540.130152024-09-13T18:40:28.020319Z"Guyot, Luc"https://zbmath.org/authors/?q=ai:guyot.luc"Yengui, Ihsen"https://zbmath.org/authors/?q=ai:yengui.ihsenSummary: It is well-known that for any integral domain \(\mathbf{R}\), the Serre conjecture ring \(\mathbf{R} \langle X \rangle\), i.e., the localization of the univariate polynomial ring \(\mathbf{R} [X]\) at monic polynomials, is a Bézout domain of Krull dimension \(\leq 1\) if \(\mathbf{R}\) is a Bézout domain of Krull dimension \(\leq 1\). Consequently, defining by induction \(\mathbf{R} \langle X_1, \dots, X_n \rangle : = (\mathbf{R} \langle X_1, \dots, X_{n - 1} \rangle) \langle X_n \rangle\), the ring \(\mathbf{R} \langle X_1, \dots, X_n \rangle\) is a Bézout domain of Krull dimension \(\leq 1\) if so is \(\mathbf{R}\). The fact that \(\mathbf{R} \langle X_1, \dots, X_n \rangle\) is a Bézout domain when \(\mathbf{R}\) is a valuation domain of Krull dimension \(\leq 1\) was the cornerstone of \textit{J. W. Brewer} and \textit{D. L. Costa}'s theorem [J. Pure Appl. Algebra 13, 157--163 (1978; Zbl 0446.13006)] stating that if \(\mathbf{R}\) is a one-dimensional arithmetical ring then finitely generated projective \(\mathbf{R} [X_1, \dots, X_n]\)-modules are extended. It is also the key of the proof of the Gröbner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain \textbf{R} of Krull dimension \(\leq 1\), any \(n \in \mathbb{N}_{> 0}\), and any finitely generated ideal \(I\) of \(\mathbf{R} [X_1, \dots, X_n]\), the ideal \(\operatorname{LT}(I)\) generated by the leading terms of the elements of \(I\) with respect to the lexicographic monomial order is finitely generated. Since the ring \(\mathbf{R} \langle X_1, \dots, X_n \rangle\) can also be defined directly as the localization of the multivariate polynomial ring \(\mathbf{R} [ X_1, \dots, X_n]\) at polynomials whose leading coefficients according to the lexicographic monomial order with \(X_1 < X_2 < \cdots < X_n\) is 1, we propose to generalize the fact that \(\mathbf{R} \langle X_1, \dots, X_n \rangle\) is a Bézout domain of Krull dimension \(\leq 1\) when \(\mathbf{R}\) is a Bézout domain of Krull dimension \(\leq 1\) to any rational monomial order, bolstering the evidence for the Gröbner Ring Conjecture in the rational case. We give an example showing that this result is no more true in the irrational case.Valuative dimension, constructive points of viewhttps://zbmath.org/1540.130162024-09-13T18:40:28.020319Z"Lombardi, Henri"https://zbmath.org/authors/?q=ai:lombardi.henri"Neuwirth, Stefan"https://zbmath.org/authors/?q=ai:neuwirth.stefan"Yengui, Ihsen"https://zbmath.org/authors/?q=ai:yengui.ihsenThe paper gives a constructive proof of the equivalence of three possible constructive definitions of the valuative dimension and shows the relation between their computational content.
Let \(\mathbf{R}\) be a commutative ring. In classical mathematics, \textit{P. Jaffard} [Mém. Sci. Math. 146, 78 p. (1960; Zbl 0096.02502)] defined the valuative dimension \(\dim_v\mathbf{R}\) in relation to the asymptotic behaviour of the Krull dimension of polynomial rings. If \(\mathbf{R}\) is integral (i.e. every element of \(\mathbf{R}\) is zero or regular), \(\dim_v\mathbf{R}\) is the supremum length of chains of valuation rings \(\mathbf{V}_0\subsetneq\cdots\subsetneq \mathbf{V}_n=\operatorname{Frac}\mathbf{R}\) containing \(\mathbf{R}\), where \(\operatorname{Frac}\mathbf{R}\) is the field of fractions of \(\mathbf{R}\). For an arbitrary ring \(\mathbf{R}\), \(\dim_v\mathbf{R}\) is the supremum of \(\dim_v\mathbf{R}/\mathfrak{p}\), where \(\mathfrak{p}\) is a prime ideal of \(\mathbf{R}\).
The paper considers the following three constructive definitions of the valuative dimension:
\begin{itemize}
\item[1.] \(\operatorname{\mathsf{vdim}}\mathbf{R}\) denotes the valuative dimension defined by \textit{H. Lombardi} and \textit{C. Quitté} [Commutative algebra: constructive methods. Finite projective modules. Translated from the French by Tania K. Roblot. Dordrecht: Springer (2015; Zbl 1327.13001)]. The definition uses the minimal pp-closure \(\mathbf{R}_{\mathrm{min}}\) to avoid the use of prime ideals when extending the definition to arbitrary rings.
\item[2.] If \(\mathbf{R}\) is integral, \(\operatorname{\mathsf{Vdim}}\mathbf{R}\) denotes the valuative dimension defined by \textit{T. Coquand} [Ann. Pure Appl. Logic 157, No. 2--3, 97--109 (2009; Zbl 1222.03072)]. The definition uses a point-free version of the space of valuations \(\operatorname{\mathsf{Val}}\mathbf{R}\) to interpret the Jaffard's definition constructively. The definition is extended to arbitrary rings by \textit{H. Lombardi} and \textit{A. Mahboubi} [in: Algebraic, number theoretic, and topological aspects of ring theory. Selected papers based on the cancelled conference on rings and polynomials, July 2020, and the fourth international meeting on integer-valued polynomials and related topics, CIRM, Luminy, France, July 19--24, 2021. Cham: Springer. 275--341 (2023; Zbl 1530.13009)].
\item[3.] \(\operatorname{\mathsf{dimv}}\mathbf{R}\) denotes the valuative dimension defined by \textit{G. Kemper} and \textit{I. Yengui} [J. Algebra 557, 278--288 (2020; Zbl 1440.13112)]. The definition uses a graded rational monomial order. It is similar to the definition of the Krull dimension by \textit{H. Lombardi} [Ann. Pure Appl. Logic 137, No. 1--3, 256--290 (2006; Zbl 1077.03039)] using the lexicographic order.
\end{itemize}
The equivalence of \(\mathsf{vdim}\), \(\mathsf{Vdim}\) and \(\mathsf{dimv}\) is proved constructively. For a detailed proof of the equivalence of \(\mathsf{vdim}\) and \(\mathsf{Vdim}\) for arbitrary rings, the paper refers to [\textit{H. Lombardi} and \textit{A. Mahboubi}, in: Algebraic, number theoretic, and topological aspects of ring theory. Selected papers based on the cancelled conference on rings and polynomials, July 2020, and the fourth international meeting on integer-valued polynomials and related topics, CIRM, Luminy, France, July 19--24, 2021. Cham: Springer. 275--341 (2023; Zbl 1530.13009)].
Reviewer: Ryota Kuroki (Tōkyō)Proper spaces are spectralhttps://zbmath.org/1540.130172024-09-13T18:40:28.020319Z"Goswami, Amartya"https://zbmath.org/authors/?q=ai:goswami.amartyaSummary: Since Hochster's work, spectral spaces have attracted increasing interest. Through this note we give a new self-contained and constructible topology-independent proof of the fact that the set of proper ideals of a ring endowed with coarse lower topology is a spectral space.The characterization of almost prime submodule on the finitely generated module over principal ideal domainhttps://zbmath.org/1540.130182024-09-13T18:40:28.020319Z"Wardhana, I. Gede Adhitya Wisnu"https://zbmath.org/authors/?q=ai:wardhana.i-gede-adhitya-wisnu"Astuti, Pudji"https://zbmath.org/authors/?q=ai:astuti.pudji"Muchtadi-Alamsyah, Intan"https://zbmath.org/authors/?q=ai:muchtadi-alamsyah.intanLet \(R\) be a commutative ring and \(M\) an \(R\)-module. For any submodule \(N\) of \(M\) let \((N:M)=\{r\in R\;|\;rM\subseteq N\}\). A proper submodule \(N\) of \(M\) is called \textit{prime} if, whenever \(r\in R\) and \(m\in M\) such that \(rm\in N\), then either \(m\in N\) or \(r\in (N:M)\). A proper submodule \(N\) of \(M\) is called \textit{almost prime} if, whenever \(r\in R\) and \(m\in M\) such that \(rm\in N-(N:M)N\), then either \(m\in N\) or \(r\in (N:M)\). In the paper under the review the authors give a new approach to the characterization of an almost prime submodule of a finitely generated module over a principal ideal domain. The main result of the paper is that the almost prime submodule must be the direct sum of submodules of the free part and the torsion part of the module, under some conditions.
Reviewer: Hamid Kulosman (Louisville)Generic local rings on a spectrum between Golod and Gorensteinhttps://zbmath.org/1540.130192024-09-13T18:40:28.020319Z"Winther Christensen, Lars"https://zbmath.org/authors/?q=ai:winther-christensen.lars"Veliche, Oana"https://zbmath.org/authors/?q=ai:veliche.oanaSummary: Artinian quotients \(R\) of the local ring \(Q = \mathsf{k} [[x, y, z]]\) are classified by multiplicative structures on \(\mathsf{A} = \operatorname{Tor}_\ast^Q(R, \mathsf{k})\); in particular, \(R\) is Gorenstein if and only if \(\mathsf{A}\) is a Poincaré duality algebra while \(R\) is Golod if and only if all products in \(\mathsf{A}_{\geqslant 1}\) are trivial. There is empirical evidence that generic quotient rings with small socle ranks fall on a spectrum between Golod and Gorenstein in a very precise sense: The algebra \(\mathsf{A}\) breaks up as a direct sum of a Poincaré duality algebra \(\mathsf{P}\) and a graded vector space \(\mathsf{V} \), on which \(\mathsf{P}_{\geqslant 1}\) acts trivially. That is, \( \mathsf{A}\) is a trivial extension, \( \mathsf{A} = \mathsf{P} \ltimes \mathsf{V} \), and the extremes \(\mathsf{A} = (\mathsf{k} \oplus \mathsf{\Sigma}\mathsf{k}) \ltimes \mathsf{V}\) and \(\mathsf{A} = \mathsf{P}\) correspond to \(R\) being Golod and Gorenstein, respectively.
We prove that this observed behavior is, indeed, the generic behavior for graded quotients \(R\) of socle rank 2, and we show that the rank of \(\mathsf{P}\) is controlled by the difference between the order and the degree of the socle polynomial of \(R\).Relative unimodular rows over polynomial ringshttps://zbmath.org/1540.130202024-09-13T18:40:28.020319Z"Chakraborty, Kuntal"https://zbmath.org/authors/?q=ai:chakraborty.kuntal"Sharma, Sampat"https://zbmath.org/authors/?q=ai:sharma.sampatAuthors' abstract: ``In this article, we prove that if \(R\) is a local ring of dimension \(d \geq 2, 1/d! \in R, I \subseteq R\) an ideal and \(v \in U m_{d+1}(R[X], I[X])\), then \(v\) can be mapped to a factorial row via a relative elementary matrix.''
Here a unimodular row is called factorial if it is of the form \((w_0^{d!},w_1,\cdots , w_d)\), where \((w_0,w_1,\cdots , w_d)\) is unimodular.
From the introduction: ``The idea of the proof is to reduce to the case \(d=2\) using Roitman's degree reduction technique. Now let \(R\) be a local ring of dimension 2. Next we identify the set \(Um_3(R[X],I[X])/E_3(R[X],I[X])\) with the relative elementary Witt group \(W_E(R[X],I[X])\). Using the multiplicative relations in the Witt group \(W_E(R[X],I[X])\) we prove that if \(1/2k\in R\), then every \(v\in Um_3(R[X],I[X])\) is a \(k^{th}\) power.''
Reviewer: Wilberd van der Kallen (Utrecht)A characterization of nonnil-projective moduleshttps://zbmath.org/1540.130212024-09-13T18:40:28.020319Z"Kim, Hwankoo"https://zbmath.org/authors/?q=ai:kim.hwankoo"Mahdou, Najib"https://zbmath.org/authors/?q=ai:mahdou.najib"Oubouhou, El Houssaine"https://zbmath.org/authors/?q=ai:oubouhou.el-houssaineRecently, \textit{W. Zhao} et al. [Commun. Algebra 50, No. 7, 2854--2867 (2022; Zbl 1487.13020)] introduced and studied new concepts of nonnilcommutative diagrams and nonnil-projective modules. They proved that an \(R\)-module that is nonnil-isomorphic to a projective module is nonnil-projective. Further they posed the following problem: Is every nonnil-projective module nonnil-isomorphic to some projective module? In this paper, author obtained some new properties of nonnil-commutative diagrams and answer this problem in the affirmative.
Reviewer: T. Tamizh Chelvam (Tirunelveli)On the generalized principally injective moduleshttps://zbmath.org/1540.130222024-09-13T18:40:28.020319Z"Gholami, Fatemeh"https://zbmath.org/authors/?q=ai:gholami.fatemeh"Habibi, Zohreh"https://zbmath.org/authors/?q=ai:habibi.zohreh"Najafizadeh, Alireza"https://zbmath.org/authors/?q=ai:najafizadeh.alirezaLet \(R\) be an associative ring with unity and let \(M\) and \(N\) be unitary right \(R\)-modules. \(M\) is said to be principally \(N\)-injective if every homomorphism from a cyclic submodule of \(N\) to \(M\) can be extended to a homomorphism from \(N\) to \(M\). Moreover, \(M\) is called \(PQ\)-injective if it is principally \(M\)-injective. The notion of a \(PQ\)-injective module is a natural generalization of a \(p\)-injective ring which has been deeply studied in [\textit{W. Li} et al., Commun. Algebra 45, No. 9, 3816--3824 (2017; Zbl 1390.18032)]. This notion has been extended in various ways by several authors over the past decades. In the article under review, the authors generalize some of the results from the case of \(p\)-injective rings in [\textit{W. Li} et al., Commun. Algebra 45, No. 9, 3816--3824 (2017; Zbl 1390.18032)] and [\textit{C. Y. Hong} et al., Commun. Algebra 28, No. 2, 791--801 (2000; Zbl 0957.16006)] to the case of \(PQ\)-injective modules. Moreover, some results related to various properties of \(PQ\)-injective modules are achieved. A number of results related to the trace and reject submodules of a module possessing some injectivity conditions over a commutative ring are given. Finally, some corrections to the results of Section 2 in [\textit{G. Puninski} and \textit{R. Wisbauer}, J. Pure Appl. Algebra 113, No. 1, 55--66 (1996; Zbl 0859.16003)] are presented.
Reviewer: Alireza Vahidi (Tehran)Duality between injective envelopes and flat covershttps://zbmath.org/1540.130232024-09-13T18:40:28.020319Z"Puuska, Ville"https://zbmath.org/authors/?q=ai:puuska.villeSummary: We establish a duality between injective envelopes and flat covers over a commutative Noetherian ring. One case of this duality states that a morphism is an injective envelope, if and only if its Matlis dual is a flat cover. We also show that if we swap injective envelopes and flat covers in this duality, neither implication is true in general.Torsion-free extensions of projective modules by torsion moduleshttps://zbmath.org/1540.130242024-09-13T18:40:28.020319Z"Fuchs, László"https://zbmath.org/authors/?q=ai:fuchs.laszlo.1|fuchs.laszlo.2Summary: We consider a generalization of a problem raised by P. Griffith on abelian groups to modules over integral domains, and prove an analogue of a theorem of M. Dugas and J. Irwin. Torsion modules \(T\) with the following property are characterized: if \(M\) is a torsion-free module and \(F\) is a projective submodule such that \(M / F \cong T\), then \(M\) is projective. It is shown that for abelian groups whose cardinality is not cofinal with \(\omega\) this is equivalent to being totally reduced in the sense of L. Fuchs and K. Rangaswamy. The problem for valuation domains is also discussed, with results similar to the case of abelian groups.2-irreducible and strongly 2-irreducible submodules of a modulehttps://zbmath.org/1540.130252024-09-13T18:40:28.020319Z"Farshadifar, F."https://zbmath.org/authors/?q=ai:farshadifar.faranak"Ansari-Toroghy, H."https://zbmath.org/authors/?q=ai:ansari-toroghy.habibollahSummary: Let \(R\) be a commutative ring with identity and \(M\) be an \(R\)-module. In this paper, we will introduce the concept of 2-irreducible (resp., strongly 2-irreducible) submodules of \(M\) as a generalization of irreducible (resp., strongly irreducible) submodules of \(M\) and investigated some properties of these classes of modules.A note on the global dimension of shifted ordershttps://zbmath.org/1540.130262024-09-13T18:40:28.020319Z"Esentepe, Özgür"https://zbmath.org/authors/?q=ai:esentepe.ozgur\textit{M. Pressland} and \textit{J. Sauter} [Glasg. Math. J. 64, No. 1, 79--105 (2022; Zbl 1511.16017)] studied a special class of tilting modules for algebras with positive dominant dimension. Specifically, for an algebra of dominant dimension \(d\), a \(k\)-tilting module is constructed for each \(k\leq d\). Such tilting modules, as well as their endomorphism algebras, are called \(k\)-shifted. One of the results found in that paper tells us that the dimension of the \(k\)-shifted algebra is less than or equal to the global dimension of the original algebra. The aim of the paper under review is to present a version of this result in the context of Cohen-Macauley representation theory (see for example [\textit{W. Rump}, J. Algebra 236, No. 2, 522--548 (2001; Zbl 0982.16008)] and [\textit{Y. Yoshino}, Cohen-Macaulay modules over Cohen-Macaulay rings. Cambridge (UK): Cambridge University Press (1990; Zbl 0745.13003)]).
In order to state in more detail the main results of this paper we will be considering the following setting. Let \(R\) be a local Cohen-Macaulay ring of Krull dimension \(d\) with canonical module \(\omega\), and \(A\) a semi-perfect module-finite \(R\)-algebra. We denote by \(CM(A)\) the class of \(A\)-modules that are maximal Cohen-Macaulay as \(R\)-modules. We will assume that \(R\) belongs to \(CM(A)\), i.e. \(A\) is an \(R\)-order. In this context, \(CM(A)\) is an exact category with enough projectives and injectives. In addition we will have minimal CM-injective resolutions and define the CM-injective dimension through these. In particular, we will be assuming that \(A\) is of CM-injective dimension \(n\) with \(0<n<\infty\), i.e. \(A\) is an \(n\)-canonical order.
A fundamental part of the paper reviewed is the so-called CM-dominant dimension of \(A\). Recall that the classical dominant dimension measures how far an algebra is from being self-injective. In the context of the paper, the CM-dominant dimension measures how far \(R\) is from being a Gorenstein order (i.e., that \( \operatorname{Hom}_R (A,\omega)\) is a projective \(A\)-module). Specifically, \(R\) is of CM-dominant dimension \(\ell\) if \(\ell\) is the smallest number such that \(I^{\ell}\) is not projective in a minimal injective resolution
\[
0\rightarrow A \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots .
\]
It is worth saying that the paper under review devotes a couple of subsections to discuss the subtleties presented by classical dominant dimension problems when translated in this context. Namely, the Nakayama Conjecture and the Auslander Correspondence are discussed.
The main results are the following for \(A\) with finite CM-dominant dimension \(\geq \ell \geq 1\). By doing a construction similar to that of Pressland and Sauter, one can construct an \(\ell\)-tilting \(A\)-module which, together with its algebra of endomorphisms, is called \(\ell\)-shifted. It turns out that the \(\ell\)-shifted algebra is an \(R\)-order and that its global dimension is less than or equal to the global dimension of \(A\).
Reviewer: Alejandro Argudín Monroy (Ciudad de México)Algebraic properties of binomial edge ideals of Levi graphs associated with curve arrangementshttps://zbmath.org/1540.130272024-09-13T18:40:28.020319Z"Karmakar, Rupam"https://zbmath.org/authors/?q=ai:karmakar.rupam"Sarkar, Rajib"https://zbmath.org/authors/?q=ai:sarkar.rajib|sarkar.rajib.1"Subramaniam, Aditya"https://zbmath.org/authors/?q=ai:subramaniam.adityaThe authors study binomial edge ideals determined by the Levi graphs of certain transversal curve arrangements in the complex projective plane. More precisely, if \(\mathcal{C} = \{C_{1},\dots, C_{k}\} \subset \mathbb{P}^{2}_{\mathbb{C}}\) is an arrangement of \(k\) smooth components such that all intersections are ordinary, then we can define the Levi graph as a bipartite that decodes the incidence structure between the curves and the intersections (i.e. the edges tell us which curve passes through a given point). Given such graphs, we can define the associated binomial edge ideals. The authors study a natural question of when such binomial edge ideals associated with transversal curve arrangements are Cohen-Macaulay. It turns out that in most cases such ideals are never Cohen-Macaulay. Then the authors study a combinatorial question when the Levi graphs associated with some transversal curve arrangements have an induced \(C_{6}\) cycle. The last part of the paper is devoted to the Castelnuovo-Mumford regularity of such ideals.
Reviewer: Piotr Pokora (Kraków)On the integral closure of radical towers in mixed characteristichttps://zbmath.org/1540.130282024-09-13T18:40:28.020319Z"Katz, Daniel"https://zbmath.org/authors/?q=ai:katz.daniel-c|katz.daniel|katz.daniel-s|katz.daniel-j"Sridhar, Prashanth"https://zbmath.org/authors/?q=ai:sridhar.prashanthSummary: We study the Cohen-Macaulay property of a particular class of radical extensions of an unramified regular local ring having mixed characteristic.Depth of \(C(X)\)-moduleshttps://zbmath.org/1540.130292024-09-13T18:40:28.020319Z"Azarpanah, F."https://zbmath.org/authors/?q=ai:azarpanah.f"Hesari, A. A."https://zbmath.org/authors/?q=ai:hesari.a-a"Salehi, A. R."https://zbmath.org/authors/?q=ai:salehi.a-rSummary: It is shown that the depth of every module over \(C(X)\), the ring of all real valued continuous functions on a topological space \(X\), is at most 1. This result is proved for modules over rings of a much more general class than the rings of continuous functions. It also turns out that many facts in the literature concerning the depths of \(C(X)\), their sub-algebras, and ideals are consequences of this main result. Some known results are generalized and some applications are given.Results on the algebraic matroid of the determinantal varietyhttps://zbmath.org/1540.130302024-09-13T18:40:28.020319Z"Tsakiris, Manolis C."https://zbmath.org/authors/?q=ai:tsakiris.manolis-cSummary: We make progress towards characterizing the algebraic matroid of the determinantal variety defined by the minors of fixed size of a matrix of variables. Our main result is a novel family of base sets of the matroid, which characterizes the matroid in special cases. Our approach relies on the combinatorial notion of relaxed supports of linkage matching fields that we introduce, our interpretation of the problem of completing a matrix of bounded rank from a subset of its entries as a linear section problem on the Grassmannian, and a connection that we draw with a class of local coordinates on the Grassmannian described by \textit{B. Sturmfels} and \textit{A. V. Zelevinsky} in [Adv. Math. 98, No. 1, 65--112 (1993; Zbl 0776.13009)].Classifying several subcategories of the category of maximal Cohen-Macaulay moduleshttps://zbmath.org/1540.130312024-09-13T18:40:28.020319Z"Saito, Shunya"https://zbmath.org/authors/?q=ai:saito.shunyaSummary: In this summary, we introduce the classification of several subcategories of a torsion-free class of the module category over a commutative noetherian ring. More precisely, we classify Serre subcategories and torsion(-free) classes of a torsion-free class in the sense of exact categories. This result extends Gabriel's classification of Serre subcategories of the module category to torsionfree classes. As an immediate consequence, we classify the Serre subcategories and the torsion(-free) classes of the category of maximal Cohen-Macaulay modules over a one-dimensional Cohen-Macaulay ring.
For the entire collection see [Zbl 1540.16001].Linear resolutions and quasi-linearity of monomial idealshttps://zbmath.org/1540.130322024-09-13T18:40:28.020319Z"Lu, Dancheng"https://zbmath.org/authors/?q=ai:lu.danchengIn this paper, the author introduces the notion of quasi-linearity and the notion of a strongly linear monomial over a monomial ideal, where a monomial ideal \(I\) is \textit{quasi-linear} if the colon ideal \((G(I) \setminus {u}) : u\) is generated by linear forms for every \(u\) in the minimal monomial generating set \(G(I)\). For monomial ideals \(I\) and \(J\), recall that \(I\) is linear over \(J\) if there exist \(r > 0\) and monomials \(u_1,\ldots, u_r\) such that \(G(I) =G(J)\cup \{u_1, \ldots, u_r\}\) and \( \langle J, u_1,\ldots, u_{i-1}\rangle : u_i\) is generated by linear forms (i.e., variables) for each \(i = 1, \ldots , r\). A monomial \(u\) is called strongly linear over a monomial ideal \(I\) generated in homogeneous degree, if \(I : u\) is generated by variables.
The author proved that quasi-linearity is necessary for a monomial ideal to have a linear resolution and, clarifies all the quasi-linear monomial ideals generated in degree \(2\). The author also proves that if a monomial \(u\) is strongly linear over the \(I\), then \(I\) has a linear resolution (respectively is quasi-linear) if and only if \(I + uP\) has a linear resolution (respectively is quasi-linear) for any (and all in the necessity case) monomial prime ideal \(P\).
Reviewer: Tongsuo Wu (Shanghai)The order of dominance of a monomial idealhttps://zbmath.org/1540.130332024-09-13T18:40:28.020319Z"Alesandroni, G."https://zbmath.org/authors/?q=ai:alesandroni.guillermoSummary: Let \(S\) be a polynomial ring in \(n\) variables over a field, and consider a monomial ideal \(M=(m_1,\dots,m_q)\) of \(S\). We introduce a new invariant, called order of dominance of \(S/M\), and denoted \(\operatorname{odom}(S/M)\), which has many similarities with the codimension of \(S/M\). We use the order of dominance to characterize the class of Scarf ideals that are Cohen-Macaulay, and also to characterize when the Taylor resolution is minimal. In addition, we show that \(\operatorname{odom}(S/M)\) has the following properties:
\begin{itemize}
\item[(i)] \(\operatorname{codim}(S/M) \leqslant \operatorname{odom}(S/M)\leqslant \mathrm{pd}(S/M)\).
\item[(ii)] \(\mathrm{pd}(S/M)=n\) if and only if \(\operatorname{odom}(S/M)=n\).
\item[(iii)] \(\mathrm{pd}(S/M)=1\) if and only if \(\operatorname{odom}(S/M)=1\).
\item[(iv)] If \(\operatorname{odom}(S/M)=n-1\), then \(\mathrm{pd}(S/M)=n-1\).
\item[(v)] If \(\operatorname{odom}(S/M)=q-1\), then \(\mathrm{pd}(S/M)=q-1\).
\item[(vi)] If \(n=3\), then \(\mathrm{pd}(S/M)=\operatorname{odom}(S/M)\).
\end{itemize}Morphisms represented by monomorphisms with \(n\)-torsionfree cokernelhttps://zbmath.org/1540.130342024-09-13T18:40:28.020319Z"Otake, Yuya"https://zbmath.org/authors/?q=ai:otake.yuyaLet \(R\) be a ring, and \(\textrm{mod}(R)\) denote the category of finitely generated right \(R\)-modules. The stable category of finitely generated right \(R\)-modules \(\underline{\bmod}(R)\) is a category with \(\mathrm{Obj}\left(\underline{\bmod}(R)\right)= \mathrm{Obj}\left(\textrm{mod}(R)\right)\), and \(\mathrm{Mor}_{{\underline{\bmod}}(R)}(M,N)= \mathrm{Hom}_{R}(M,N)/ \mathcal{P}_{R}(M,N)\) for every \(M,N\in\mathrm{Obj}\left(\textrm{mod}(R)\right)\) where \(\mathcal{P}_{R}(M,N)\) is the set of \(R\)-homomorphisms \(M\rightarrow N\) that factor through a finitely generated projective right \(R\)-module. It is folklore that the Auslander transpose defines an additive functor \(\mathrm{Tr}:\underline{\bmod}(R)\rightarrow \underline{\bmod}(R^{\mathrm{op}})\).
The author defines a condition \((\textrm{T}_{n})\) for any \(n\geq 1\) as follows. An \(R\)-homomorphism \(f:M\rightarrow N\) is said to satisfy the condition \((\textrm{T}_{n})\) if
\[
\mathrm{Ext}^{i}_{R^{\mathrm{op}}}\left(\mathrm{Tr}(f),R\right):\mathrm{Ext}^{i}_{R^{\mathrm{op}}}\left(\mathrm{Tr}(M),R\right)\rightarrow\mathrm{Ext}^{i}_{R^{\mathrm{op}}}\left(\mathrm{Tr}(N),R\right)
\]
is an isomorphism for every \(1\leq i\leq n-1\), and is injective for \(i=n\).
It has been previously shown that an \(R\)-homomorphism \(f:M\rightarrow N\) satisfies the condition \((\textrm{T}_{1})\) if and only if \(f\) is represented by monomorphisms, in the sense that there is an \(R\)-homomorphism \(g:M\rightarrow P\) with \(P\) projective such that \(\binom{f}{g}:M\rightarrow N\oplus P\) is a monomorphism. The authors studies the condition \((\textrm{T}_{n})\) for any \(n\geq 1\) and provides various connections to the notion of \(n\)-torsionfreeness generalizing several well-known results in the literature.
Reviewer: Hossein Faridian (Clemson)Embeddings into modules of finite projective dimensions and the \(n\)-torsionfreeness of syzygieshttps://zbmath.org/1540.130352024-09-13T18:40:28.020319Z"Otake, Yuya"https://zbmath.org/authors/?q=ai:otake.yuyaSummary: Let \(R\) be a commutative noetherian ring. In this article, we find out close relationships between the module \(m\) being embedded in a module of projective dimension at most \(n\) and the \((n+1)\)-torsionfreeness of the \(n\)th syzygy of \(M\). As an application, we consider the \(n\)-torsionfreeness of syzygies of the residue field \(k\) over a local ring \(R\).
For the entire collection see [Zbl 1540.16001].Unique maximal Betti diagrams for Artinian Gorenstein \(k\)-algebras with the weak Lefschetz propertyhttps://zbmath.org/1540.130362024-09-13T18:40:28.020319Z"Richert, Ben"https://zbmath.org/authors/?q=ai:richert.benGiven a Hilbert function \(\underline{h}\) for a standard graded algebra over a field, it was shown by \textit{A. M. Bigatti} [Commun. Algebra 21, No. 7, 2317--2334 (1993; Zbl 0817.13007)] and by \textit{H. A. Hulett} [Commun. Algebra 21, No. 7, 2335--2350 (1993; Zbl 0817.13006)] (in characteristic zero) and by \textit{K. Pardue} [Ill. J. Math. 40, No. 4, 564--585 (1996; Zbl 0903.13004)] (in arbitrary characteristic) that the lex segment ideal has the largest Betti numbers among all ideals with Hilbert function \(\underline{h}\). Specializing the kind of algebra leads to new kinds of obstacles, since lex segment ideals may not arise in the class of algebras being considered. One kind of algebra that has attracted a lot of attention for many different problems is the class of artinian Gorenstein algebras. Here even the possible Hilbert functions are not entirely known, although many papers have been written to narrow down the possibilities. However, an important special case occurs when the algebra has the Weak Lefschetz Property (WLP). Then it turns out that the Hilbert functions are completely known, thanks mostly to work of \textit{T. Harima} [Proc. Am. Math. Soc. 123, No. 12, 3631--3638 (1995; Zbl 0857.13013)]. These are the so-called SI-sequences (for Stanley and Iarrobino), which are sequences of integers \((1,h_1,\dots,h_{e-1},1)\) (where \(e\) is the socle degree) that are symmetric and satisfy a certain growth condition. So the problem addressed here is to find a sharp upper bound for the Betti numbers of an artinian Gorenstein algebra with WLP whose Hilbert function \(\underline{h}\) is a given SI-sequence. This problem was solved by \textit{J. Migliore} and \textit{U. Nagel} [Adv. Math. 180, No. 1, 1--63 (2003; Zbl 1053.13006)], and the current paper gives a new and simpler proof of this result. The original proof involved a technical geometric construction (using generalized stick figures and Gorenstein liaison). The author here points out that ``the economy and computability obtained by our approach comes at a steep cost, because the important geometric intuition and insight inherent in the generalized stick figures used in Migliore and Nagel's work is lost in our machinery.'' Nevertheless, the elegance of the new approach is very nice!!
Reviewer: Juan C. Migliore (Notre Dame)Cohomology of flag supervarieties and resolutions of determinantal idealshttps://zbmath.org/1540.130372024-09-13T18:40:28.020319Z"Sam, Steven V."https://zbmath.org/authors/?q=ai:sam.steven-v"Snowden, Andrew"https://zbmath.org/authors/?q=ai:snowden.andrew-wThe paper explores the coherent cohomology of generalized flag supervarieties and finds a connection between these cohomology groups and the free resolutions of generalized determinantal ideals. In particular, this paper focuses on the case of supergrassmannians, computing the cohomology of the structure sheaf and relating it to the singular cohomology of a Grassmannian and the syzygies of a determinantal variety.
It uses the following spectral sequence as a key to connect the cohomology of supervarieties with the free resolutions of determinantal ideals
\[
E_1^{p,q}=H^{p+q}(X_{\mathrm{ord},\wedge^p(I/I^2)})\Rightarrow H^{p+q}(X,\mathcal{O}_X).
\]
The main and most important achievement of this paper is generalization of Grothendieck-Springer to supergrassmannians. Just as the Grothendieck-Springer resolution smooths out singularities in the nilpotent cone, generalization case relates the geometry of super Grassmannians to ordinary Grassmannians, smoothing out and clarifying the complex structure of the supervarieties.
This works also provides a conceptual explanation of the Pragacz-Weyman result, showing that the syzygies of determinantal ideals admit an action of the general linear supergroup.
Reviewer: Fereshteh Bahadorykhalily (Shiraz)Injective dimension of cofinite modules and local cohomologyhttps://zbmath.org/1540.130382024-09-13T18:40:28.020319Z"Asghari, Fardin"https://zbmath.org/authors/?q=ai:asghari.fardin"Naghipour, Reza"https://zbmath.org/authors/?q=ai:naghipour.reza"Sedghi, Monireh"https://zbmath.org/authors/?q=ai:sedghi.monirehLet \(R\) be a commutative noetherian ring, \(\mathfrak{a}\) an ideal of \(R\), and \(M\) an \(R\)-module. For any \(i \geq 0\), the \(i\)th local cohomology module of \(M\) with respect to \(\mathfrak{a}\) is given by
\[
H^{i}_{\mathfrak{a}}(M) \cong \underset{n\geq 1}\varinjlim \mathrm{Ext}^{i}_{R}\left(R/ \mathfrak{a}^{n},M\right).
\]
Hartshorne defines an \(R\)-module \(M\) to be \(\mathfrak{a}\)-cofinite if \(\mathrm{Supp}_{R}(M)\subseteq \mathrm{Var}(\mathfrak{a})\) and \(\mathrm{Ext}^{i}_{R}\left(R/ \mathfrak{a},M\right)\) is a finitely generated \(R\)-module for every \(i\geq 0\).
Now suppose that \((R,\mathfrak{m})\) is local, and \(M\) is \(\mathfrak{a}\)-cofinite. In this paper, the authors show that if \(M\) has finite injective dimension, then the inequalities
\[
\dim(R/\mathfrak{a}) \leq \mathrm{id}_{R}(M) \leq \mathrm{depth}(\mathfrak{m},R),
\]
hold, and if \(\mathfrak{m}M \neq M\), then \(\mathrm{id}_{R}(M) = \mathrm{depth}(\mathfrak{m},R)\). This generalizes the classical Bass formula for injective dimension. As an application, they obtain some results on the injective dimension of local cohomology modules. In addition, they show that \(R\) is a Cohen-Macaulay ring if it admits a Cohen-Macaulay \(R\)-module of finite projective dimension.
Reviewer: Hossein Faridian (Clemson)Cohen-Macaulay dimension for complexeshttps://zbmath.org/1540.130392024-09-13T18:40:28.020319Z"Mashhad, Fatemeh Mohammadi Aghjeh"https://zbmath.org/authors/?q=ai:mashhad.fatemeh-mohammadi-aghjehLet \(R\) be a commutative noetherian local ring. The projective dimension of an \(R\)-module is a well-known and widely studied numerical invariant in classical homological algebra whose finiteness can characterize regularity of \(R\). However, there exist several refinements and extensions of this dimension that provide more subtle information. Namely, complete intersection dimension, G-dimension, and Cohen-Macaulay dimension that can be used to characterize complete intersection rings, Gorenstein rings, and Cohen-Macaulay rings, respectively. These homological dimensions satisfy the following inequalities
\[
\textrm{CM-dim}_{R}(M) \leq \textrm{G-dim}_{R}(M) \leq \textrm{CI-dim}_{R}(M) \leq \textrm{pd}_{R}(M)
\]
with equality to the left of any finite quantity. Each of these homological dimensions has been extended to complexes of \(R\)-modules. The focus of this paper is on the exploration of Cohen-Macaulay dimension within the category of homologically finite \(R\)-complexes. Since the Cohen-Macaulay dimension is defined via quasi-deformations, it is interesting to be able to study this dimension through resolutions. Accordingly, the author presents a significant theorem that allows the computation of the Cohen-Macaulay dimension for a homologically finite complex of \(R\)-modules using its syzygies. As a crucial application of this theorem, she demonstrates that any homologically finite \(R\)-complex \(X\) of finite Cohen-Macaulay dimension possesses a finite Cohen-Macaulay resolution, i.e. a bounded \(R\)-complex \(G\) of finitely generated \(R\)-modules that is isomorphic to \(X\) in the derived category of \(R\) and consists of modules of Cohen-Macaulay dimension zero.
Reviewer: Hossein Faridian (Clemson)A cotorsion pair induced by the class of Gorenstein \((m,n)\)-flat moduleshttps://zbmath.org/1540.130402024-09-13T18:40:28.020319Z"Yang, Qiang"https://zbmath.org/authors/?q=ai:yang.qiang.3Summary: In this paper, we introduce the notion of Gorenstein \((m,n)\)-flat modules as an extension of \((m,n)\)-flat left \(R\)-modules over a ring \(R\), where \(m\) and \(n\) are two fixed positive integers. We demonstrate that the class of all Gorenstein \((m,n)\)-flat modules forms a Kaplansky class and establish that \((\mathcal{GF}_{m,n}(R)\), \(\mathcal{GC}_{m,n}(R))\) constitutes a hereditary perfect cotorsion pair (where \(\mathcal{GF}_{m,n}(R)\) denotes the class of Gorenstein \((m,n)\)-flat modules and \(\mathcal{GC}_{m,n}(R)\) refers to the class of Gorenstein \((m,n)\)-cotorsion modules) over slightly \((m,n)\)-coherent rings.The Auslander-Reiten conjecture for normal ringshttps://zbmath.org/1540.130412024-09-13T18:40:28.020319Z"Kimura, Kaito"https://zbmath.org/authors/?q=ai:kimura.kaitoSummary: In this article, we consider the Auslander-Reiten conjecture, which is a celebrated long-standing conjecture in ring theory. One of the main results of this article asserts that the conjecture holds for an arbitrary normal ring.
For the entire collection see [Zbl 1540.16001].On a generalization of a result of Peskine and Szpirohttps://zbmath.org/1540.130422024-09-13T18:40:28.020319Z"Puthenpurakal, Tony J."https://zbmath.org/authors/?q=ai:puthenpurakal.tony-jThe author introduced \textit{Peskine-Szpiro ideal}. The paper provided an elementary proof of the following theorem.
{Theorem 1.2} Let \((R, \mathfrak{m})\) be a regular local ring of dimension d containing a field \(K\). Let \(I\) be a Cohen-Macaulay ideal of height \(g.\) The following conditions are equivalent.
\begin{itemize}
\item [(i)] \(I\) is a \textit{Peskine-Szpiro ideal} of \(R,\)
\item[(ii)] For any prime ideal \(\mathfrak{p}\in V(I),\) the Bass number
\[
\mu_i(\mathfrak{p},H^g_I(R))=\begin{cases} 1, \text{ if } i=\mathrm{ht}\mathfrak{p}-g,\\
0, \text{ otherwise.} \end{cases}
\]
\end{itemize}
Reviewer: Tri Nguyen (Biên Hòa)Cohomological dimension of DG-moduleshttps://zbmath.org/1540.130432024-09-13T18:40:28.020319Z"Rao, Yanping"https://zbmath.org/authors/?q=ai:rao.yanping"Liu, Zhongkui"https://zbmath.org/authors/?q=ai:liu.zhong-kui"Yang, Xiaoyan"https://zbmath.org/authors/?q=ai:yang.xiaoyanLet \(R\) be a commutative Noetherian ring with identity, and \(\mathfrak{a}\) an ideal of \(R\). The \textit{cohomological dimension} of an \(R\)-module \(M\) with respect to the ideal \(\mathfrak{a}\), denoted \(\text{cd}_R(\mathfrak{a},M)\), is defined as the largest integer \(i\) for which the \(i\)th local cohomology module \(\text{H}_{\mathfrak{a}}^i(M)\) is nonzero. In [Commun. Algebra 32, No. 11, 4375--4386 (2004; Zbl 1093.13011)], \textit{M. T. Dibaei} and \textit{S. Yassemi} extended this notion to complexes. For a complex \(X\), they defined the \textit{cohomological dimension} of \(X\) with respect to \(\mathfrak{a}\) by
\[
\text{cd}_R(\mathfrak{a},X):=\sup \{\text{cd}_R(\mathfrak{a},\text{H}^n(X))+n \mid n\in \mathbb{Z} \}.
\]
Also, for a bounded above complex \(X\), they proved the inequality
\[
\sup \mathbf{R}\Gamma_{\mathfrak{a}}(X)\leq \text{cd}_R(\mathfrak{a},X)
\]
holds, with equality occurring when all cohomology modules \(\text{H}^n(X)\) are finitely generated.
The authors of the present paper establish the above result in the context of DG-modules over DG-rings. Let \(A\) be a commutative Noetherian non-positive DG-ring. (In particular, \(\overline{A}:=\text{H}^i(A)\) is a commutative Noetherian ring with identity.) Let \(\overline{\mathfrak{a}}\) be an ideal of \(\overline{A}\) and \(X\) a non-exact bounded above DG-module with finitely generated cohomologies. The authors show that
\[
\sup\mathbf{R}\Gamma_{\overline{\mathfrak{a}}}(X)=\sup \{\text{cd}_{\overline{A}}(\overline{\mathfrak{a}},\text{H}^n(X))+n \mid n\in \mathbb{Z} \}
\]
\[
= \sup \{\text{cd}_{\overline{A}}(\overline{\mathfrak{a}},\overline{A}/\mathfrak{p})+ \sup X_{\mathfrak{p}}\mid\mathfrak{p}\in \text{Spec}~ \overline{A} \}.
\]
Reviewer: Kamran Divaani-Aazar (Tehran)Uniformly \(S\)-Noetherian ringshttps://zbmath.org/1540.130442024-09-13T18:40:28.020319Z"Chen, Mingzhao"https://zbmath.org/authors/?q=ai:chen.mingzhao"Kim, Hwankoo"https://zbmath.org/authors/?q=ai:kim.hwankoo"Qi, Wei"https://zbmath.org/authors/?q=ai:qi.wei"Wang, Fanggui"https://zbmath.org/authors/?q=ai:wang.fanggui"Zhao, Wei"https://zbmath.org/authors/?q=ai:zhao.wei.8Let \(R\) be a commutative ring with identity, \(S\) a multiplicative subset of \(R\) and \(M\) an \(R-\)module. We say that \(M\) is uniformly \(S-\)Noetherian if there exists \(s\in S\) such that for any submodule \(N\) of \(M\) there is a finitely generated submodule \(F\) of \(N\) such that \(sN\subseteq F\). Then \(R\) is said to be an uniformly \(S-\)Noetherian ring if it is uniformly \(S-\)Noetherian as an \(R-\)module. In this paper, the authors give, among many other things, an Eakin-Nagata-Formanek theorem for uniformly \(S-\)Noetherian module. Indeed, the following assertions are equivalent (1) \(M\) is uniformly \(S-\)Noetherian. (2) There exists \(s\in S\) such that any ascending chain \(M_1\subseteq M_2\subseteq\ldots\) of submodules of \(M\) is stationary with respect to \(s\); i.e, there exists \(k\geq 1\) such that \(sM_n\subseteq M_k\) for any \(n\geq k\). (3) There exists \(s\in S\) such that any nonempty set \(\{M_{\lambda}\}_{\lambda\in\Lambda}\) of submodules of \(M\) has a maximal element \(M_0\in\{M_{\lambda}\}_{\lambda\in\Lambda}\) with respect to \(s\); i.e, if \(M_0\subseteq M_j\) for some \(M_j\in\{M_{\lambda}\}_{\lambda\in\Lambda}\), then \(sM_j\subseteq M_0\). The particular case of rings is deduced. In addition, the authors study the uniformly \(S-\)Noetherian properties on several constructions such as trivial extension, pullback and amalgamation algebra.
Reviewer: Ali Benhissi (Monastir)Realizations of semilocal \(\ell\)-groups over \(k[x_1,x_2,\dots, x_n]\)https://zbmath.org/1540.130452024-09-13T18:40:28.020319Z"Paudel, Lokendra"https://zbmath.org/authors/?q=ai:paudel.lokendraSummary: \textit{A. M. de Souza Doering} and \textit{Y. Lequain} [J. Algebra 211, No. 2, 711--735 (1999; Zbl 0940.12005)] introduced a weak approximation theorem for dependent valuation rings and they proved that every finitely generated lattice-ordered group can be realized as the group of divisibility of a semilocal Bézout overring of a polynomial ring over a field \(k\) in infinitely many variables, where each of the valuation rings appearing in the finite intersection has residue field \(k\). Moreover, they proved that every semilocal lattice-ordered group admits a lexico-cardinal decomposition form. In this work, we focus on realizing the semilocal \(\ell\)-group over a polynomial ring in finitely many variables. We prove that every semilocal lattice-ordered group having a finite rational rank can be realized as the group of divisibility of a Bézout overring of \(k[x_1,x_2,\dots, x_n]\) up to lexico-cardinal decomposition, where \(k\) is a field and \(x_1,x_2,\dots, x_n\) are indeterminates over \(k\) and \(n\) depends on the group. As a corollary, we prove that every semilocal \(\ell\)-group either finitely generated or divisible with finite rational rank is realizable over \(k[x_1,x_2,\dots, x_n]\) where each of the valuation rings appearing in the finite intersection has residue field \(k\).Half-factorial real quadratic ordershttps://zbmath.org/1540.130462024-09-13T18:40:28.020319Z"Pollack, Paul"https://zbmath.org/authors/?q=ai:pollack.paulAn integral domain \(D\) is called half-factorial (\(HFD\)) if all factorizations into irreducibles of any non-unit and non-zero \(a\in D\) have the same number of factors. In 2001 \textit{J. Coykendall} [J. Algebra 235, No. 2, 417--430 (2001; Zbl 0989.11058)] considered \(HFD\)-orders in quadratic fields, showed that \(Z[\sqrt{-3}]\) is the only such imaginary order, gave a necessary and sufficient condition for real quadratic \(HFD\)-orders of prime index and conjectured that there infinitely many quadratic \(HFD\)-orders.
The author establishes this conjecture by showing that there at most \(45\) real quadratic fields with class-number \(1\) or \(2\) and fundamental unit of norm \(-1\) which contain only finitely many \(HFD\)-orders (Proposition 4). He shows moreover that under \(GRH\) the number of primes \(p\le x\) with \(p\equiv3\) mod \(4\) such that \(Z[p\sqrt2]\) is a \(HFD\)-order equals \(Ax/2\log x,\) where
\[
A=\prod_{q\ \text{prime}}\left(1-\frac1{q(q-1)}\right)
\]
is the Artin's constant.
Reviewer: Władysław Narkiewicz (Wrocław)A sharp bound for the resurgence of sums of idealshttps://zbmath.org/1540.130472024-09-13T18:40:28.020319Z"van Kien, Do"https://zbmath.org/authors/?q=ai:kien.do-van"Nguyen, Hop D."https://zbmath.org/authors/?q=ai:nguyen.dang-hop"Thuan, Le Minh"https://zbmath.org/authors/?q=ai:thuan.le-minhThe resurgence and asymptotic resurgence of an ideal measure the (asymptotic) difference between the ordinary and symbolic powers of the ideal. In this paper, the authors show that if \(A\) and \(B\) are standard graded polynomial rings over \(\Bbbk\) and \(I \subseteq A\) and \(J \subseteq B\) are non-zero proper homogeneous ideals, then the following inequalities hold:
\[
\max \{\rho(I), \rho(J)\} \leq \rho(I+J) \leq \max \left\{\rho(I), \rho(J), \frac{2(\rho(I)+\rho(J))}{3}\right\}
\]
The upper bound is an improvement on the result of \textit{S. Bisui} et al. [Collect. Math. 72, No. 3, 605--614 (2021; Zbl 1470.13039)]. Moreover, the authors completely resolve two conjectures proposed by Bisui, Ha, Jayanthan, and Thomas.
Reviewer: Truong Hoàng Lê (Hà Nội)Application of Lagrange inversion to wall-crossing for Quot schemes on surfaceshttps://zbmath.org/1540.130482024-09-13T18:40:28.020319Z"Bojko, Arkadij"https://zbmath.org/authors/?q=ai:bojko.arkadijThe goal of this paper is to give a new proof of the equality between tow some formal power series using only Lagrange inversion. By these techniques, the author's work on Quot scheme becomes independent of the existing literature.
Reviewer: Ali Benhissi (Monastir)On the recursive and explicit form of the general J.C.P. Miller formula with applicationshttps://zbmath.org/1540.130492024-09-13T18:40:28.020319Z"Bugajewski, Dariusz"https://zbmath.org/authors/?q=ai:bugajewski.dariusz"Bugajewski, Dawid"https://zbmath.org/authors/?q=ai:bugajewski.dawid"Gan, Xiao-Xiong"https://zbmath.org/authors/?q=ai:gan.xiao-xiong"Maćkowiak, Piotr"https://zbmath.org/authors/?q=ai:mackowiak.piotrFor \(a\in\mathbb{C}\) and \(n\in\mathbb{N}^*\), \((^a_n)=\frac{a(a-1)\ldots (a-n+1)}{n!}\). Let \(B_a(z)=1+(^a_1)z+(^a_2)z^2+\ldots\in\mathbb{C}[[z]]\) be a formal binomial series and \(f(z)=b_1z+b_2z^2+b_3z^3+\ldots\in\mathbb{C}[[z]]\) be any nonunit formal power series. Put \(B_a\circ f(z)=c_0+c_1z+c_2z^2+c_3z^3+\ldots\in\mathbb{C}[[z]]\). By the well known J. C. P. Miller formula, we have \(c_0=1\) and for \(n\geq 1\), \( c_n=\frac{1}{n}\displaystyle\sum_{k:0}^{n-1}\big[a(n-k)-k\big]c_kb_{n-k}\). In the paper under review, the authors replace the nonunitness hypothesis by a necessary and sufficient condition for the existence of the composition \(B_a\circ f\). Then they establish a general recurrence algorithm allowing the calculation of the coefficients of \(B_a\circ f\) when it exists. They also pass from the case of one variable to the multivariable case. Their techniques allow them to explicit the inverse of polynomials and formal power series.
Reviewer: Ali Benhissi (Monastir)Generic generalized diagonal matriceshttps://zbmath.org/1540.130502024-09-13T18:40:28.020319Z"Nguyen, Vinh"https://zbmath.org/authors/?q=ai:nguyen.vinh-dinh|nguyen.vinh-hung|nguyen.vinh-phu|nguyen.vinh-q|nguyen.vinh-tan|nguyen.vinh-v|nguyen.vinh-hao|nguyen.vinh-kha|nguyen.vinh-duc"Simper, Hunter"https://zbmath.org/authors/?q=ai:simper.hunterSummary: Generalized diagonal matrices are matrices that have two ladders of entries that are zero in the upper right and bottom left corners. The minors of generic generalized diagonal matrices have square-free initial ideals and the associated Stanley-Reisner complex can be viewed as a higher order complex of a certain poset. With this description, we characterize the configuration of ladders that yield Cohen-Macaulay ideals. In the special case where both ladders are triangles, we show that the corresponding complex is vertex decomposable.The valuation pairing on an upper cluster algebrahttps://zbmath.org/1540.130512024-09-13T18:40:28.020319Z"Cao, Peigen"https://zbmath.org/authors/?q=ai:cao.peigen"Keller, Bernhard"https://zbmath.org/authors/?q=ai:keller.bernhard"Qin, Fan"https://zbmath.org/authors/?q=ai:qin.fanIn this article, the authors introduce a valuation pairing on upper cluster algebras, and apply the valuation pairing to obtain many results concerning factoriality, \(d\)-vectors, \(F\)-polynomials, and cluster Poisson variables.
Let \(\mathbb{A}\) be the set of cluster variables in an upper cluster algebra \(\mathcal{U}\). The \textit{valuation pairing} is a map
\begin{align*}
(-||-)_v:\mathbb{A}\times \mathcal{U}& \longrightarrow\mathbb{N}\cup\{\infty\}\\
(A,M)&\longmapsto \max\{s\in \mathbb{N}\mid M/A^s\in \mathcal{U}\}.
\end{align*}
This valuation pairing satisfies the following properties (Proposition 3.3):
\begin{itemize}
\item \((A||M)_v=\infty\) if and only if \(M=0\).
\item \((A||M+L)_v\geq \min\{(A||M)_v,(A||L)_v\}\).
\item For any \(s\geq 0\), \((A||A^sM)_v=s+(A||M)_v\).
\end{itemize}
The authors also introduce the following local factorization in an upper cluster algebra \(\mathcal{U}\): for any non-zero element \(M\in \mathcal{U}\) and any cluster \(t=\{A_{1;t},\dots, A_{n;t}\}\), we can factor \(M=NL\) with \(N\) being a cluster monomial in \(t\) and \((A_{k;t}||L)_v=0\) for all \(k=1,\dots, n\).
They then prove in Theorem 3.7 that for a full rank upper cluster algebra \(\mathcal{U}\), the local factorization of any non-zero element \(M\) with respect to any cluster \(t\) is unique.
One important question in commutative algebra is to address the factoriality of an algebra. The authors prove in Theorem 4.9 that \(\mathcal{U}\) is factorial (UFD) if and only if the cluster variables in one cluster are prime (i.e., generating a prime principal ideal).
Recall that the exchange matrix \(B\) of a seed is an \(n\times m\) matrix where \(n\) is the number of cluster variables in the seed and \(m\) is the number of mutable cluster variables in the seed. The exchange matrix \(B\) is said to be \textit{primitive} if each column vector is primitive (in the sense that the gcd of the entries is \(1\)). We say \(\mathcal{U}\) is primitive if it admits a primitive exchange matrix. The authors then prove in Theorem 4.13 that if \(\mathcal{U}\) is primitive and full-rank, then \(\mathcal{U}\) is factorial (UFD), and the numerator polynomial of any cluster variable in the Laurent expansion with respect to any given seed is irreducible. Since any upper cluster algebra with principal coefficients is primitive and full rank, this theorem holds for any upper cluster algebra with prinicipal coefficients. Moreover, the authors prove in Corollary 4.21 that for a primitive and full rank upper cluster algebra \(\mathcal{U}\), its cluster algebra \(\mathcal{A}\) is factorial (UFD) if and only if \(\mathcal{A}=\mathcal{U}\).
In cluster algebra III, Berenstein, Fomin, and Zelevinsky proved the star fish lemma, stating that if an upper cluster algebra \(\mathcal{U}\) is full rank, then it is equal to the intersection of the Laurent polynomial rings associated with an initial seed together with all its once-mutated neighboring seeds (\(m+1\) in total where \(m\) is the number of mutable vertices). In this article, the authors prove the \textit{ray fish lemma} (Theorem 4.23), stating that if \(\mathcal{U}\) is primitive and full rank, and \(t_0\) and \(t\) are two seeds that share no common mutable cluster variables, then \(\mathcal{U}=\mathcal{L}_{t_0}\cap \mathcal{L}_t\) where \(\mathcal{L}_{t_0}\) and \(\mathcal{L}_t\) are the Laurent polynomial rings associated with the seeds \(t_0\) and \(t\).
By the Laurent phenomenon, we can fix an initial cluster \(t_0=\{A_1,\dots, A_n\}\) and expand any non-zero element \(M\) as \(P_M(A_1,\dots, A_n)/\prod_{i=1}^n A_i^{d_i}\). The polynomial \(P_M\) is called the \textit{numerator polynomial} of \(M\) with respect to the initial cluster and the vector \(d_M:=(d_1,\dots, d_n)\) is called the \textit{\(d\)-vector} of \(M\) with respect to the initial cluster. The authors prove in Theorem 5.1 that if \(\mathcal{U}\) is full rank, then \(d_k=(A_k||P_M)_v-(A_k||M)_v\). As a corollary, if \((A_k||M)_v=0\), then \(d_M\) is uniquely determined by \(P_M\). Note that \((A_k||M)_v=0\) holds whenever \(M\) is a cluster monomial that does not contain the initial cluster variables, and thus the above corollary is in particular applicable to such a cluster monomial.
For a fixed initial cluster \(t_0=\{A_1,\dots, A_n\}\), one can define for each mutable index \(i\) a Laurent monomial \(\hat{X}_i=\prod_j A_j^{b_{ij}}\). Then any cluster monomial \(M\) can be factored as \(\left(\prod_i A_i^{g_i}\right)F_M(\hat{X}_1,\dots, \hat{X}_m)\); the vector \(g_M:=(g_1,\dots, g_n)\) is called the \textit{\(g\)-vector} of \(M\) with respect to the initial cluster and the polynomial \(F_M\) (in variables \(X_1,\dots, X_m\)) is called the \textit{\(F\)-polynomial} with respect to the initial cluster. In Theorem 6.1, the authors prove that in an (upper) cluster algebra with principal coefficients, cluster monomials in non-initial cluster variables are uniquely determined by their \(F\)-polynomials. In theorem 6.2, they further prove that the \(F\)-polynomial of a non-initial cluster variable is irreducible (as a polynomial in \(X_1,\dots, X_m\)).
The variables \(X_1,\dots, X_n\) are also known as the cluster Poisson variables of Fock and Goncharov, and they obey their own set of cluster mutation rules. Moreover, one can construct an upper cluster algebra with universal coefficients by merging the cluster mutation rules for the \(A\)-variables and the \(X\)-variables together:
\[
\mu_k(A_i)=\left\{\begin{array}{ll} \frac{X_{k;t}}{1+X_{k;t}}\frac{\prod_{b_{jk;t}>0}A_{j;t}^{b_{jk;t}}}{A_{k;t}}+\frac{1}{1+X_{k;t}}\frac{\prod_{b_{jk;t}<0}A_{j;t}^{-b_{jk;t}}}{A_{k;t}} & \text{if \(i=k\),}\\
A_i & \text{if \(i\neq k\).} \end{array}\right.
\]
Note that this is not Reading's cluster algebra with universal coefficients for finite type cluster algebras; rather it is what Fock and Goncharov called the symplectic double. Let \(t_0\) and \(t\) be two cluster seeds of an upper cluster algebra with universal coefficients. Suppose the mutation of \(t_0\) at \(j\) produces a new cluster variable \(\frac{X_{j;t_0}}{1+X_{j;t_0}}N_1+\frac{1}{1+X_{j;t_0}}N_2\) and the mutation of \(t\) at \(k\) produces a new cluster variable \(\frac{X_{k;t}}{1+X_{k;t}}M_1+\frac{1}{1+X_{k;t}}M_2\). The authors prove in Theorem 7.5 that the cluster Poisson variables \(X_{j;t_0}=X_{k;t}\) if and only if \((A_{j;t_0},N_1,N_2)=(A_{k;t},M_1,M_2)\). As a corollary, they prove that the cluster Poisson seeds that contain a particular cluster Poisson variable form a connected subgraph of the exchange graph.
Reviewer: Daping Weng (Davis)Twist automorphisms and Poisson structureshttps://zbmath.org/1540.130522024-09-13T18:40:28.020319Z"Kimura, Yoshiyuki"https://zbmath.org/authors/?q=ai:kimura.yoshiyuki"Qin, Fan"https://zbmath.org/authors/?q=ai:qin.fan"Wei, Qiaoling"https://zbmath.org/authors/?q=ai:wei.qiaolingSummary: We introduce (quantum) twist automorphisms for upper cluster algebras and cluster Poisson algebras with coefficients. Our constructions generalize the twist automorphisms for quantum unipotent cells. We study their existence and their compatibility with Poisson structures and quantization. The twist automorphisms always permute well-behaved bases for cluster algebras. We explicitly construct (quantum) twist automorphisms of Donaldson-Thomas type and for principal coefficients.From Frieze patterns to cluster categorieshttps://zbmath.org/1540.130532024-09-13T18:40:28.020319Z"Pressland, Matthew"https://zbmath.org/authors/?q=ai:pressland.matthewCluster algebras were originally introduced and examined by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)] to study \textit{G. Lusztig}'s theory of total positivity [Prog. Math. 123, 531--568 (1994; Zbl 0845.20034)] and canonical bases [\textit{G. Lusztig}, J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008)]. The utilization of cluster algebras has extended to various realms within mathematics, including combinatorics, representation theory, algebraic geometry, mathematical physics, Teichmüller theory, mirror symmetry, integrable systems, as indicated in [\textit{B. Keller}, Lond. Math. Soc. Lect. Note Ser. 375, 76--160 (2010; Zbl 1215.16012)] and the accompanying references. In the current paper, the author delivers a comprehensive and insightful overview of the impact of cluster algebras on the representation theory of finite-dimensional algebras. This overview commences with an exploration of the combinatorial aspects of frieze patterns, a concept initially introduced and examined by \textit{H. S. M. Coxeter} [Acta Arith. 18, 297--310 (1971; Zbl 0217.18101)] and \textit{J. H. Conway} and \textit{H. S. M. Coxeter} [Math. Gaz. 57, 87--94 (1973; Zbl 0285.05028)] circa 1970. These patterns are characterized by an infinite sequence of positive integers manifesting numerous combinatorial properties. The author elucidates how the frieze entries can serve as cluster variables, thus contributing to the generation of cluster algebra. Subsequently, the author elucidates the process of categorifying cluster algebras, a concept established representationally by \textit{P. Caldero} et al. [Algebr. Represent. Theory 9, No. 4, 359--376 (2006; Zbl 1127.16013); Trans. Am. Math. Soc. 358, No. 3, 1347--1364 (2006; Zbl 1137.16020)], as well as by \textit{A. B. Buan} et al. [Adv. Math. 204, No. 2, 572--618 (2006; Zbl 1127.16011)]. The cluster category introduces a fresh outlook on the combinatorial aspects of cluster algebras. The author also provides a detailed exposition in the appendix regarding the bounded derived category, a crucial component for delineating the cluster category. Additionally, the author offers illustrative examples and recommends good references for further reading about the related concepts.
For the entire collection see [Zbl 1524.22003].
Reviewer: Alireza Nasr-Isfahani (Isfahan)Regularity of edge ideals via suspensionhttps://zbmath.org/1540.130542024-09-13T18:40:28.020319Z"Banerjee, Arindam"https://zbmath.org/authors/?q=ai:banerjee.arindam.1|banerjee.arindam.2|banerjee.arindam"Nevo, Eran"https://zbmath.org/authors/?q=ai:nevo.eranLet \(G\) be a simple graph. Denote by \(I(G)\) the edge ideal of \(G\) over an arbitrary field \(\mathrm{k}\). The authors studied the following conjecture [\textit{A. Banerjee} et al., Springer Proc. Math. Stat. 277, 17--52 (2019; Zbl 1418.13014)]
\[
\operatorname{reg} (I(G)^s) \le \operatorname{reg} (I(G)) + 2s - 2.
\]
The main result of the paper establishes the conjecture for \(s = 2\). From that, they deduced the conjecture for all \(s\) when \(G\) is a bipartite graph. The main idea of the proof is based on the following result [\textit{A. Banerjee}, J. Algebr. Comb. 41, No. 2, 303--321 (2015; Zbl 1318.13001)] \[
\operatorname{reg} (I(G)^s) \le 2s - 2 + \max \left \{ \operatorname{reg} (I(G)^s : e_1 \cdots e_{s-1}) \mid e_j \text{ are edges of } G \right \}.
\]
When \(s = 2\), it reduces to prove that \(\operatorname{reg} (I(G)^2 :e) \le \operatorname{reg} (I(G))\) for all edges \(e\) of \(G\). In this case, \(I(G)^2:e\) can be described explicitly. Then a simple argument based on the Mayer-Vietoris sequence gives the desired conclusion.
When \(G\) is a bipartite graph, then they proved that
\[
I(G)^{s+1}: (e_1\cdots e_s) = (I(G)^s : e_1 \cdots e_{s-1})^2 : e_s.
\]
Hence, the conclusion follows from induction and the case \(s = 2\). This equality is not true when \(G\) is not bipartite.
This result has been generalized in two recent articles. [\textit{S. A. S. Fakhari}, Math. Scand. 129, No. 1, 39--59 (2023; Zbl 1516.13006)] proved that the conjecture holds for an exponent \(s\) as long as \(G\) does not have an induced odd cycle of length \(\le 2s - 1\). [\textit{N. C. Minh} et al., J. Comb. Theory, Ser. A 190, Article ID 105621, 30 p. (2022; Zbl 1493.13026)] proved the conjecture for \(s = 2\) and \(s = 3\) for arbitrary graphs.
Reviewer: Thanh Vu (Hà Nội)Closed binomial edge idealshttps://zbmath.org/1540.130552024-09-13T18:40:28.020319Z"Peeva, Irena"https://zbmath.org/authors/?q=ai:peeva.irena-vSummary: We prove a conjecture by \textit{V. Ene} et al. [Nagoya Math. J. 204, 57--68 (2011; Zbl 1236.13011)] that the Betti numbers of the binomial edge ideal \(J_G\) of a closed graph \(G\) coincide with the Betti numbers of its lex initial ideal \(M_G\). We describe the Betti numbers of the ideal \(M_G\).Reflexive modules over the endomorphism algebras of reflexive trace idealshttps://zbmath.org/1540.130562024-09-13T18:40:28.020319Z"Endo, Naoki"https://zbmath.org/authors/?q=ai:endo.naoki"Goto, Shiro"https://zbmath.org/authors/?q=ai:goto.shiroSummary: In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of
the second author et al. [J. Algebra 379, 355--381 (2013; Zbl 1279.13035), Theorem 5.1] and
\textit{T. Kobayashi} [Algebr. Represent. Theory 25, No. 5, 1061--1070 (2022; Zbl 1496.13016), Theorem 1.3]
concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, Arf local rings contain only finitely many Ulrich ideals once the normalization is a local ring.Remarks on almost Gorenstein ringshttps://zbmath.org/1540.130572024-09-13T18:40:28.020319Z"Endo, Naoki"https://zbmath.org/authors/?q=ai:endo.naoki"Matsuoka, Naoyuki"https://zbmath.org/authors/?q=ai:matsuoka.naoyukiThis paper explores the relationship between the almost Gorenstein properties of graded rings and local rings. The authors show that if \(R\) is an almost Gorenstein graded ring, then the localization \(R_M\) at the graded maximal ideal \(M\) is also almost Gorenstein as a local ring. While the converse generally does not hold, it is true for one-dimensional graded domains under certain mild conditions.
Reviewer: Truong Hoàng Lê (Hà Nội)Rings with an elementary abelian \(p\)-group of unitshttps://zbmath.org/1540.130582024-09-13T18:40:28.020319Z"Chebolu, Sunil K."https://zbmath.org/authors/?q=ai:chebolu.sunil-k"Corry, Jeremy"https://zbmath.org/authors/?q=ai:corry.jeremy"Grimm, Elizabeth"https://zbmath.org/authors/?q=ai:grimm.elizabeth"Hatfield, Andrew"https://zbmath.org/authors/?q=ai:hatfield.andrewSummary: What are all rings \(R\) for which \(R^\times\) (the group of invertible elements of \(R\) under multiplication) is an elementary abelian \(p\)-group? We answer this question for finite-dimensional commutative \(k\)-algebras, finite commutative rings, modular group algebras, and path algebras. Two interesting byproducts of this work are a characterization of Mersenne primes and a connection to Dedekind's problem.Polyfunctions over commutative ringshttps://zbmath.org/1540.130592024-09-13T18:40:28.020319Z"Specker, Ernst"https://zbmath.org/authors/?q=ai:specker.ernst"Hungerbühler, Norbert"https://zbmath.org/authors/?q=ai:hungerbuhler.norbert"Wasem, Micha"https://zbmath.org/authors/?q=ai:wasem.michaFor a commutative ring \(R\) and a subring \(S\) of \(R\), a specific number denoted \(s(S,R)\) is defined as:
\[
s(S ; R):=\min \left\{m \in \mathbb{N}, \exists p \in S[x], \operatorname{deg}(p)<m, \forall x \in R \Rightarrow p(x)=x^m\right\}.
\]
In particular \(s(R,R)\) is denoted by \(S(R)\). This paper classifies all finite commutative rings with unit element, which satisfy \(s(R)= |R|\). This allowed authors to give two alternatives proofs of the Rédei-Szele Theorem. For an infinite commutative ring \(R\) with unit element which satisfy \(S(R) < \infty\) and \(R^{'}\) a subring of \(R\) generated by \(1\), an upper bound for \(|R^{'}|\) and for \(S(R^{'},R)\) is provided.
Reviewer: Joël Kabore (Ouagadougou)The \(h\)-vectors of the edge rings of a special family of graphshttps://zbmath.org/1540.130602024-09-13T18:40:28.020319Z"Higashitani, Akihiro"https://zbmath.org/authors/?q=ai:higashitani.akihiro"Shibu Deepthi, Nayana"https://zbmath.org/authors/?q=ai:shibu-deepthi.nayanaSummary: The \(h\)-vectors of homogeneous rings are one of the most important invariants that often reflect ring-theoretic properties. On the other hand, there are a few examples of edge rings of graphs whose \(h\)-vectors are explicitly computed. In this paper, we compute the \(h\)-vector of a special family of graphs, by using the technique of initial ideals and the associated simplicial complex.Gröbner bases of radical Li-Li type ideals associated with partitionshttps://zbmath.org/1540.130612024-09-13T18:40:28.020319Z"Ren, Xin"https://zbmath.org/authors/?q=ai:ren.xin"Yanagawa, Kohji"https://zbmath.org/authors/?q=ai:yanagawa.kohjiSummary: For a partition \(\lambda\) of \(n\), the \textit{Specht ideal} \(I_\lambda\subset K[x_1,\dots,x_n]\) is the ideal generated by all Specht polynomials of shape \(\lambda\). In their unpublished manuscript, Haiman and Woo showed that \(I_\lambda\) is a radical ideal, and gave its universal Gröbner basis (\textit{S. Murai} et al. [Commun. Algebra 50, No. 12, 5430--5434 (2022; Zbl 1502.13048)] published a quick proof of this result). On the other hand, an old paper of \textit{S.-Y. R. Li} and \textit{W.-C. W. Li} [Combinatorica 1, 55--61 (1981; Zbl 0524.05037)] studied analogous ideals, while their ideals are not always radical. The present paper introduces a class of ideals generalizing both Specht ideals and \textit{radical} Li-Li ideals, and studies their radicalness and Gröbner bases.Metric algebraic geometryhttps://zbmath.org/1540.130622024-09-13T18:40:28.020319Z"Breiding, Paul"https://zbmath.org/authors/?q=ai:breiding.paul"Kohn, Kathlén"https://zbmath.org/authors/?q=ai:kohn.kathlen"Sturmfels, Bernd"https://zbmath.org/authors/?q=ai:sturmfels.berndThe book under review grew out from lecture notes for an Oberwolfach Seminar on Metric Algebraic Geometry -- a summer school for PhD students and postdocs -- held May 29 to June 2, 2023. It consists of 15 chapters each of about twelve to fourteen pages of length and each with three to four sections. Together the sections form an impressive collection of essays on applicable algebraic geometry centering around distance problems in various forms.
The objective is, so the authors, to reintegrate two areas once the same -- differential geometry and algebraic geometry -- with the intent to develop practical tools for optimization, statistics, but also 21st century applications like data science and machine learning. Understanding distances, volumes and angles and, for example, being able to minimize the Euclidean distance from an algebraic variety to a given data point or for doing topological data analysis computing the homology of a submanifold using curvature and bottlenecks and sample points is important in applications.
In doing so, the diverse chapters present inumerous examples illuminating cited theorems for the proofs of which the readers frequently will have to consult the original articles and even background texts in Algebraic Geometry, Differential Geometry, or Statistics and Probability Theory. Many different software tools are employed, namely specific packages in Macaulay2, SAGE, MAGMA, Julia, Mathematica, and Maple. The emphasis is however on an at least intuitive understanding.
The final sentences of the preface say the book adresses a wide audience of researchers and students who will find it useful for seminars and self-study; it can serve as a text for a one semester course at a graduate level. The key prerequisite is a solid foundation in undergraduate mathematics especially algebra and geometry. Course work in statistics, computer science and numerical analysis as well as experience with mathematical software are helpful as well.
There seem to be a few mistakes -- e.g. the inflection points in Figure 6.3 do not seem to be correctly positioned, the formalization of \(f_{i,\theta}\) in equation (10.1) seems not quite correct, and the number of variables in Proposition 14.10 should be \(n=1\) -- but at the whole the book seems to be carefully crafted. The text has no exercises as such but understanding the details will be training enough. As the book is also an e-book the digital `find' facilities will easily overcome the lack of an index.
In order that no important topic to which these unique notes offer access is forgotten, the following review is quite detailed. It is done in the format: `Chapter n, title ..., Section n.1, title ..., Section n.2, title, ...'.
Chapter 1, Historical Snapshots: after informing about the general hypotheses that the book makes, it presents in
Section 1.1, Polars, a highly interesting theorem due to Roger Cotes published in 1722 which led to the concept of the polar of an algebraic curve of any degree and which was generalized by \textit{G. Salmon} [Dublin: Hodges, Foster and Co. (1873; JFM 05.0340.03)]. The definitions there given are metric, dealing with oriented distances while modern algebraic geometry defines polar curves without any relation to metric considerations but rather via differential operators; a connection found by Salmon himself. Also, such theorems suitably generalized are important in contemporary applied mathematics. See [\textit{A. J. Sommese} and \textit{C. W. Wampler II}, The numerical solution of systems of polynomials. Arising in engineering and science. River Edge, NJ: World Scientific (2005; Zbl 1091.65049)].
Section 1.2, Foci, familiarizes the reader with foci of general plane algebraic curves \(C\) viewn in projective plane \(\mathbb P^2\): a point \(\mathbf{f}\) is a focus of \(C\) in the sense of Pluecker, showing the foci of an ellipse as a very special case. Next dual curve \(C^{\vee}\) of a plane curve \(C\) in \(\mathbb P^2\), there degree and number of complex foci are discussed. Beautiful colored pictures of curves that arise from modifications of the definition of the (classical) ellipse by foci are next offered. Cartesian ovals, Cassini ovals, Lemniscates, etc.
Section 1.3, Envelopes. These are the curves that are tangent to each of a suitable family (for example a one dimensional algebraic family) of lines in the plane. An example is given of a cubic curve and the envelope of its diameters as well as the evolute (the envelope of the normals) of a Cassini oval. The chapter concludes with caustics by reflection and refraction as examples of envelopes. The whole chapter is descriptive and explains as far as this is possible in a limited space; it enriches the narrative with historical details and motivating connections to later chapters. For mathematical details cited the reader is relegated frequently to [Salmon, loc. cit]. An extensive modern study is [\textit{R. Piene} et al., ``Return of the plane evolute'', Preprint, \url{arXiv:2110.11691}].
Chapter 2, Critical Equations:
Section 2.1, Euclidean Distance degree. The problem is to minimize the Euclidean distance of a given point \(u\in \mathbb R^n\) outside an irreducible algebraic variety \(X\) (having applications in mind called a \textit{model}) defined by polynomials \(f_1,f_2, \dots, f_k \in \mathbb R[x_1, \dots, x_n]\) to \(X\). That is we wish to compute \(\min_{x\in X} \|x-u\|^2\). To do so ideal theoretic constructions involving the Jacobian of the \(f_i\), certain minors of an augmented Jacobian and a saturation process are involved resulting in the the \textit{critical ideal} \(C_{X,u}\) whose variety in \(\mathbb C^n\) is typically finite, consisting of the critical points. It contains in particular the minimizers to the Euclidean Distance Problem (ED problem) mentioned. The cardinality of \(V(C_{X,u})\) is the the EDdegree\((X)\). It is an important measure for the difficulty of solving the ED problem by exact algebraic methods. The chapter draws heavily on [\textit{J. Draisma} et al., Found. Comput. Math. 16, No. 1, 99--149 (2016; Zbl 1370.51020)]; Short codes using the Macaulay2 software for ideal computations and Julia for homotopy continuation are also presented in examples involving the Trott curve \(144(x^4+y^4)-225(x^2+y^2) +350 x^2 y^2+81=0\) and the cardioid. At the end of the section the concept of a \(\Lambda\)-weighted Euclidean norm is introduced to better deal with genericity requirements in later sections.
Section 2.2, Low-Rank Matrix Approximations. Such are considered as a special case of the set-up in the previous section. We are given a subspace \(\mathcal L\) of the space of \(m\times n\) matrices and a \(m\times n\) matrix \(U\in \mathcal L\) and wish to find a matrix \(A\in \mathcal L\) of rank \(\leq r\), minimizing a weighted Euclidean distance. The family of all such matrices can be viewn as an algebraic variety \(\mathcal L_{\leq r}\). The famous Eckhard-Young theorem is a very special case. The general case to find \(\text{EDdegree}(\mathcal L_{\leq r})\) where now \(\mathcal L_{\leq r}=\{A\in \mathcal L: \text{rank}(A)\leq r\}\) is quite hard and phenomena occurring are illustrated with Hankel- and circulant matrices.
Section 2.3, Invitation to Polar Degrees. A deeper understanding for the ED degree is here laid down looking at the algebro-geometric roots. These center around the following Theorem 2.13 of Draisma et al. [loc. cit.]: If a variety \(X\) intersects as well the isotropic quadric as the hyperplane at infinity transversally, then its EDdegree equals the sum of the polar degrees of its projective closure. An important corollary is that the generic ED degree of a variety is the sum of the polar degrees of its projective closure. The conormal variety \(N_X\) associated to \(X\). is used for explaining duality phenomena in programming; see Theorem 2.23. The cohomology ring \(H^*(\mathbb P^n \times \mathbb P^n, \mathbb Z)\) is \(\mathbb Z[s,t]/\langle s^n+1, t^n+1 \rangle\) and as \(N_X\subset \mathbb P^n \times \mathbb P^n\), we have a representation of the class \([N_X]\) in that ring as a binary form in \(s,t\) of degree \(n+1=\text{codim}(N_X)\) in it. Its coefficients \(\delta_i(X)\) are nonnegative integers; this is one possibility to define the polar degrees. This and above result is used to derive with a Macaulay2 computation more elegantly than before the result of an exammple in the section before. It is finally shown that the primal dual set-up of conormal varieties allows an elegant formulation of the critical equations for the original ED problem.
Chapter 3, Computations: This chapter now, fittingly, presents the methods available to compute the solutions to polynomial systems. The authors consider Groebner bases and approximate zeros as data structures for representing solutions for \(F(x)=0\).
Section 3.1, Groebner Bases. The section presents first the classical material on the topic like lexicographic monomial order, the definition of Groebner bases with leading terms, standard monomials, the elimination theorem and the theorem that in case of finite varieties \(V(I)\), the number of standard monomials with respect to the Groebner basis GB(\(I\)) of \(I\) equals the number of solutions. Next are presented more sophisticated theorems like that in case of that \(V(I)\) has components of positive dimension, then a certain saturation of \(I\) can be used to describe the solution. There follow theorems on parametrized polynomial systems. In these the basic order \(x_1>x_2> \cdots >x_n> p_1> \cdots >p_k\) is lexicographically extended and \(p_1, \dots,p_k\) are thought of as parameters, usually symbolic coefficients. For \(q\in \mathbb C^k\) the homomorphism \(\mathbb C[x,p]\ni f(x,p)\stackrel{\phi_q}{\mapsto} f(x,q)\in \mathbb C[x]\) is considered and conditions are given in Proposition 3.10 for when given \(G\) is a Groebner basis for an ideal \(I\) in \(\mathbb C[x,p]\) then \(\phi_q(G)\) is a Groebner basis for ideal \(\phi_q(I) \in \mathbb C[x]\).
Section 3.2, The Parameter Continuation Theorem. This theorem is found in [\textit{A. Morgan} and \textit{A. Sommese} Appl. Math. Comput. 29, No. 2, 123--160 (1989; Zbl 0664.65049)]: Suppose we have a system of polynomial \(F(x,p)=(f_1(x,p), \dots, f_n(x,p))\) with \(f_i(x,p)\in \mathbb C[x_1, \dots, x_n; p_1, \dots, p_k]\). For every \(p\in \mathbb C^k\) we may consider the regular zeros of \(F(x,p)\). Then there exists a subvariety \(\Delta\) of \(\mathbb C^k\) (called discriminant) such that for all \(p\not\in \Delta\) the number of regular zeros of \(F(x,p)\) is the same finite number. This is proved following [\textit{P. Piwek}, ``Solvable and non-solvable finite groups of the same order type'', Preprint, \url{arXiv:2403.02197}] using the ideal theoretic theorems of the section before and illustrated by examples using Macaulay2.
Section 3.3, Polynomial Homotopy Continuation. Though this is quite a different and much more numerical approach than Groebner bases for solving systems \(F(x)=0\), these still provide the theoretical underpinnings for PHC. For example the parameter continuation theorem before is an essential ingredient in the proof of the correctness of the method which roughly speaking to solve \(F(x)=0\) involves an ordinary differential equation applied to a parameter homotopy to connect \(G(x)=F(x,q)\) with known solutions of \(G(x)=0\) to \(F(x)=F(x,p)\). The output of this is an approximate zero for \(F(x)\) by which one means a point \(z\in \mathbb C^n\) with the property that the sequence of Newton iterates \(z_{k+1}=z_k-JF(z_k)^{-1} F(z_k)\), initiated with \(z_0=z\) converges to a zero of \(F\). As a corollary one gets an `almost Bezout' theorem: a general system \(F(x)=(f_1(x), \dots, f_n(x))\) is of \(n\) polynomials of degrees \(d_1,d_2, \dots,d_n\), has \(d_1d_2 \cdots d_n\) isolated zeros. See [Sommese and Wampler II, loc. cit]. Next sparse quadratic polynomial systems \(F(x)=(f_1(x), \dots, f_n(x))\) are considered. Such a system is one in which for each \(i\) all monomial occurring in \(f_i\) stem from a fixed predetermined set \(\{x^\alpha| \alpha\in A_i\}\), where \(A_1, \dots,A_n \subset \mathbb Z_{\geq 0}^n\), \(i=1, \dots, n\). Let \(P_i=\text{conv}A_i\). Using the famous the Bernstein-Kusnirenko-Hovanski theorem on the number of zeros of \(F\) subtleties when trying to solve systems by Homotopy continuation are pointed out. A relevant article is [\textit{B. Huber} and \textit{B. Sturmfels} Math. Comput. 64, No. 212, 1541--1555 (1995; Zbl 0849.65030)] and [\textit{D. J. Bates} et al., ``Numerical Nonlinear Algebra'', Preprint, \url{arXiv:2302.08585}] and, reviewer thinks, interesting also will be here: [\textit{B. Sturmfels}, Am. Math. Mon. 105, No. 10, 907--922 (1998; Zbl 0988.52021)].
Chapter 4, Polar Degrees: These are fundamental for assessing the algebraic complexity of polynomial optimization problems like e.g. the Euclidean Distance (ED) problem, which in modified guise will recur as a problem in terms of the Wasserstein distance and the Kullback Leibler distance. Polar degrees are here defined via nontransversal intersections, Schubert varieties, the Gauss map; and conormal varieties. Projective duality and the connections to Chern classes are discussed.
Section 4.1, Polar varieties. The aim in section 4.2 will be to show that the family of polar degrees \(\delta_i(X)\) of a projective variety \(X\) coincides in reverse order with the family of certain other degrees \(\mu_i(X)\) (also called polar degrees) defined with the help of polar varieties. The preparation for this is done here. The polar variety \(P(X,V)\) of a projective variety \(X\) in \(\mathbb P^n\) with respect to a subspace \(V\) is formally defined and from it numbers \(\mu_i(X)=\deg(P(X,V))\) are derived depending on the dimension of \(V\). In one simple case, \(V=v\) (one point), \(X\) a surface, \(P(X,v)\) is the contour how the eye sees a surface. Using formulas for the dimension of intersection and join of projective spaces, one gets a link between non-transversel intersection of \(X\) by \(V+p\) at a regular point \(p\) -- this is a concept entering in the definition of \(P(X,V)\) -- and Schubert varieties \(\Sigma_m(V)\). The (partial) Gauss map \(X\ni p\stackrel{\gamma_X}{\rightarrow} T_pX\in \text{Gr}(m, \mathbb P^n)\) is introduced and with this one finds \(P(X,V)=\overline{\gamma_X^{-1}(\Sigma_m(V))}\); So \(P(X,V)\) is the closure of the pullback of the Schubert variety under the Gauss map.
Section 4.2, Projective duality. Primal-dual approaches in polynomial optimization can be understood via the duality theory of projective algebraic geometry. If \(X\) is a projective variety, then \(X^{\vee}\) (the dual of \(X\)) is the projection of the conormal variety onto the 2nd factor; so it parametrizes all tangent hyperplanes: \(X^{\vee}=\overline{\{H^v\in \mathbb (P^n)^* : \exists p\in \text{Reg}(X): T_pX\subseteq H \}}\). This generalizes the definition of the dual of a plane curve. If \(X\) is a projective variety and \(p\in \text{Reg}(X)\), then we have: the hyperplane \(H\) is tangent to \(X\) in \(p\) iff the hyperplane \(p^{v}\) is tangent to \(X^{\vee}\) in \(H^{\vee}\). This instance of biduality is illustrated with the twisted cubic. As the highlight of this section it is shown that the multidegrees \(\delta_j(X)=\# (N_X\cap (L_1\times L_2))\) with \(\dim L_1=n+1-j\) and \(\dim L_2=j\) where \(L_1, L_2\) are generic projective subspaces coincide with the polar degrees in reverse order: \(\delta_j(X)=\mu_i(X)\) iff \(i+j=\dim X+1\). Finally a Segre variety whose polar degrees \(\delta_j(X)\) were computed in Chapter 2 is revisited to exemplify the theorem. Relevant literature in this and the next section seems to be [\textit{A. Holme}, Manuscr. Math. 61, No. 2, 145--162 (1988; Zbl 0657.14033)] and [\textit{J. B. Carrell} (ed.) et al., Proceedings of the 1984 Vancouver conference in algebraic geometry, held July 2-12, 1984, University of British Columbia, Vancouver, British Columbia, Canada. American Mathematical Society (AMS), Providence, RI (1986; Zbl 0575.00008)].
Section 4.3, Chern classes. These are discussed because they are topological invariants that are associated with vector bundles on smooth varieties or manifolds and they occur in algebraic geometry, differential geometry, and algebraic topology which in turn furnish mathematical foundation of data science. Chern classes are objects derived from degeneracy locusses of vector bundles; these latter are certain determinantal varieties. What interests in our context is that Chern classes, denoted \(c_o(\mathcal E), \dots, c_r(\mathcal E)\) where \(\mathcal E\) is a rank \(r\) vector bundle have degrees which are nonnegative integers. They are linked to other notions of degrees for optimization problems in metric algebraic geometry. For example under proper conditions one has the following formula linking polar degrees to degrees of Chern classes: \(\mu_i(X)= \sum_{k=0}^i (-1)^k \binom{m-k+1}{m-i+1} \deg(c_k(X))\). Some arguments here are again from [Draisma et al, loc. cit.].
Chapter 5, Wasserstein Distance: We assume given a norm \(\|.\|\) whose unit ball \(B=\{x\in \mathbb R^n: \|x\|\leq 1 \}\) is a centrally symmetric polytope. As in chapter 2, the topic of the current one is to find given \(u\in \mathbb R^n\) a point \(x\in X\) which minimizes \(\|x-u\|\) subject to \(x\in X\). Section 5.1 presents general results on such norms, while sections 5.2, 5.3 focus on a special class of norms, inducing the so called Wasserstein distance, that arise from optimal transport theory.
Section 5.1, Polyhedral norms. The highlight of this section is as follows. A theorem of \textit{T. Ö. Çelik} et al [J. Symb. Comput. 104, 855--873 (2021, Zbl. 1460.62205)] says: If \(L\subset \mathbb R^n\) is a general affine-linear space of codimension \(i-1\) and \(\ell\) a general linear form, then the number of critical points of \(\ell\) on \(L\cap X\) is the polar degree \(\delta_i(X)\) (introduced in chapters 2 and 4). This is used after a number of reformulations of the above optimization problem to see that (just as happened analogously with the EDdegree), the polar degrees of \(X\) (here together with the \(f\)-vector of \(B\)) determine the total number of its critical points.
Section 5.2, Optimal transport and Independence models. This section centers around the notion of the Wasserstein distance between probability distributions on finite spaces like \([n]=\{1,2, \dots,n \}\). The set \(\Delta_{n-1}=\{(\nu_1, \ldots,\nu_n)\in \mathbb R_{\geq 0}^n : \nu_1+\cdots +\nu_n =1\}\) of distributions is considered and a nonnegative symmetric matrix \((d_{ij})\) implementing a metric (i.e. \(d_{ik}\leq d_{ij}+d_{jk}\)) is fixed. The Wasserstein distance between \(\mu,\nu \in \Delta_{n-1}\), \(W_d(\mu,\nu)\) is given by the optimal value of \(\max \sum_i (\nu_i-\mu_i) z_i\) subject to \(|z_i-z_j| \leq d_{ij}\) for all occurring \(i,j\). Next, it is shown that the dual of the Lipschitz polytope \(P_d\) pertaining to metric \(d\) is precisely the unit ball that defines \(W_d\). A discrete statistical model is a set \(\mathcal M= X\cap \Delta_{n-1}\), where \(X\subset \mathbb R^n\) is an affine variety. An example is the surface \(\{(\nu_1,\nu_2,\nu_3,\nu_4): \nu_1+\nu_2+\nu_3+\nu_4 =1, \nu_1\nu_4=\nu_2\nu_3\}\). The task is to compute \(W_d(\mu,\mathcal M) =\min_{\nu\in \mathcal M} W_d(\mu, \nu)\). The results \(W_d(\mu,\mathcal M)\) for the example are given as published in [\textit{T. Ö. Çelik} et al., Lect. Notes Comput. Sci. 11989, 364--381 (2020; Zbl 07441083)].
Section 5.3, Wasserstein meets Segre-Veronese. The earlier examples are worked out and extended for the case that \(\mathcal M\subset \Delta_{n-1}\) is an independence model for discrete random variables, given by tensors of rank one. (Tensors are the subject of Chapter 12.) By \((m)_r\) is denoted a multinomial distribution with \(m\) possible outcomes and \(r\) trials. The model \(\mathcal M=((m_1)_{r_1}, \dots, (m_k)_{r_k})\). Its elements are \(k\) independent multinomial distributions. To these models correspond in algebraic geometry Segre Veronese varieties. For example if \(\mathcal M=(2_2,2_1)\) then \(\mathcal M\) is the image of the map \([0,1]^2 \ni (p,q) \mapsto (p^2 q, 2p(1-p)q, (1-p)^2q, p^2(1-q), 2p(1-p)(1-q),(1-p)^2 (1-q))=(p^2, 2p(1-p),(1-p)^2)\otimes (q,1-q)\). In algebraic statistics, the independence models are replaced by their complex Zariski closure in projective space; see [\textit{S. Sullivant}, Algebraic statistics. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1408.62004)] So the map is seen as an embedding of \(\mathbb P^1 \times \mathbb P^1\) into \(\mathbb P^5\). Luca Sodomaco has computed the Chern classes of the Segre-Veronese varieties \(\mathcal M\). From there and chapter 4, or see results like those in [\textit{K. Kozhasov} et al., ``On the minimal algebraic complexity of the rank-one approximation problem for general inner products'', Preprint, \url{arXiv:2309.15105}] one gets upper bounds for Wasserstein distance problems using polar degrees. See Proposition 5.21.
Chapter 6, Curvature: This chapter, after reviewing the curvature of plane curves and more sophisticated notions connected to it in Section 6.1, turns to a shorter Section 6.2 where notions like the second fundamental form are considered for algebraic varieties and culminates with presenting in Section 6.3 Herrman Weyl's formulas for the volume of tubes.
Section 6.1, Plane curves. A smooth algebraic curve \(C\subset \mathbb R^2\) is supposed to be given in its implicit representation \(f(x_1,x_2)=0\) via an irreducible polynomial \(f\in \mathbb R[x_1,x_2]\). Elementary differential geometry is first recalled; in particular formulas are given for the normalized normal and tangent vectors, \(N(x)\) and \(T(x)\), \(x\in C\), as well as for curvature \(c(x)\) and radius of curvature. The evolute of a curve \(C\) as defined in Chapter 1 is then understood as the Zariski closure of all centers of curvature of \(C\). Next a later formula, namely \(c(x)=\|\nabla f(x) \|^{-1} \cdot T(x) H(x) T(x)^{\mathsf{T}}\) with \(H\) the Hessian of \(f\) is used in order to compute the inflection points (i.e. curvture=0 points) of the Trott curve with Julia. It illustrates a theorem of Felix Klein that at most a third of the complex inflection points of a curve can be real. That a general curve of degree \(d\) has \(3d(d-2)\) complex inflection points is proved as Theorem 6.7 with the help of Bezout's theorem and the Parameter Continuation Theorem 3.18. It is also shown that the evolute of a general plane curve of degree \(d\) has degree \(3d(d-1)\); for the points of critical curvature an algebraic equation is given and it is shown that a general plane curve \(C\) of degree d has \(2d(3d-5)\) points of critical curvature over \(\mathbb C\). In this proof enter recent results of [\textit{R. Piene} et al., ``Return of the plane evolute'', Preprint, \url{arXiv:2110.11691}].
Section 6.2, Algebraic varieties. The main goal here is to study the curvature of a smooth algebraic variety of any dimension. This largely means adapting general differential geometry to the case of algebraic varieties. Now every choice of a normal and tangent vector to \(X\) in a point \(x\in X\) defines a curvatures; so curvature is now a function \(c=c(x,t,v)\) where \(t\in T_xX\) and \(v\in N_xX\). Supposing the Zariski closure of \(X\) (usually again called \(X\)) is irreducible and has (prime) ideal \(I(X)=\text{ideal}(f_1, \dots, f_k)\), a formula \(c(x,t,v)= \frac{1}{\|v\|} t^{\mathsf{T}} \left(\sum_{i=1}^k w_i H_i \right) t\) where \(H_i\) is the Hessian of \(f_i\), \(t\in T_xX \), \(v=J(x)w(x)\in N_xX\) is deduced and shown to be well defined although \(w:X\rightarrow \mathbb R^k\) is a nonunique smooth function. One also writes \(\Pi_v(t)\) for \(c(x,t,v)\) and calls this the second fundamental form. Associated to this quadratic form is the linear Weingarten map \(L_v: T_xX\rightarrow T_xX\) given by \(L_v(t)=P_x\left(\sum_{i=1}^k w_i H_i t \right) \), where \(P_x:\mathbb R^n \rightarrow T_xX\) is an orthogonal projection. The eigenvalues of \(L_v\) are the principal curvatures. The maximum over all principal curvatures when \(x\) ranges over \(X\) can be found among the critical points of the \(\Pi_v(t)\). The rest of the section is dedicated to a report on the number of umbilics (points with two equal principal curvatures) for surfaces \(S\subset \mathbb R^3\). This concerns the number of complex umbilics on a degree \(d\) surface, a 1865 result of Salmon, and results in [\textit{P. Breiding} et al., ``Critical curvature of algebraic surfaces in three-space'', Preprint, \url{arXiv:2206.09130}] concerning real umbilics and real critical curvature points on general quadric surfaces.
Section 6.3. Volumes of Tubular Neighbourhoods. This section is dedicated to the study of volumes of sets of the form \(\text{Tube}(X,\epsilon)=\{u\in \mathbb R^n: d(u,X)<\epsilon\}\) for \(X\) real algebraic varieties. So it is the set of points in \(\mathbb R^n\) that have from \(X\) an Euclidean distance not more than \(\epsilon\). If \(\mathcal N X\) is the normal bundle of \(X\) let \(\mathcal N_\epsilon X =\{(x,v)\in \mathcal N X: \|v\|< \epsilon\}\) and consider the map \(\varphi_\varepsilon : \mathcal N_\epsilon X \ni (x,v)\stackrel{\varphi_\epsilon }{\mapsto} x+v \in \text{Tube}(X,\epsilon)\). With this the precise definition of the \textit{reach} of \(X\) is \(\tau(X) := \sup\{\epsilon:\varphi_\varepsilon \text{ is a diffeomorphism} \}\). It is the topic of the next chapter. Here it serves as a bound for \(\epsilon\). Namely the arguments present the principal ideas that establish the tube formula [\textit{H. Weyl}, Am. J. Math. 61, 461--472 (1939; Zbl 0021.35503)]: If \(X\) is smooth compact of dimension \(m\) and \(\epsilon <\tau(X)\), then \(\text{vol}(\text{Tube}(X,\epsilon)) =\sum_{0\leq 2i \leq m} \kappa_{2i}(X) \epsilon^{n-m+2i}\), where the \(\kappa_{2i}(X)\), called principal curvature coefficients are computed by integration over the sum of the principal \(2i\times 2i\) minors of the Weingarten map. Three illustrations of this formula are given. Most notably the volume of an \(\epsilon\)-tube of a surface \(S=\{f(x)=0\}\subset \mathbb R^3\) of Euler characteristic \(\chi (S)\) is computed. Here the Gauss Bonnet theorem is invoked.
Chapter 7, Reach and Offset: Here are studied notions important in many applications like the medial axis, bottlenecks and offset hypersurfaces. The first two are intuitive, certain real images of the last one as well. Section 7.3 establishes a link to the differential geometry of Chapter 6.
Section 7.1, Medial Axis and Bottlenecks. Given a set \(X\subset \mathbb R^n\), the medial axis \(\text{Med}(X)\) is the set of all points of \(\mathbb R^n\) which have from at least two distinct points of \(X\) the same distance. If \(X\) is semialgebraic, then so is \(\text{Med}(X)\) and if \(X\) is a smooth real variety, then the distance from \(X\) to \(\text{Med}(X)\) equals the reach \(\tau(X)\). The algebraic medial axis is the Zariski closure \(M_X\) of \(\text{Med}(X)\). For example if \(X={y-x^2=0}\) is the parabola, then \(\text{Med} X=\{(0,u_2): u_2>\frac{1}{2}\}\), but \(M_X=\{u_1=0\}\). \(\tau(X)= \frac{1}{2}\). A set \(\{x,y\}\) of two distinct points of a smooth variety is a bottleneck if \(x-y\) is at the same time perpendicular to the tangent spaces at \(x\) and at \(y\). By [\textit{S. Di Rocco} et al., SIAM J. Appl. Algebra Geom. 4, No. 1, 227--253 (2020; Zbl 1505.14122)] a general plane curve of degree \(d\) has \(\frac{1}{2}(d^4-5d^2+4d)\) complex bottlenecks. Short inviting codes are indicated of Macaulay2 to compute a medial axis and Julia is used to compute the 36 real bottlenecks of the alltogether 96 existing bottlenecks of the Trott curve. Buy the book to find on page 84 the long proof of a short formula to compute the reach \(\tau(X)\) of a smooth variety using the smallest width of a bottleneck and the maximal curvature of \(X\)!
Section 7.2, Offset Hypersurfaces. This section is in its first part based on [\textit{E. Horobeţ} and \textit{M. Weinstein}, Comput. Aided Geom. Des. 74, Article ID 101767, 14 p. (2019; Zbl 1505.55017)]. It first offers the formal and rather technical definition of an offset hypersurface Off\(_X\) associated to an irreducible variety \(X\) in \(\mathbb R^n\) (identified with its Zariski closure in \(\mathbb C^n\)). It is then shown that Off\(_X\) has codimension 1 and is in fact the zero set of a certain polynomial \(g_X(u,\epsilon)\). In other words \(\text{Off}_X =V(g_X)\subset \mathbb C^n \times \mathbb C\). Concerning the meaning of \(\text{Off}_X\), when one intersects it with the hyperplane \(\epsilon=r\), one gets the set \(\text{Off}_{X,r}\) the real locus of which contains the boundary of the tubular neighbourhood Tube\((X,r)\) and so loosely speaking is a surface at distance \(r\) from \(X\). In fact concerning the offset polynomial \(g_X\) for a general point \(u\in \mathbb C^n\) the zeros of the polynomial function \(\epsilon\mapsto g_X(u,\epsilon)\) are the complex numbers \(\pm\sqrt{\|x-u\|^2} \), where \(x\) ranges over all ED critical points for \(u\) on \(X\). The offset polynomial (also known as ED polynomial) was studied by [\textit{G. Ottaviani} and \textit{L. Sodomaco}, Comput. Aided Geom. Des. 82, Article ID 101927, 20 p. (2020; Zbl 1453.65046)]
Section 7.3, Offset Discriminant. From the definitions before it follows that if \(u\) lies on the medial axis of variety \(X\subset \mathbb R^n\) then the polynomial \(g_X(u,\epsilon)\) must have a double root. This motivates the study the discriminant of this polynomial with respect to \(u\), that is \(\delta_X(u)= \text{Disc}_\epsilon g_X(u,\epsilon)\). Its zeroset \(\Delta_X^{\text{Off}} = V(\delta_X)\). It is shown based on [Horobeţ and Weinstein, loc. cit.] that the real locus of \(\Delta_X^{\text{Off}}\) consists of the real points of the varieties \(X\), \(M_X\) (algebraic medial axis) and \(\Sigma_X\) (evolute of \(X\)). It is also shown that the second fundamental form of variety \(X\) can be computed from the offset polynomial.
Chapter 8, Voronoi Cells: Let \(X\subseteq \mathbb R^n\) and \(y\in X\). The Voronoi cell \(\text{Vor}_X(y)\) is the set of all \(u\in \mathbb R^n\) which are nearer to \(y\) than to any other point in \(X\). More precisely \(u\in \text{Vor}_X(y)\) iff \(d(u,X)=d(u,y)\).
Section 8.1, Voronoi Basics. If \(X\) is a finite set then \(\text{Vor}_X(y)\) is a polyhedron with \(|X|-1\) facets. The emphasis in the current chapter are Voronoi cells associated to real algebraic varieties. The case of taking samples from or near an algebraic curve and determining the associated Voronoi cells was studied by [\textit{M. Brandt} and \textit{M. Weinstein}, ``Voronoi cells in metric algebraic geometry of plane curves'', Preprint, \url{arXiv:1906.11337}] who inclusively posted an instructive video that makes one smile. If variety \(X\) has codimension \(c\) Vor\(_X(y)\) is a convex semialgebraic full-dimensional subset of the \(c\)-dimensional affine normal space \(N_X(y)=y+N_yX\). The interest of the authors is in the algebraic boundary \(\partial_{\mathrm{alg}} (\text{Vor}_X(y))\) of a Voronoi cell \(\text{Vor}_X(y)\), defined as the Zariski closure of the topological boundary \(\partial \text{Vor}_X(y)\). Information is provided in particular for \(\text{Vor}_X(y)\) and its (algebraic) boundary in the the cases \(X\) is a curve or surface (i.p. quadric) in 3-space.
Section 8.2, Algebraic Boundaries. This section and the next is based on [\textit{D. Cifuentes} et al. J. Symb. Comput. 109, 351--366 (2022; Zbl 1477.13054)] Here a Groebner basis method is described for finding the Voronoi boundaries \(\partial_{\mathrm{alg}} (\text{Vor}_X(y))\) of a given variety. The algorithm described takes as input \(y\) and the ideal of \(X\) and yields as an output an ideal \(\text{Vor}_I(y)\) defining an affine variety in \(\mathbb C^n\) containing the algebraic Voronoi boundary \(\partial_{\mathrm{alg}} (\text{Vor}_X(y))\) and actually generically being equal to it. Notions involved in the algorithm are the ring \(R=\mathbb Q[x_1, \dots, x_n, u_1, \dots, u_n]\) for computations, \((c+1)\times(c+1)\) minors of an augmented Jacobian \(\mathcal{A J}(x,u)\), the Euclidean normal bundle; saturation; ideal intersection. The main results are discussed using the case \(n=2\) and the cuspidal cubic given by \(I=\mathrm{ideal}(x_1^3-x_2^2)\). Experiments with Macaulay2 lead to precise conjectures for the Voronoi degree \(\deg(\partial_{\mathrm{alg}} (\text{Vor}_{\langle f\rangle }(y)))\) of a generic hypersurface of degree \(d\) and of a projective such hypersurface. \;\;
Section 8.3, Degree Formulas. Results on the degrees \(\nu_X(y)\) are presented for curves and surfaces \(X\) (identified with their Zariski closures) in complex projective space \(\mathbb P^n\). It is always assumed that \(X\) is in general position and \(y\) a general point of \(X\). For example if \(X\) is a smooth surface of degree \(d\), then \(\nu_X(y)=3d+\chi(X)+4g(X)-11\), where \(\chi(X)\) is the Euler characteristic of \(X\) and \(g(X)\) the genus of the curve obtained by intersecting \(X\) with a general quadratic hypersurface. The proofs can be found in [Cifuentes et al., loc. cit.].
Section 8.4, Voronoi meets Eckart-Young. This section considers the Voronoi question in the case of \(X\) being the variety of all real \(m\times n\) matrices of rank\(\leq r\). The two norms used are the operator norm and the Euclidean or Frobenius norm. Using the Eckart-Young theorem if \(V\) is in \(X\) of rank \(r\), the Voronoi cell \(\text{Vor}_X(V)\) is shown to be a ball of radius equal to the \(r\)-th singular value in the operator norm on the space of \((m-r)\times (n-r)\)-matrices. (The Voronoi-cells are viewn as living in the normal space \(N_X(V)\).) The section also establishes a relation between the relative size of Voronoi cells and condition numbers and points out phenomena that can happen as we consider different norms on the variety \(X\) of symmetric \(n\times n\) matrices. This section borrows from the article by Draisma et al. already cited.
Chapter 9, Condition Numbers: These numbers measure how much the output value of a function changes for a small change in input argument.
Section 9.1, Errors in Numerical Computations. The classical example showing how minimal perturbations of a matrix can have dramatic effect on matrix inversion is presented. This motivates the concept of condition numbers which helps to understand and measure such effects. Understanding (with numerical analysts) by a computational problem simply a function \(\phi: M\rightarrow N\) where \(M,N\) are semialgebraic, Rice's concept of the absolute condition number [\textit{J. R. Rice}, Siam J. Numer. Anal, 3: 287--310 (1966; Zbl 0143.37101)] of \(\phi\) at \(p\in M\), denoted \(\kappa[\phi](p)\) and the relative condition number \(\kappa_{\mathrm{REL}}[\phi](p) = \kappa[\phi](p)\frac{\|p\|}{\|\phi(p)\|}\) are presented and the relevance of the latter for floating point number systems -- which are also formally defined -- are explained. In the setting of metric algebraic geometry Rice's condition number is \(\kappa[\phi](p)=\max_{t\in T_pM, \|t\|=1} \|D_p\phi(t)\|\). Condition numbers for the problem to find one real root of a 1-variable polynomial and for the ED minimization problem are determined [\textit{P. Breiding} and \textit{N. Vannieuwenhoven}, SIAM J. Optim. 31, No. 1, 1049--1077 (2021; Zbl 1462.90133)]. In this case \(\phi: \mathbb R^n \setminus \text{Med}(X) \rightarrow X\) is the closest-point-to-variety \(X\) function (in the sense of Chapter 2).
Section 9.2, Matrix Inversion and Eckart-Young. Here the condition numbers of matrix inversion are shown to be \(\kappa[\text{inv}](A)=\sigma_n^{-2}\) and \(\kappa_{\mathrm{REL}}[\text{inv}](A)=\sigma_1/\sigma_n\) where the \(\sigma_i\) are the singular values of \(A\) and the operator norm is used. It is shown that Turing' s condition number [\textit{A. M. Turing}, Q. J. Mech. Appl. Math. 1, 287--308 (1948; Zbl 0033.28501)] is the squared inverse distance from matrix \(A\) to the variety of singular matrices. Its relation to Voronoi cells is also mentioned. Macaulay2 is used to compute an offset hypersurface for the variety of \(2\times 2\) matrices of determinant 0. A proof of the famous Eckart-Young theorem is presented.
Section 9.3, Condition Number Theorems. These are theorems that connect condition numbers with distances to ill-posed problems. See [\textit{J. W. Demmel}, Numer. Math. 51, 251--289 (1987; Zbl 0597.65036)]. Condition number theorems for solving systems of polynomial equations are proved. Endow the vector space \(\mathcal H_d\) of real homogeneous polynomials of degree \(d\) in variables \(x=(x_0, \dots, x_n)\) with the Bombieri-Weyl inner product. Extend this inner product naturally to pairs of system of polynomials by \(\langle F,G \rangle_{BW} =\sum_{i=1}^m \langle f_i,g_i\rangle_{BW}\). One has herewith a notion of a norm of a system and distance between two systems of \(m\) polynomials. Given \(F\) one selects a root \(\mathbf{a}=a(F)\) where \(a\) is the root-selection function defined in a neighbourhood of \(\mathbf{a}\). Upon diferentiating the equation \(F(a(F))=0\), the authors find the condition number \(\kappa [a](F)\) for solving \(n\) polynomial equations in \(\mathbb P^n\) as being \(\sigma_n(JF(\mathbf{a})(I_{n+1}-\mathbf{a}\mathbf{a}^{\mathsf{T}})))^{-1}\). The section determines furthemore the distances \(\text{dist}_{BW}(F,\sigma(x))\) and \(\text{dist}_{BW}(F,\sigma)\), where given \(\mathbf{d}\), for \(x\in \mathbb P_{\mathbb R}^n\) the local discriminant \(\Sigma(x)=\{F\in \mathcal H_{\mathbf{d}}: F(x)=0 \text{ and rank} JF(x)<m \}\), while the discriminant \(\Sigma\) as such is the union of the local discriminants \(\Sigma(x)\) as \(x\) runs over \(\mathbb P^n\). Find also a Macaulay2 code by which the discriminant of a ternary cubic was determined to be a polynomial in 2040 terms of degree 12 in the 10 symbolic coefficients of the cubic.
Section 9.4, Distance to the Discriminant. This section deals with sparse quadratic systems (see Section 3.3). In this case there exists an essentially unique irreducible polynomial \(\Delta(\mathbf{c})\) which vanishes whenever the equations (via the coefficients encoded in \(\mathbf{c}\)) have a double root in \((\mathbb C^*)^n\). The discriminant \(\Sigma=\{\mathbf c\in \mathbb R^N: \Delta(c)=0\}\). To the \(n\)-uple of collections \((\mathcal{A}_i)_{i=1}^n\) associated is a toric variety \(X\) whose dual \(X^{\vee}\) is precisely \(\Sigma\). This is known as the Cayley trick [\textit{I. M. Gelfand} et al., Discriminants, resultants, and multidimensional determinants. Boston, MA: Birkhäuser (1994; Zbl 0827.14036); \textit{E. Cattani} et al., Math. Z. 274, No. 3--4, 761--778 (2013; Zbl 1273.13051)]. The discussions result in Theorem 9.26: The Euclidean distance from the coefficient vector \(\mathbf{u}\) of a polynomial system to the discriminant \(\Sigma\) equals the smallest norm \(\|x\|\) among all points \(\mathbf{x}\) in the toric variety \(X=\Sigma^\vee\) that are critical for the distance to \(\mathbf{u}\).
Chapter 10, Machine Learning: One of the principal goals of machine learning is to learn in an automated way functions that represent the relationship between data points: If we are given a data set \(\mathcal D=\{(x_1,y_1), \dots, (x_d,y_d)\} \subset \mathbb R^n\times \mathbb R^m\) with input data \(x_i\) and output data \(y_i\) the goal is to find a function \(f:\mathbb R^n \rightarrow \mathbb R^m\) such that \(f(x_i)\approx y_i\). In practice this means we wish to minimize a loss \(\ell_D(f)= \frac{1}{d} \sum_{i=1}^d l(f(x_i),y_i)\) where the loss function \(l\) could be for example given as the squared euclidean distance or the Wasserstein distance of Kullback -Leibler divergence. These explanations refer to supervised learning. In unsupervised learning we have a dataset \(\mathcal D= \{x_1,x_2, \dots, x_d\}\) and wish for example to learn a variety \(X\subset \mathbb R^n\) which represents the data.
Section 10.1, Neural Networks. Mathematically a feed forward neural network is a composition of functions \(f=f_{L,\theta}\circ f_{L-1,\theta}\circ \cdots \circ f_{1,\theta}\) where each \(f_{i,\theta}\) is a map from some \(\mathbb R^j\) to some \(\mathbb R^k\) such that the composition makes sense and is of the form \((\sigma\circ \alpha_1, \cdots, \sigma\circ \alpha_k)\), where each \(\alpha_i:\mathbb R^j \rightarrow \mathbb R\) is linear and is expressed by the parameters \(\theta= (\theta_1, \dots,\theta_N)\in \mathbb R^N\). \(\sigma: \mathbb R\rightarrow \mathbb R\) is called activation function and might depend on \(i\). Possible activation functions are e.g. \(\sigma_i(z)=\sigma(z)=\max\{0,z\}\) (called Rectified Linear Units, ReLU) or the identity function. The image of the map \(\mathbb R^N\ni \theta \stackrel{\mu}{\mapsto} f_{L,\theta}\circ \cdots \circ f_{1,\theta}\) is the \textit{neuromanifold} \(\mathcal M\). Using the loss function \(\ell_D\) one minimizes by training the function \(\mathcal L= \ell_D\circ \mu\). The section refers that every piecewise linear function with finitely many pieces arises from a fully connected ReLU [\textit{R. Arora} et al., ``Understanding deep neural networks with rectified linear units'', Preprint, \url{arXiv:1611.01491}]; that ReLU end-to-end functions can be interpreted as tropical rational functions [\textit{L. Zhang} et al., ``Tropical geometry of deep neural networks'', Preprint, \url{arXiv:1805.07091}; \textit{G. Montufar} et al., SIAM J. Appl. Algebra Geom. 6, No. 4, 618--649 (2022; Zbl. 07673258)]; and that the mystery why training neural networks results typically in nice minima was e.g. was examined in [\textit{D. Mehta} et al., IEEE Trans. Pattern Anal. Mach. Intel. 44, No. 9, 5664--5680 (2021)]. The rest of the section focuses on results for linear fully connected networks -- so on cases where the activation function is linear. In such cases the neuromanifold is a determinantal variety, namely it is of the form \(\mathcal M=\{W\in \mathbb R^{k_l\times k_0}: \text{rank} W\leq \min\{k_0,k_1, \dots, k_L\}\}\), where the \(k_i\) reflect the dimensions of the in and output spaces of the \(f_{i,\theta}\). In [\textit{M. Trager} et al., ``Pure and Spurious Critical Points: a Geometric Study of Linear Networks'', Preprint, \url{arXiv:1910.01671}] is shown that in linear fully connected networks under mild conditions the function \(\mathbb L\) has non-global local minima if and only if \(\ell_{\mathbb D}|_{\mathrm{Reg}}(\mathcal{M}\) has nonglobal local minima. This is used to analyse special cases in which bad minima can or cannot occur. Finally convergence analyses for optimization algorithms used in the training of neural networks are reported. Algebraic invariants of the gradient flow are here important. See [\textit{G. Nguegnang} et al., ``Convergence of gradient descent for learning linear neural networks'', Preprint, \url{arXiv:2108.02040}].
Section 10.2, Convolutional Networks. In linear convolutional networks the activation functions are still identities but the linear maps \(\alpha_{i,\theta}\) are chosen to be generalized Toeplitz matrices namely convolutions: roughly a \(k\times n\) matrix of this type has the property that the first row is of the form \((*,*,\dots,*, 0,0,\dots 0)\) and the \(i\)-th row equals the first row shifted to the right by \((i-1)s\) places. \(s\) is the stride. It is shown that the neuromanifold of such networks is parametrized by polynomial multiplication and is semialgebraic. The degrees and dimension of the Zariski closure of such manifolds are determined and their singular locuses can be described. They often stand in relation with Segre embeddings. The results reported in this section stem from [\textit{K. Kohn} et al., ``Function Space and Critical Points of Linear Convolutional Networks'', Preprint, \url{arXiv:2304.05752}; SIAM J. Appl. Algebra Geom. 6, No. 3, 368--406 (2022; Zbl 1514.68254)].
Section 10.3, Learning Varieties. This section deals with unsupervised learning as explained in the introduction to the chapter. We assume \(d\) data points in \(\mathcal D=\{x_1, \dots,x_d\}\) assumed to lie (forgetting noise) on a real algebraic variety \(X\). We do not know anything of the variety but wish to learn it. \(X\) is assumed irreducible. Strategies are discussed to learn \(I_{\mathcal D}\) which is meant to be a proxy to \(I(X)\) with respect to a certain loss function. Using a selection of monomials from which to linearly combine the polynomials \(f\in I(X)\),Vandermonde matrices and singular value decomposition, a strategy is described to learn \(X\). More such proposals are found in [\textit{P. Breiding} et al., Rev. Mat. Complut. 31, No. 3, 545--593 (2018, Zbl 1481.13040)]. The section concludes with a discussion of how machine learning can be helpful or not helpful for theoretical research in pure and applied algebraic geometry.
Chapter 11, Maximum Likelihood: The chapter marks a return to the question of minimizing the distance to a family of probability distributions. This time by means of the Kullback-Leibler divergence: Minimizing the KL divergence is equivalent to maximum likelihood estimation. In Section 11.2 the MLdegree(X) is introduced as an analogue to the EDdegree(X). A connection to physics (scattering) is found in Section 11.3. In Section 11.4 the ML degree for Gaussian models is introduced. This chapter draws on the numerous papers by Sturmfels and coworkers whose titles contains the word `likelihood.'
Section 11.1, KL Divergence. First, the authors look at discrete models. Define \(\Delta_n^0=\{p=(p_0,p_1,\dots,p_n)\in \mathbb R_{>0}^{n+1}: |p|=1 \}\). For \(p,q\in \Delta_n^0\) the the KL divergence is defined by \(D_{KL}(q||p)=-\sum_0^n q_i\ln (\frac{q_i}{p_i})\). Although \(D_{KL}\) does not define a metric (it is not even symmetric) it is a nonnegative quantity that is 0 iff \(q=p\). The entropy of a distribution \(q\) is \(H =-\sum_0^n q_i \ln q_i\). The summands, and hence \(H\), are \(\geq 0\) since the \(q_i\leq 1\). \(H\) is the expected value of the information content of the event \(Z=i\). The idea is that the probability of \(Z=i\) determines its information content: If \(q_i=\text{prob}(Z=i)\) is small then its information content \(-\ln q_i\) is large. If we have \(N\) samples each in one of the states \(i\) and write \(u_i=\# \text{ of samples in state \(i\) }\),\; the function \(\ell_u\) is defined by \(\Delta_n^0 \ni p\mapsto \ell_u (p) :=\sum_1^n u_i \ln p_i\). Performing an ML-estimation for the model \(X\) means solving Maximize \(\ell_u (p) \text{ subject to } p\in X\). Note \(|u|=N\). Then this maximization is equivalent to minimizing \(D_{KL}(\frac{u}{N}||p) \text{ subject to } p\in X\). The independence model for two binary random variables is a quadratic surface mentioned in Section 5.3. That example is continued. Lagrange multipliers are applied in order to minimize the KL divergence from a \(2\times 2\) matrix u to the quadratic surface \(X\). It turns out that the optimal solution is an algebraic function of \(u\). Namely \(\hat{p}_0= |u|^{-2}(u_0+u_1)(u_0+u_2)\), and similar formulae for \(\hat{p}_1,\hat{p}_2,\hat{p}_3\).
Section 11.2, Maximum Likelihood Degree. Algebraicities like In the previous example make such problems often amenable to algebraic geometry. In the present section \(X\subseteq \mathbb{P}_{\mathbb R}^n\) is a fixed real projective variety and the algebraic geometry of the function \(u\mapsto \hat p\) is studied. Important will be the very affine variety \(X^0\). The maximum likelihood degree (ML degree) of the variety \(X\) is defined to be the number of complex critical points of the optimization problem above. The MLdegree(X) is shown to be the degree of a critical ideal which is determined following ideas already seen earlier for the EDdegree. In one example again Bezout' s theorem is used. Further is found an upper bound for \(\text{MLdegree}(X)\) with \(X\) a closed subvariety of an algebraic torus, expressing it via the Euler characteristic; a theorem describing the algebraic structure \(\hat p\) of a model \(X\subseteq \Delta_n^0\) of ML degree one as a consequence of a theorem of (Fields medallist) June Huh; models with rank constraints on matrices and tensors. These are particularly important in applications. One of these occurred as a neuromanifold.
Section 11.3, Scattering equations. This section treats a connection between algebraic statistics and particle physics, see [\textit{B. Sturmfels} and \textit{S. Telen}, Algebr. Stat. 12, No. 2, 167--186 (2021; 1515.62137)]. It relies on a model proposed by Cachazo and coworkers. The CEGM model is the \((k-1)(m-k-1)\)-dimensional manifold \(X^0=\text{Gr}(k-1, \mathbb P^{m-1})^0 /(\mathbb C^*)^m\). It has a configuration space which is a very affine variety (i.e. a closed subvariety of an algebraic torus) here given in detail. The scattering potential for the CEGM model is the function \(\ell_u= \sum_{i_1<i_2< \ldots <i_k} u_{i_1\cdots i_k} \ln p_{i_1\cdots i_k}\) where \(u= u_{i_1\cdots i_k}\) is the data vector (Mandelstam invariants) and \(p_{i_1\cdots i_k}\) are the minors of matrix \(M_{k,m}\). The number of complex solutions to the scattering equations (of form \((\frac{\partial \ell_u}{\partial x_{ij} })=0\)) can be determined up to enourmous numbers still lying within current software capabilities; in this case Julia. See also [\textit{D. Agostini} et al., Adv. Math. 414, Article ID 108863, 39 p. (2023; Zbl 1535.14106)]. The section shows that the ML degree of very affine varieties appears in contexts beyond algebraic statistics.
Section 11.4, Gaussian Models. Statistical models for Gaussian random variables are considered. Given a mean vector \(\mu\in \mathbb R^n\) and a covariance matrix \(\Sigma\in PD_n\) (positive semidefinite matrices) the associated Gaussian distribution \(f_{\mu,\Sigma}(x) =(\sqrt{(2\pi)^n \det \Sigma})^{-1} \exp(-\frac{1}{2}(x-\mu)^{\mathsf{T}}\Sigma (x-\mu))\). One fixes then a model \(Y\subset \mathbb R^n \times PD_n\) defined by polynomial equations in \((\mu,\Sigma)\). Given \(N\) samples \(U^{(1)}, \dots, U^{(N)}\in \mathbb R^n\), the sample mean \(\overline{U}\) and the sample covariance matrix \(S\) define the log-likelihood \(\ell(\mu,\Sigma)=-\frac{N}{2}(\ln \det(\Sigma) + \text{trace}(S\Sigma^{-1}) + (\overline{U}-p)^{\mathsf{T}} \Sigma^{-1} (\overline{U}-p))\). The task is to maximize this function subject to \((\mu,\Sigma)\in Y\). The case \(Y=\mathbb R^n \times X\) with \(X\) some subvariety of \(S^2(\mathbb R^n)\) is treated here. The critical equations for this problem can be written as polynomials because the partial derivatives of logarithms are rational functions. The finite number of complex solutions to these equations is the MLdegree of the statistical model \(X^{-1}\). In the remainder of the section the cases \(X=\mathcal L\) and \(X^{-1}=\mathcal L\) are treated where \(\mathcal L\) is a sufficiently generic linear subspace of \(S^2(\mathbb R^n)\) (symmetric \(n\times n\) matrices). A rather surprising result linking the problem to the number of quadrics in \(\mathbb P^{n-1}\) passing through general points and hyperplanes, due to [\textit{M. Michalek} et al. SIAM J. Appl. Algebra Geom. 5, No. 1, 60--85 (2021; Zbl 1461.62228)], is cited.
Chapter 12, Tensors: For \(n_1,\dots,n_d\) positive integers a tensor is a map \([n_1]\times \cdots \times [n_d] \rightarrow \mathbb R\), and therefore can be identified with an element in \(\mathbb R^{n_1\times \cdots \times n_d}=\{A=(a_{i_1, \dots,i_d})_{1\leq i_1\leq n_1, \dots, 1\leq i_d\leq n_d}| a_{i_1, \dots,i_d}\in \mathbb R \}\). The number \(d\) is the order of the tensor. So for \(d=1\) we speak of vectors; for \(d=2\) of matrices.
Section 12.1, Tensors and their rank. For tensors \(A,B\) of the same order \(d\) one defines the Euclidean inner product \(\langle A,B \rangle = \sum a_{i_1,i_2, \dots, i_d} b_{i_1,i_2, \dots, i_d} \), and hence has a norm \(\|A\|=\sqrt{\langle A,A\rangle}\). A way to create tensors of order \(d\) from vectors \(v_1\in \mathbb R^{n_1}, \dots, v_d \in \mathbb R^{n_d}\) is given by forming the outer product \(v_1\otimes \cdots \otimes v_d = ((v_1)_{i_1} (v_2)_{i_2} \cdots (v_d)_{i_d})_{1\leq i_1\leq n_1, \dots, 1\leq i_d\leq n_d}\) . Such a tensor is symmetric if for all \(\pi \in \mathfrak S_d\) we have \(v_{\pi(1)}\otimes \cdots \otimes v_{\pi(d)} =A\) and the vector space of symmetric tensors of order \(d\) is denoted \(S^d(\mathbb R^n)\). If \(A\in S^d(\mathbb R^n)\) it determines a homogeneous polynomial of degree \(d\) namely \(F_A(x) =\sum a_{i_1,i_2, \dots, i_d} x_{i_1} \cdots x_{i_d} \in \mathbb R[x_1, \dots, x_n]_d= \langle A,x^{\otimes d} \rangle\) and we have an isomorphism \(S^d(\mathbb R^n) \ni A \mapsto F_A \in \mathbb R[x_1, \dots, x_n]_d\). It holds \(\|A\|=\|F_A\|_{BW}\). One defines for suitably formatted matrices and vectors the action \((M_1, \dots, M_d). (v_1\otimes \cdots \otimes v_d):= (M_1v_1)\otimes \cdots \otimes (M_dv_d)\) and extends linearly. It is called multilinear multiplication and one has for example \((M_1, M_2).A= M_1AM_2^{\mathsf{T}}\). The inner product of rank one tensors (i.e. tensor products of vectors) of the same size is \(\langle v_1\otimes\cdots \otimes v_d, w_1\otimes\cdots\otimes w_d \rangle= \langle v_1,w_1\rangle \cdots \langle v_d,w_d\rangle\). The set \(\mathcal S_{n_1,n_2, \dots, n_d}= \{v_1\otimes \cdots \otimes v_d : v_i\in \mathbb R^{n_i} \}\) is a variety called a Segre variety (special cases of which occured earlier). For symmetric tensors one defines similarly the Veronese variety. Given a tensor \(A\in \mathbb R^{n_1\times n_2\times \cdots n_d}\), one defines \(\text{rank}(A)=\min\{r\geq 0: \text{ there exist \(A_1, \dots, A_r \in \mathcal S_{n_1,n_2, \dots, n_d}\) such that \(A=A_1+\cdots + A_r\) }\}\). Letting \(\mathcal R_r= \mathcal R_{r,n_1, \dots, n_d} =\{A\in R^{n_1\times n_2\times \cdots n_d} : \text{rank}(A)\leq r\}\), one has that rank 1 tensors are varieties, but the sets \(\mathcal R_r\) for \(r\geq 2\) are not varieties; see [\textit{Vin de Silva} and \textit{Lek-Heng Lim}, SIAM J. Matrix Anal. Appl. 30, No. 3, 1084--1127 (2008; Zbl 1167.14038) ]. Finally a result saying that a tensor \(A\in R^{n_1\times n_2\times \cdots n_d}\) has often a unique rank \(r\) decomposition is presented; see [\textit{L. Chiantini} et al., SIAM J. Matrix Anal. Appl. 35, No. 4, 1265--1287 (2014; Zbl 1322.14022)].
Section 12.2, Eigenvectors and Singular vectors. Here are treated the questions given a tensor \(A\in R^{n_1\times n_2\times \cdots n_d}\), to find the rank 1 tensors \(v_1\otimes v_2 \otimes \cdots \otimes v_d \in \mathcal S_{n_1,n_2, \dots, n_d}\) nearest to \(A\) and similarly the symmetric rank 1 tensor nearest to \(A\). From this the authors deduce via directional derivatives equations of form \(A\bullet \otimes_{j\neq k} v_j = \sigma_k v_k\), \(\sigma_k\in \mathbb C\) and \(\bullet\) is an abbreviation for a certain complicated action of \(A\). This yields \(d\) critical equations for \(k=1, \dots,d\). If these \(d\) equations are valid for \((v_1, \dots, v_d)\) and some \(\sigma_k\), then this \(d\)-uple is called a singular vector-tuple. The critical points of the ED-problem for the Segre variety are the tuples of singular vectors. A formula in [\textit{S. Friedland} and \textit{G. Ottaviani}, Found. Comput. Math. 14, No. 6, 1209--1242 (2014; Zbl 1326.15036)] gives the number of singular vector \(d\)-uples if \(A \in \mathbb R^{n_1\times n_2\times \cdots \times n_d}\) is a general tensor . For the case that \(A\in (\mathbb R^n)^{\otimes d}\) one considers the approximation by symmetric tensors \(v^{\otimes d}\) and the critical equations are encoded in \(A\bullet v^{\otimes (d-1)} = \lambda v\). In that case see [\textit{D. Cartwright} and \textit{B. Sturmfels}, Linear Algebra Appl. 438, No. 2, 942--952 (2013)] for the number of complex eigenvectors for \(A\). Analogous questions for rank 2 varieties `do not make sense' since \(\mathcal R_r\) for \(r\geq 2\) is not a closed set.
Section 12.3, Volumes of rank one varieties. The metric geometry of Segre and Veronese varieties is examined. Given \(n_1, \dots,n_d\) and \(N=n_1\cdots n_d -1\) let \(\mathcal S_{\mathbb P}= \{v_1\otimes \cdots \otimes v_d \in \mathbb P^N: v_i\in \mathbb P^{n_i-1}, 1\leq i\leq d \}\) and let \(\mathcal V_{\mathbb P}= \{v^{\otimes d}\in \mathbb P^N: v_i \in \mathbb P^{n-1} \}\), \(N=n^d-1\) The aim is to compute volumes of \(\mathcal S_{\mathbb P}\) and \(\mathcal V_{\mathbb P}\). Let \(\pi: \mathbb S^{2n-1} \rightarrow \mathbb P^{n-1}\) be the projection that sends a vector \(a\in \mathbb S^{2n-1}\) to its class in complex projective space. It is explained why for \(m\)-dimensional real volume of a measurable subset \(U\subset \mathbb P^{n-1}\) the definition \(\text{vol}_m(U)=\frac{1}{2\pi} \text{vol}_{m+1}(\pi^{-1}(U))\) makes sense. Using this definition are then calculated the \(m\) dimensional volume and the degree of the Segre variety \(\mathcal S_{\mathbb P}\) and the \(2(n-1)\)-dimensional volume of \(\mathcal V_{\mathbb P}\). An intimate connection with Howard' s Kinematic formula, see [\textit{R. Howard}, The kinematic formula in Riemannian homogeneous spaces. Providence, RI: American Mathematical Society (AMS) (1993; Zbl 0810.53057)] is pointed out.
Chapter 13, Computer Vision: The aim here is to present techniques to reconstruct a 3D object from images taken by unknown cameras. In recent years classical methods as found in [\textit{R. Hartley} and \textit{A. Zisserman}, Multiple view geometry in computer vision. With foreword by Olivier Faugeras. 2nd edition. Cambridge: Cambridge University Press (2004; Zbl 1072.68104)] were enriched by ideas and algorithms from algebraic geometry. This led to the aerea of Algebraic Vision [\textit{J. Kileel} and \textit{K. Kohn}, \url{arXiv 2210.1143}].
Section 13.1, Multiview varieties. Mathematically a pinhole camera is a \(3\times 4\) matrix \(A\) delivering a surjective projection \(C:\mathbb P^3 \stackrel{A}{\rightarrow } \mathbb P^2\). So \(C(x)=Ax\). The standard camera is given by \(A=[I_3|0]\) and it is called calibrated if it is of the form \(A=[R|t]\) with \(R\in SO(3), t\in \mathbb R^3\). Various, say \(m\), cameras together define the joint camera map \(X\ni ((C_1, \dots,C_m),x) \stackrel{\Phi}{\rightarrow } (C_1(x), \dots, C_m(x)) \in Y\), where \(X\) and \(Y\) are `appropriate spaces of objects', often \(X=\mathbb P^3\) and \(Y=(\mathbb P^2)^m\), in which case we speak of objects \(x\in X\) as single points and \(Y\) as parametrizing \(m\)-uples of points. But also, say, a map \(\Phi: (\mathbb{PR}^{3\times 4})^2 \times (\mathbb P_\mathbb R^3)^k \rightarrow (\mathbb P_ \mathbb R ^2)^k \times (\mathbb P_ \mathbb R^2)^k\) given by \((A_1,A_2, x_1, \dots, x_2) \mapsto (A_1 x_1, \dots, A_1 x_k, A_2 x_1, \dots, A_2 x_k)\) is conceivable. It corresponds to \(m=2\) cameras observing \(k\) points. In any case the task is to determine the unknown cameras and the positions of the points. The section focuses from here on on the more modest task of reconstruction in the case that the cameras are known and the multiview variety \(\mathcal M_C\) concerns a single point in space. Then \(\mathcal M_C\) is hence the Zariski closure of the image of the map \(\mathbb P^3 \ni x \stackrel{\Phi_C}{\mapsto} (A_1x,A_2 x, \dots, A_m x)\). The variety \(\mathcal M_C\) is characterized by the rank condition for a certain matrix \(M_A=M_A(y_1, \dots, y_m)\) which in the case \(m=2\) is quadratic and then its determinant is a bilinear form \(\det M_A=y_2^{\mathsf{T}} F y_1\). The so-called fundamental matrix \(F\) has still a role to play below. In computer vision one understands by triangulation the task of 3D reconstruction from a given camera configuration. The measurements \(y=(y_1, \dots, y_m)\) are typically noisy and thus do not lie on the variety \(\mathcal M_C\) but close to it. One seeks to compute the point \(\tilde y\in \mathcal M_C\) closest to \(y\) and then computes its fiber. The interest in the concept of ED degree originated from such questions; see [\textit{J. Draisma} et al., Found. Comput. Math. 16, No. 1, 99--149 (2016; 1370.51020)]. In this paper, based on Groebner basis calculations to find the ED degree of \(\mathcal M_C^0 = \mathcal M_C\cap (\mathbb R^2)^m\) the formula \(\frac{9}{2} m^3 -\frac{21}{2}m^2 +8m -4\) was conjectured. This was later proved by [\textit{L. Maxim} et al. (2020; Zbl 1434.13049)]. The section ends with explaining the Groebner basis computation that solves the triangulation problem for small \(m\).
Section 13.2, Grassmann tensors. In the current section a class of tensors is introduced that generalizes the fundamental matrix \(F\) of the previou section. The approach was intro duced by [\textit{R. Hartley} and \textit{F. Schaffalitzky}, Int. J. Comput. Vis. 83, No. 3, 274--293 (2009; Zbl 1477.68473)]. Now \(m\geq 2\) cameras and higher dimensional projections \(C_i:\mathbb P^N\rightarrow \mathbb P^{n_i}\) are considered. They are of relevance to model basic dynamics. It is studied when the multiview variety \(\Gamma_{C,c}:=\) Zarisky closure of \(\{(L_1, \dots, L_m)\in \text{Gr}(n_1-c_1, \mathbb P^{n_1}) \times \cdots \times \text{Gr}(n_m-c_m, \mathbb P^{n_m}) \text{ for which there exists a point \(x\in \mathbb P^N\) so that } C_1(x)\in L_1, \dots, C_m(x) \in L_m \}\) is a hypersurface. This is so if and only if \(c_1+\cdots + c_m =N+1\) and in this case the defining equation of \(\Gamma_{C,c}\) is a multilinear in the Pluecker coordinates of the Grassmannians. The proof of these claims is preceded by a number of examples.
Section 13.3, 3D reconstruction from unknown cameras. This topic is now taken up again. Here the state-of-the-art 3D-reconstruction algorithms work with problems that are as small as possible (minimal problems). The pairs \((m,k)\) with \(m=\)number of cameras, \(k=\)number of observed points are determined for minimal problems in the cases of projective cameras and calibrated cameras. The algebraic degree of minimal problems is the number of complex solutions in case of generic data. These degrees where partially already found (presumably in different guises) in the 1860s. See [\textit{O. Hesse}, J. Reine Angew. Math. 62, 188--192 (1863)] and [\textit{R. Sturm}, Clebsch Ann. I, 533--574 (1869; JFM 02.0428.02)]. The practical usage of minimal problems (which must be solved many times over) is explained and motivates the citations of the last two formal mathematical statements in the section. They concern the parametrization of \(\Gamma_{C,c}\) in case it is a hypersurface, by means of its Grassmann tensor (see previous sections)
Chapter 14, Volumes of Semialgebraic Sets: A semialgebraic set is a boolean combination of sets of the form \(S=\{x\in \mathbb R^n: f(x)\geq 0\}\) where \(f\in \mathbb R[x_1,x_2, \dots, x_n]\). The current chapter focuses on the highly accurate computation of the volumes \(\text{vol}(S)\) of sets like \(S\). In case \(g(x)\) is a rational function then the value of \(\int_S g(x)dx\) is called a period integral. The first section is quite elementary, the other two sections are much more advanced.
Section 14.1, Calculus and Beyond. This section reviews the typical multivariable calculus techniques for computing \(\text{vol}(S)=\int_S dx\) by means of a nice example: the elliptotope which is the set \(S=\{(x,y,z): 2xyz-x^2-y^2-z^2+1\geq 0 \}\). It is the logo of the Nonlinear Algebra group in the Max Planck Institute for Mathematics in the Sciences in Leipzig (of which the third author is director). In general it is much more difficult to accurately evaluate such integrals and such questions led to elliptic and abelian integrals.
Section 14.2, D-modules. Defines the Weyl algebra \(\mathbb C\langle x_1, \dots,x_n; \partial_1, \dots,\partial_n \rangle\) as the free \(\mathbb C\)-algebra generated by variables \(x_1, \dots,x_n; \partial_1, \dots,\partial_n\) modulo the relation \(\partial_i x_i - x_i\partial_i -1\). (This latter relation embodies the product rule.) The D-module most often considered is a space \(M\) of infinitely differentiable functions. The natural action of \(D\) on \(M\) is denoted \(P\bullet f\) with \(P\in D, f\in M\). The annihilator of \(f\in M\) is \(\text{Ann}_D(f)=\{P\in D: P\bullet f=0\}\). An \(n\)-variable function \(f\in M\) is holonomic if for each \(k\in \{1,2, \dots, n\}\) there exists a nonzero operator \(P_k = \sum_0^{m_k} p_{ik}(x)\partial_k^i \in D\cap \mathbb C[x_1,\dots, x_n, \partial_k]\) which annihilates \(f\). Rational functions in \(x_1, \dots, x_n\) are holonomic. Compositions of holonomic one-variable functions are holonomic [\textit{A.-L. Sattelberger} and \textit{B. Sturmfels}, ``D-Modules and Holonomic Functions'', Preprint, \url{arXiv:1910.01395}]. If \(f,g\) are holonomic then \(f+g\) and \(f\cdot g\) are holonomic. An example using Macaulay2 to compute an annihilator \(\text{Ann}_D(r)\) of a rational function is given; then the set \(S=\{(x,y)\in \mathbb R^2: q(x,y):= x^4+y^4 + \frac{1}{100} xy-1\leq 0\}\) which represents a TV-screen-similar region is considered. With a Mathematica package due to C. Koutschan a holonomic representation \(P\) for the length of the fibers of the projection \(\text{pr}_1:S\rightarrow \mathbb R\), i.e. for the function \(v(x)=\ell(\text{pr}_1^{-1}(x)\cap S)\) is found. If \(x_0, x_1\) are the branch points of the TV-screen -- these are informally defined as the points where the lengths \(v(x)\) are 0 -- then we have to find \(\text{vol}(S)=w(x_1)\) where \(w(x)= \int_{x_0}^x v(t)dt\) is holonomic again by a general theorem. For finding a unique \(w\) which must be annihilated by \(P\partial\) a number of constraints -- values for \(w\) and \(w'\) at certain points -- have to be imposed. Then a SAGE package is used for computing local series solutions to \(P\partial w=0\). The TV-screen area can now be calculated to any degree of precision. (50 digits are indicated: considerably more than the 4 digit approximation found by sampling 10000 points with the Metropolis-Hastings algorithm.) The authors then report on the work of [\textit{P. Lairez} et al., in: Proceedings of the 44th international symposium on symbolic and algebraic computation, ISSAC '19, Beijing, China, July 15--18, 2019. New York, NY: Association for Computing Machinery (ACM). 259--266 (2019; Zbl 1467.14139)]. Apart of the holonomic apparatus, here the additional buzzword are periods; certain types of integrals. Apart of the definition given in the current book, see e.g. [\textit{M. Kontsevich} and \textit{D. Zagier}, in: Mathematics unlimited -- 2001 and beyond. Berlin: Springer. 771--808 (2001; Zbl 1039.11002)] As an example \(\text{vol}(f\leq 0)\) is computed where \(f(x,y,z)=x^4+y^4+z^4+ \frac{x^3 y}{20}- \frac{xyz}{20}- \frac{yz}{100}+ \frac{z^2}{50} -1\). This value is indicated to about 500 places `with all digits guaranteed to be accurate', the authors say. Such precision is precious often in experimental mathematics, especially for discovering formulas.
Section 14.3 SPD Hierarchies. This is a very different method for computing volumes, which however offers deep algebraic structure. Given a compact semialgebraic set \(S\subset B=[-1,1]^n \subset \mathbb R^n\) one assumes the moments \(b_\alpha =\int_B x^\alpha dx = \int_B x_1^{\alpha_1}x_2^{\alpha_2} \cdots x_n^{\alpha_n} dx_1dx_2 \cdots dx_n\) known but the moments \(m_\alpha =\int_S x^\alpha dx\) are unknown. Here \(\alpha\in \mathbb Z_{\geq 0}^n\). These moments will be the decision variables. We note that \(m_0=\text{vol}(S)\) and we can formulate the problem via the following infinite dimensional linear program: Maximize the integral \(\int_S d\mu\) where \(\text{support}(\mu)\subset S\) and \(\text{support}(\hat \mu)\subset B\) and \(\mu+\hat \mu= \lambda\), the Lebesgue measure on \(B\). In a first step towards solving this, one formulates a surrogate problem which is countably infinite dimensional using moment sequences \(\mathbf{m}=(m_\alpha)\) and \(\hat {\mathbf{m}}=(\hat m_\alpha)\) of the unknown measures \(\mu\) and \(\hat \mu\). In the next step assume \(S=\{x\in \mathbb R^n: f(x)\geq 0\}\), where \(f(x)= \sum_\alpha c_\alpha x^\alpha\). One fixes an integer \(d\geq \deg f\) and constructs three symmetric matrices of format \(\binom{n+d}{d}\times \binom{n+d}{d}\) with rows /columns indexed by elements \(\alpha,\beta \in \mathbb N^n\) so that \(|\alpha|,|\beta|\leq d\). One defines matrices \(M_d(\mathbf{m})=(m_{\alpha+\beta})\), \(M_d(\hat {\mathbf{m}})=(\hat m_{\alpha+\beta})\), and \(M_d(f\mathbf{m})=(\sum_\gamma m_{\alpha+\beta+ \gamma})\). One considers then the following finite-dimensional now semidefinite program: Maximize \(m_0\) subject to \(m_\alpha+\hat m_\alpha= b_\alpha\), for all \(\alpha \in \mathbb N^n\) with \(|\alpha|\leq d\), where the matrices \(M_d(\mathbf{m}), M_d(\hat {\mathbf{m}}), M_{d'}(f \mathbf{m})\) are positive definite. (Here \(d' = d-\lceil \deg(f)/2\rceil\), meaning that in matrix \(M_{d' }(.)\) we may assume \(|\alpha|,|\beta| \leq d'\)) The point of this program is that it was proved in [\textit{D. Henrion} et al., SIAM Rev. 51, No. 4, 722--743 (2009; Zbl 1179.14037) ] as \(d\rightarrow \infty\) that the solutions \(m_0\) converge to \(\text{vol}(S)\). The quality of the convergence -- avoiding Gibbs phenomena -- can be improved using constraints based on Stokes' theorem.
Chapter 15, Sampling: The emphasis in Section 15.1 is on aspects of topological data analysis. In Section 15.2 algorithms are discussed that permit sampling with density guarantees. The ability to rapidly solve intersections of real algebraic varieties with linear subspaces and finding ED critical points is here important. Section 15.3 introduces Markov kernels on a variety \(X\). The state space \(X\) is here continuous and requires partially unfamiliar analysis. The Metropolis-Hastings algorithm plays a role. Sampling from uniform distribution is done/explained and, surprisingly, Chow forms creep in.
Section 15.1, Homology from finite samples. Given a real variety \(X\) in \(\mathbb R^n\) the aim is to compute a finite set \(S\subset X\) called a sample which can be used to explore properties of the variety. \(S\) is an \(\epsilon\)-sample if every point of \(X\) has a distance\(<\epsilon\) from some point of \(S\). This is to say \(X\subset U:=\bigcup_{s\in S} B_\epsilon (s)\). A theorem of [\textit{P. Niyogi} et al., Discrete Comput. Geom. 39, No. 1--3, 419--441 (2008; Zbl 1148.68048)] says how sampling yields useful results in topological data analysis: if \(\sqrt{\epsilon} < \sqrt{\frac{3}{20}} \tau(X)\) (reach of X, see Section 7), then \(X\) is a deformation retract of \(U\) and so the homology of \(X\) equals that of \(U\). The notion of a 2-bottleneck is generalized to a \(k\)-bottleneck which in turn originates a quantity called weak feature size of \(X\), written \(\text{wfs}(X)\). If \(\epsilon < \text{wfs}(X)\), then the homology groups \(H_0\) and \(H_1\) of \(X\) coincide with that of the relatively easily construced 2-dimensional Vietoris-Rips complex at scale \(\epsilon\). Theorems in earlier chapters allow to compute the \(\epsilon\)s.
Section 15.2, Sampling with density guarantees. The section presents two sampling algorithms that, given a real algebraic variety \(X\subset \mathbb R^n\), an \(\epsilon >0\) and a box \(R=[a_1,b_1]\times \cdots \times [a_n, b_n]\), yield an \(\epsilon\)-sample \(S\subset X\cap R\). First it is constructed a box \(R\) which when \(X\) is compact contains \(X\). The idea then is to sample points \(u\in R\) and to collect the ED critical points with respect to \(u\). The algorithm is described in pseudocode. It is due to [\textit{E. Dufresne} et al., ``Sampling real algebraic varieties for topological data analysis'', Preprint, \url{arXiv:1802.07716}] and its correctness was proved there. [\textit{S. Di Rocco}, Math. Comput. 91, No. 338, 2969--2995 (2022; Zbl 1495.13041)] published another algorithm which uses as input apart of \(\epsilon\) also the width of the smallest bottleneck of \(X\) (which first has to be found). Over the previous algorithm it has apparently the benefit that points in certain connected components are collected. In both algorithms the chapters on ED critical points are relevant. The work of [Niyogi et al., loc. cit.] also yields a sampling mechanism which yields with high probability \(\epsilon\)-samples.
Section 15.3. Markov chains on varieties. The so-called Markov Chain Monte Carlo (MCMC) methods present a popular class of methods for sampling from probability distributions. In case of finite state spaces they amount to random walks on finite graphs. But here the question is treated for the case that the state space is a real algebraic variety \(X\) in \(\mathbb R^n\). With a \(\sigma\)-algebra \(\mathcal A\) of measurable sets on the variety \(X\), a Markov kernel which is a map \(p:X\times \mathcal A \rightarrow [0,1]\) so that \(p(x,\cdot)\) is a probability measure for all \(x\in X\), while \(p(\cdot,A)\) is a measurable function for all \(A\in \mathcal A\). See [\textit{S. P. Meyn} and \textit{R. L. Tweedie}, Markov chains and stochastic stability. Berlin: Springer-Verlag (1993; Zbl 0925.60001)]. A Markov process with starting point \(x_0\) is a stochastic process \(x_0,x_1,x_2, \dots\) where the probability that the next \(k\) points lie in certain measurable sets \(A_1, \dots,A_k\) is given by a probability \(\text{prob}(x_k\in A_k, \dots, x_1\in A_1 | x_0=x)\) expressible by a certain iterated integral over the Markov kernel. The probability law of the \(k\)-th state depends only on the position of the \(k-1\)-st state and not on earlier ones. But while in the discrete setting the Markov property can be expressed via conditional probability, in the current setting this is impossible because it would involve conditioning over events of probability 0. Next are presented examples computed with Julia: Markov chains on \(\mathbb{R}^2\) and a Markov chain on a surface. A probability distribution \(\pi\) on \(X\) is called stationary for a kernel \(p\) if it satisfies \(\int_{x\in X} p(x,A)\pi(dx)= \pi(A)\) for all \(A\in \mathcal A\). The idea of MCMC methods for sampling from a probability distribution \(\pi\) is to set up a Markov process on \(X\) with stationary distribution. It is explained that this convergence - with respect to the total variation distance between measures \(\mu,\nu\) defined by \(d_{\text{TV}}(\mu,\nu)= \sup_{A\in\mathcal A} |\mu(A)-\nu(A)|\) -- is achieved via irreducibility and aperiodicity. Theorem 15.15 says that in this case if \(\mu^k(x,\cdot): A\mapsto \text{Prob}(x_k\in A| x_0=x)\) is the probability law of the \(k\)-th state and \(\pi\) is stationary, then for almost all \(x\in X\), \(\lim_{k\rightarrow \infty} d_{\text{TV}}(\pi, \mu^k (x,\cdot))=0\).
Section 15. 4, Chow goes to Monte Carlo. The problem of sampling from a variety \(X\) using the Markov chain Monte Carlo paradigm is here studied. This involves intersecting \(X\) with random linear spaces. (Chow forms encode a variety by means of linear subspaces that intersect it.) The cited Theorem 15.15 has the consequence that if a chain designed for stationary distribution \(\pi\) runs long enough we will get that the points in the process have a distribution close to \(\pi\). The Metropolis-Hastings algorithm is indicated which upon an input of a probability measure \(\pi\) on \(X\) with density \(\phi(y)\) (i.e. \(\pi(dx)=\phi(x)dx\) with \(dx\) Lebesgue measure) and Markov kernel \(p(x,A)\) on \(X\) with density \(q(x,y)\) will produce a Markov chain on \(X\) with stationary distribution \(\pi\). Among examples for its use is one method due to Hauenstein and Kahle [81] for sampling points near a variety. This again is done with Julia. Finally two ChowMCMC algorithms by [\textit{P. Breiding} and \textit{O. Marigliano}, SIAM J. Math. Data Sci. 2, No. 3, 683--704 (2020; 1486.14076)] and [\textit{T. Lelièvre} et al., IMA J. Numer. Anal. 43, No. 2, 737--788 (2023)] are described which remove the difficulty to have to input into Metropolis-Hastings algorithm the proposal density \(q\).
Reviewer: Alexander Kovačec (Coimbra)Non-commutative resolutions for the discriminant of the complex reflection group \(G(m, p,2)\)https://zbmath.org/1540.140032024-09-13T18:40:28.020319Z"May, Simon"https://zbmath.org/authors/?q=ai:may.simonSummary: We show that for the family of complex reflection groups \(G = G(m, p\),2) appearing in the Shephard-Todd classification, the endomorphism ring of the reduced hyperplane arrangement \(A(G)\) is a non-commutative resolution for the coordinate ring of the discriminant \(\Delta\) of \(G\). This furthers the work of Buchweitz, Faber and Ingalls who showed that this result holds for any true reflection group. In particular, we construct a matrix factorization for \(\Delta\) from \(A(G)\) and decompose it using data from the irreducible representations of \(G\). For \(G(m, p,2)\) we give a full decomposition of this matrix factorization, including for each irreducible representation a corresponding maximal Cohen-Macaulay module. The decomposition concludes that the endomorphism ring of the reduced hyperplane arrangement \(A(G)\) will be a non-commutative resolution. For the groups \(G(m,1,2)\), the coordinate rings of their respective discriminants are all isomorphic to each other. We also calculate and compare the Lusztig algebra for these groups.Hilbert-Poincaré series of matroid Chow rings and intersection cohomologyhttps://zbmath.org/1540.140092024-09-13T18:40:28.020319Z"Ferroni, Luis"https://zbmath.org/authors/?q=ai:ferroni.luis"Matherne, Jacob P."https://zbmath.org/authors/?q=ai:matherne.jacob-p"Stevens, Matthew"https://zbmath.org/authors/?q=ai:stevens.matthew"Vecchi, Lorenzo"https://zbmath.org/authors/?q=ai:vecchi.lorenzoThe authors study the Hilbert series of four objects arising in the Chow-theoretic and Kazhdan-Lusztig framework of matroids. These are, respectively, the Hilbert series of the Chow ring, the augmented Chow ring, the intersection cohomology module, and its stalk at the empty flat. The authors develop an explicit parallelism between the Kazhdan-Lusztig polynomial of a matroid and the Hilbert-Poincaré series of its Chow ring. This extends to a parallelism between the Z-polynomial of a matroid and the Hilbert-Poincaré series of its augmented Chow ring. This suggests to bring ideas from one framework to the other. Two main motivations are the real-rootedness conjecture for all of these polynomials, and the problem of computing them.
Uniform matroids are a case of combinatorial interest, where the authors link the resulting polynomials with certain real-rooted families appearing in combinatorics such as the Eulerian and the binomial Eulerian polynomials, and thus, they prove the real-rootedness of the Hilbert series of the augmented Chow rings of uniform matroids. In addition, they prove a version of a conjecture of Gedeon in the Chow setting: uniform matroids maximize coefficient-wise these polynomials for matroids with fixed rank and cardinality. By relying on the nonnegativity of the coefficients of the Kazhdan-Lusztig polynomials and the semi-small decompositions of Braden, Huh, Matherne, Proudfoot, and Wang, the authors strengthen the unimodality of the Hilbert series of Chow rings, augmented Chow rings, and intersection cohomologies to \(\gamma\)-positivity.
Reviewer: Mee Seong Im (Annapolis)Khovanskii-finite rational curves of arithmetic genus 2https://zbmath.org/1540.140212024-09-13T18:40:28.020319Z"Ilten, Nathan"https://zbmath.org/authors/?q=ai:ilten.nathan-owen"Mokhtar, Ahmad"https://zbmath.org/authors/?q=ai:mokhtar.ahmad-sLet \(X\) be a projective variety over an algebraically closed field \(K\) with homogeneous coordinate ring \(S\). A valuation on \(S\) is called \textit{Khovanskii-finite} if it is homogeneous and the value semigroup is finitely generated of rank equal to \(\dim X + 1\). To any such Khovanskii-finite valuation \(\nu\), in [Math. Ann. 356, No. 3, 1183--1202 (2013; Zbl 1273.14104)], \textit{D. Anderson} provides a general construction of a degeneration of \(X\) to the toric variety whose homogeneous coordinate ring is the semigroup algebra associated with the value semigroup of \(\nu\). Such a construction offers a systematic method and a way to understand the degeneration process from a projective variety to a toric variety.
The question of interest in this case is whether every projective variety \(X = \mathrm{Proj}\; S\) admits a Khovanskii-finite valuation on its homogeneous coordinate ring \(S\). In [\textit{N. Ilten} and \textit{M. Wrobel}, J. Comb. Algebra 4, No. 2, 141--166 (2020; Zbl 1444.14089)], it is shown that such an existence is not guaranteed for all varieties, except, in characteristic zero, for rational curves of arithmetic genus \(0\) or \(1\). Thus the existence of a Khovanskii-finite valuation on \(S\) depends on the behaviour of the variety in terms of, for instance, its arithmetic genus and degree.
This article provides a complete answer to the aforementioned question by determining the existence of Khovanskii-finite valuations and their applications for rational curves of arithmetic genus two, as well as relating them to toric degenerations. More specifically, the article gives a partially detailed description of the locus of degree \(n+2\) rational curves in \(\mathbb P^n\) of arithmetic genus two that admit a Khovanskii-finite valuation, along with a description of an effective method for determining if a rational curve of arithmetic genus two defined over a number field admits a Khovanskii-finite valuation. This provides a criterion for deciding if such curves admit a toric degeneration. Finally, it is shown that rational curves with a single unibranch singularity are always Khovanskii-finite if their arithmetic genus is sufficiently small.
Reviewer: Ivan Kimuli Philly (Arua)Noether's problemshttps://zbmath.org/1540.140312024-09-13T18:40:28.020319Z"Futorny, Vyacheslav"https://zbmath.org/authors/?q=ai:futorny.vyacheslav-m"Schwarz, João"https://zbmath.org/authors/?q=ai:schwarz.joao-fernandoSummary: The purpose of these notes is to give an introduction to the
classical (commutative) and the noncommutative Noether's problems and
show their intrinsic connection. We also discuss the subrings of invariants
by finite groups of families of noncommutative Galois algebras, including
the Weyl algebras and the generalized Weyl algebras.
For the entire collection see [Zbl 1486.16001].The first Euler characteristic and the depth of associated graded ringshttps://zbmath.org/1540.140622024-09-13T18:40:28.020319Z"Ozeki, Kazuho"https://zbmath.org/authors/?q=ai:ozeki.kazuhoSummary: The homological property of the associated graded ring of an ideal is an important problem in commutative algebra. ln this talk, we explore the structure of the associated graded ring of \(\mathfrak{m}\)-primary ideals in the case where the first Euler characteristic attains almost minimal value in a Cohen-Macaulay local ring.
For the entire collection see [Zbl 1540.16001].Inseparable maps on \(W_n\)-valued local cohomology groups of nontaut rational double point singularities and the height of \(K3\) surfaceshttps://zbmath.org/1540.140752024-09-13T18:40:28.020319Z"Matsumoto, Yuya"https://zbmath.org/authors/?q=ai:matsumoto.yuyaSummary: We consider rational double point singularities (RDPs) that are nontaut, which means that the isomorphism class is not uniquely determined from the dual graph of the minimal resolution. Such RDPs exist in characteristic 2, 3, and 5. We compute the actions of Frobenius and other inseparable morphisms on \(W_n\)-valued local cohomology groups of RDPs. Then we consider RDP \(K3\) surfaces admitting nontaut RDPs. We show that the height of the \(K3\) surface, which is also defined in terms of the Frobenius action on \(W_n\)-valued cohomology groups, is related to the isomorphism class of the RDP.Limit points and additive group actionshttps://zbmath.org/1540.140862024-09-13T18:40:28.020319Z"Arzhantsev, Ivan"https://zbmath.org/authors/?q=ai:arzhantsev.ivan-vThe author deals with effective \(\mathbb{G}_m\)-actions on normal affine varieties over an algebraically closed field of characteristic zero. The main theorem states that there is a compatible \(\mathbb{G}_a\)-action on \(X\) if and only the original \(\mathbb{G}_m\)-action is parabolic, i.e., if the \(\mathbb{G}_m\)-fixed points form a prime divisor in \(X\).
A regular \(\mathbb{G}_a\)-action on \(X\) is called compatible with a given \(\mathbb{G}_m\)-action if \(\mathbb{G}_m\) normalizes \(\mathbb{G}_a\) within \(\mathrm{Aut}(X)\) such that generic \(\mathbb{G}_a\)-orbits coincide with the closures of generic \(\mathbb{G}_m\)-orbits. Hence, the points of generic \(\mathbb{G}_m\)-orbits become equivalent to the limit points of the respective orbits.
In the special case of \(X\) being an affine toric variety, the situation is illustrated by refering to well-known facts. The parabolic subgroups \(\mathbb{G}_m\hookrightarrow\mathrm{Aut}(X)\) are in one-to-one correspondence with the rays of the defining polyhedral cone, and the associated prime divisors provide the \(\mathbb{G}_m\)-fixed points. Moreover, the Demazure roots correspond to additive subgroups \(\mathbb{G}_a\hookrightarrow\Aut(X)\) providing compatible actions.
Reviewer: Klaus Altmann (Berlin)A theorem of Gordan and Noether via Gorenstein ringshttps://zbmath.org/1540.140882024-09-13T18:40:28.020319Z"Bricalli, Davide"https://zbmath.org/authors/?q=ai:bricalli.davide"Favale, Filippo Francesco"https://zbmath.org/authors/?q=ai:favale.filippo-francesco"Pirola, Gian Pietro"https://zbmath.org/authors/?q=ai:pirola.gian-pietroIn the 1850's [J. Reine Angew. Math. 42, 117--124 (1851; ERAM 042.1147cj); J. Reine Angew. Math. 56, 263--269 (1859; ERAM 056.1491cj)], \textit{O. Hesse} claimed that any hypersurface \(X \subset \mathbb P^N\) with vanishing hessian is a cone. Subsequently in 1876 \textit{P. Gordan} and \textit{M. Nöther} [Math. Ann. 10, 547--568 (1876; JFM 08.0064.05)] showed that this is not true for \(N \geq 4\). In \(\mathbb P^4\) they gave a complete description of the hypersurfaces with vanishing hessian that are not cones. A nice treatment of this topic can be found in the book by \textit{F. Russo} [On the geometry of some special projective varieties. Cham: Springer (2016; Zbl 1337.14001)]. In this context hessians were studied by
J. Watanabe in 2000 and by \textit{T. Maeno} and \textit{J. Watanabe} [Ill. J. Math. 53, No. 2, 591--603 (2009; Zbl 1200.13031)], connecting it to the Lefschetz properties for artinian Gorenstein algebras via Macaulay's inverse systems. One class of such hypersurfaces that have been carefully studied (and is further studied in this paper) with an eye to the Lefschetz properties is that of Perazzo hypersurfaces. See for instance the paper of L. Fiorindo, E. Mezzetti and R. Miró-Roig [\textit{L. Fiorindo} et al., J. Algebra 626, 56--81 (2023; Zbl 1516.14066)], where they study the connections between the Lefschetz properties and the Hilbert function for the associated artinian Gorenstein algebras. In the paper under review, the authors start by recalling the Gordan-Noether theorem and its connection to the Lefschetz properties. They take a different point of view for \(\mathbb P^4\), giving a direct proof of the fact that all artinian Gorenstein algebras \(R\) of codimension \(\leq 4\) have the property that there exists a linear form \(\ell\) such that \(\times \ell : [R]_1 \rightarrow [R]_{e-1}\) is an isomorphism (where \(e\) is the socle degree). From this they deduce the Gordan-Noether theorem. The authors use this point of view to prove that in \(S = k[x_0,\dots,x_4]\), if \(I\) is an ideal generated by a regular sequence of 5 quadrics then \(S/I\) satisfies the Strong Lefschetz Property (SLP), and in particular \(\times \ell^3 : [R]_1 \rightarrow [R]_4\) is an isomorphism. For example, this holds for Jacobian rings associated to smooth cubic threefolds. As a consequence, their work gives a new proof of a result of U. Nagel and the reviewer related to the Weak Lefschetz Property for complete intersections of quadrics in general.
Reviewer: Juan C. Migliore (Notre Dame)Taylor polynomials of rational functionshttps://zbmath.org/1540.141092024-09-13T18:40:28.020319Z"Conca, Aldo"https://zbmath.org/authors/?q=ai:conca.aldo"Naldi, Simone"https://zbmath.org/authors/?q=ai:naldi.simone"Ottaviani, Giorgio"https://zbmath.org/authors/?q=ai:ottaviani.giorgio-maria"Sturmfels, Bernd"https://zbmath.org/authors/?q=ai:sturmfels.berndGiven two polynomials \(P\) and \(Q\) whose constant term is 1, the rational function \(P/Q\) has a Taylor series expansion with constant term 1. Truncating that series at terms of degree \(m\), one obtains the \(m\)th Taylor polynomial of \(P/Q\). A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed.
In one variable, Taylor varieties are given by rank constraints on Hankel matrices (i.e. matrices which are constant along ascending skew-diagonals from left to right). Inversion of the natural parametrization is known as Padé approximation.
The authors study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters.
The authors explain this with Fröberg's Conjecture in commutative algebra.
Reviewer: Vladimir P. Kostov (Nice)An atlas for the pinhole camerahttps://zbmath.org/1540.141182024-09-13T18:40:28.020319Z"Agarwal, Sameer"https://zbmath.org/authors/?q=ai:agarwal.sameer"Duff, Timothy"https://zbmath.org/authors/?q=ai:duff.timothy"Lieblich, Max"https://zbmath.org/authors/?q=ai:lieblich.max"Thomas, Rekha R."https://zbmath.org/authors/?q=ai:thomas.rekha-rThe authors of this paper study pinhole cameras by means of some algebro-geometric objects that effectively describe many of the problems studied in 3D computer vision, such as reconstruction. They begin by explaining that the standard model of a pinhole camera in computer vision is a surjective linear projection \(\mathbb P^3 \dashrightarrow \mathbb P^2\). Such a map is determined by a \(3\times 4\) matrix \(A\) of rank \(3\), up to a scalar.
The study focuses on the case \(m\) matrices \(\mathbf A=(A_1,\dots, A_m)\) are simultaneously considered acting on \(n\) points \(\mathbf q=(q_1,\dots,q_n)\) of \(\mathbb P^3\) and, hence, producing \(mn\) points \(\mathbf p=(A_iq_j)\) in \(\mathbb P^2\). From a universal perspective, the cameras \(\mathbf A\) and the points \(\mathbf q\) are treated as unknowns. Therefore, given integers \(m,n\geq 1\), a map of the following type is taken:
\[
(\mathbb P^{11})^m \times (\mathbb P^3)^n \dashrightarrow (\mathbb P^2)^m,
\]
and for every nonempty Zariski open \(U\subset (\mathbb P^{11})^m \times (\mathbb P^3)^n\), where the map is defined, the closure \(\Gamma^{m,n}_{\mathbf A, \mathbf q, \mathbf p}\) of the graph of the map restricted to \(U\) is considered. The vanishing ideal of this variety is denoted by \(I^{m,n}_{\mathbf A, \mathbf q, \mathbf p}\).
On the algebraic variety \(\Gamma^{m,n}_{\mathbf A, \mathbf q, \mathbf p}\), three natural geometric operations are considered: coordinate projection (and hence elimination of the variables \(\mathbf A\) or \(\mathbf q\) or \(\mathbf p\)), coordinate specialization and a combination of the projection and the specialization. In all three cases non-trivial varieties are obtained along with their vanishing ideals. These varieties, represented by their ideals, are taken as nodes of a graph that is called the ``atlas for the pinhole camera''. The edge between two varieties is the operation among the above types that connects one variety to another.
The authors focus on a square that appears on the left side of this atlas (as presented in Figures 1 and 2) and has nodes made of the four ideals \(I^{m,n}_{\mathbf A, \mathbf q, \mathbf p},\) \(I^{m,n}_{\bar{\mathbf A}, \mathbf q, \mathbf p}\) (obtained by specializing \(\mathbf A\)), \(I^{m,n}_{\mathbf A, \mathbf p}\) (obtained by eliminating \(\mathbf q\)) and \(I^{m,n}_{\bar{\mathbf A}, \mathbf p}\) (obtained by specializing \(\mathbf A\) and eliminating \(\mathbf q\)).
Using the words of the authors in section 2.2, ``the main results of this paper may be divided into three progressively stronger categories: geometric results giving set-theoretic equations for the varieties of interest, ideal-theoretic results which characterize the vanishing ideals of these varieties, and, strongest of all, results about the Gröbner bases of these vanishing ideals.'' Depending on the specific scope, some hypotheses of genericity are introduced and discussed.
Each of the four above ideals is first carefully studied in the case \(n=1\) and then results on the general case \(n\geq 1\) are obtained.
The paper finally presents a list of challenging open problems. This paper is very pleasant to read.
Reviewer: Francesca Cioffi (Napoli)Can a ground-based vehicle hear the shape of a room?https://zbmath.org/1540.141192024-09-13T18:40:28.020319Z"Boutin, Mireille"https://zbmath.org/authors/?q=ai:boutin.mireille"Kemper, Gregor"https://zbmath.org/authors/?q=ai:kemper.gregorThe authors work on a nice, practical problem that is modelled in a way that permits a solution by computational algebra: ``reconstructing the position of walls and other planar surfaces using echoes''. It is a continuation of their previous [\textit{M. Boutin} and \textit{G. Kemper}, SIAM J. Appl. Algebra Geom. 4, No. 1, 123--140 (2020; Zbl 1433.51010)] where they considered all six degrees of freedom for a vehicle carrying microphones. However, maintaining stable angles in the air, in order to obtain positive results, can be challenging, so this article concerns a version with less degrees of freedom (a wheeled vehicle on flat ground, or a hovering drone).
The article is very readable. It utilizes relatively basic geometrical, matrix and group-theoretic arguments to reduce the results to calculations with ideals of many variables, that are put, upon further reduction, within the realm of current computers and software (they used MAGMA for a short ideal computation to finish their argument). Their bibliography on similar problems, not only of the mathematical kind, seems useful.
Reviewer: David Sevilla (Merida)Proceedings of the 55th symposium on ring theory and representation theory, Osaka Metropolitan University, Osaka, Japan, September 6--9, 2021https://zbmath.org/1540.160012024-09-13T18:40:28.020319ZThe articles of this volume will be reviewed individually. For the preceding symposium see [Zbl 07912833].
Indexed articles:
\textit{Aihara, Takuma}, On trivial tilting theory, 1-3 [Zbl 07869703]
\textit{Aoki, Toshitaka; Escolar, Emerson G.; Tada, Shunsuke}, On interval global dimension of posets: a characterization of case 0, 4-10 [Zbl 07869704]
\textit{Asai, Sota; lyama, Osamu}, Faces of interval neighborhoods of silting cones, 12-19 [Zbl 07869705]
\textit{Hiramatsu, Naoya}, The spectrum of the category of maximal Cohen-Macaulay modules, 20-25 [Zbl 07869706]
\textit{Itaba, Ayako}, Quantum projective planes and Beilinson algebras of 3-dimensional quantum polynomial algebras for Type S', 26-31 [Zbl 07869707]
\textit{Kimura, Kaito}, The Auslander-Reiten conjecture for normal rings, 32-35 [Zbl 1540.13041]
\textit{Koshio, Ryotaro; Kozakai, Yuta}, On inductions and restrictions of support \(\tau\)-tilting modules over group algebras, 36-39 [Zbl 07869709]
\textit{Matsuno, Masaki}, Classification of twisted algebras of 3-dimensional Sklyanin algebras, 40-44 [Zbl 07869710]
\textit{Matsuno, Masaki; Saito, Yu}, The classification of 3-dimensional cubic AS-regular algebras of Type P,S,T and WL, 45-50 [Zbl 07869711]
\textit{Minamoto, Hiroyuki}, Quiver Heisenberg algebras and the algebra \(B(Q)\), 51-57 [Zbl 07869712]
\textit{Nakajima, Yusuke}, Wall-and-chamber structures of stability parameters for some dimer quivers, 58-65 [Zbl 07869713]
\textit{Nakamoto, Kazunori; Torii, Takeshi}, The moduli of 4-dimensional subalgebras of the full matrix ring of degree 3, 66-73 [Zbl 1540.14026]
\textit{Nakamura, Tsutomu}, Govorov-Lazard type theorems, big Cohen-Macaulay modules, and Cohen-Macaulay hearts, 74-76 [Zbl 07869715]
\textit{Ogawa, Yasuaki}, \(K_0\) of weak Waldhausen extriangulated categories, 77-84 [Zbl 07869716]
\textit{Otake, Yuya}, Embeddings into modules of finite projective dimensions and the \(n\)-torsionfreeness of syzygies, 85-88 [Zbl 1540.13035]
\textit{Ozeki, Kazuho}, The first Euler characteristic and the depth of associated graded rings, 89-96 [Zbl 1540.14062]
\textit{Saito, Shunya}, Classifying several subcategories of the category of maximal Cohen-Macaulay modules, 97-101 [Zbl 1540.13031]
\textit{Sakai, Arashi}, A classification of \(T\)-structures by a lattice of torsion classes, 102-106 [Zbl 07869720]
\textit{Takahashi, Ryo}, Resolving subcategories of derived categories, 107-111 [Zbl 07869721]
\textit{Usui, Satoshi}, Periodic dimensions of modules and algebras, 112-115 [Zbl 07869722]The multiplication formulas of weighted quantum cluster functionshttps://zbmath.org/1540.170102024-09-13T18:40:28.020319Z"Chen, Zhimin"https://zbmath.org/authors/?q=ai:chen.zhimin"Xiao, Jie"https://zbmath.org/authors/?q=ai:xiao.jie.2|xiao.jie"Xu, Fan"https://zbmath.org/authors/?q=ai:xu.fan|xu.fan.1Summary: By applying the property of Ext-symmetry and the affine space structure of certain fibers, we introduce the notion of weighted quantum cluster functions and prove their multiplication formulas associated to abelian categories with Ext-symmetry and 2-Calabi-Yau triangulated categories with cluster-tilting objects.Fundamentals of abstract algebrahttps://zbmath.org/1540.200012024-09-13T18:40:28.020319Z"DeBonis, Mark J."https://zbmath.org/authors/?q=ai:debonis.mark-jPublisher's description: Fundamentals of Abstract Algebra is a primary textbook for a one year first course in Abstract Algebra, but it has much more to offer besides this. The book is full of opportunities for further, deeper reading, including explorations of interesting applications and more advanced topics, such as Galois theory. Replete with exercises and examples, the book is geared towards careful pedagogy and accessibility, and requires only minimal prerequisites. The book includes a primer on some basic mathematical concepts that will be useful for readers to understand, and in this sense the book is self-contained.
Features
\begin{itemize}
\item Self-contained treatments of all topics
\item Everything required for a one-year first course in Abstract Algebra, and could also be used as supplementary reading for a second course
\item Copious exercises and examples
\end{itemize}Row-factorization matrices in Arf numerical semigroups and defining idealshttps://zbmath.org/1540.201182024-09-13T18:40:28.020319Z"Süer, Meral"https://zbmath.org/authors/?q=ai:suer.meral"Yeşil, Mehmet"https://zbmath.org/authors/?q=ai:yesil.mehmetLet \(S\) be a numerical semigroup, that is, a subset of \(\mathbb{N}\), the set of non-negative integers, such that \(0\in S\), \(S+S=S\), and \(\mathbb{N}\setminus S\) has finitely many elements. Every numerical semigroup admits a minimal set of generators: \(S^*\setminus (S^*+S^*)\), where \(S^*=S\setminus\{0\}\). The cardinality of the minimal set of generators of \(S\) is known as the embedding dimension of \(S\).
The multiplicity of \(S\) is the least positive integer belonging to \(S\). As two minimal generators cannot be congruent modulo the multiplicity of \(S\), the cardinality of the minimal generating set of \(S\) is at most the multiplicity of \(S\). Numerical semigroups attaining this upper bound are known as numerical semigroups with maximal embedding dimension.
A numerical semigroup \(S\) is Arf if for every \(x,y,z\in S\), with \(x\ge y\ge z\), the integer \(x+y-z\in S\). Arf numerical semigroups have maximal embedding dimension.
Non-negative integers not belonging to a numerical semigroup of \(S\) are known as gaps of \(S\). Maximal gaps with respect to the order induced by \(S\), \(a\le_S b\) if \(b-a\in S\), are the pseudo-Frobenius numbers of \(S\). The number of pseudo-Frobenius numbers is the type of the semigroup. It is well known that the type of a numerical semigroup with maximal embedding dimension is the multiplicity of the semigroup minus one (this actually characterizes numerical semigroups with maximal embedding dimension).
If \(f\) is a pseudo-Frobenius number of a numerical semigroup \(S\) and \(n\) is a minimal generator of \(S\), then the maximality of \(f\) implies that \(f+n\in S\) and so, it has an expression in terms of the minimal generators of \(S\). A row-factorization matrix associated to \(f\) encodes this information in a matrix, having one row per each minimal generator of \(S\).
Let \(S\) be a numerical semigroup minimally generated by \(\{n_1,\dots,n_e\}\). Let \(\mathbb{K}\) be a field and let \(x_1,\dots,x_e,t\) be unknowns and let \(\varphi: \mathbb{K}[x_1,\dots,x_e]\to \mathbb{K}[t]\) determined by \(x_i\mapsto t^{n_i}\) for all \(i\in \{1,\dots,e\}\). The kernel of \(\varphi\) is known as the ideal associated to \(S\) and it is denoted by \(I_S\). The ideal \(I_S\) is generic if its minimal (binomial) generators have full support (all the variables \(x_1,\dots,x_e\) appear in the binomial).
The authors provide the expressions of the row factorizations matrices of some families of Arf numerical semigroups and then apply this information to detect when an Arf numerical semigroup has a generic defining ideal.
The authors did not seem to notice that as Arf numerical semigroups have maximal embedding dimension, one can compute a minimal presentation just using [\textit{J. C. Rosales}, Semigroup Forum 52, No. 3, 307--318 (1996; Zbl 0853.20041)], where it is shown that the relations involved in a minimal presentation are factorizations of sums of two minimal generators.
Also, the authors seemed to miss the fact that a numerical semigroup \(S\) with embedding dimension three is generic if and only if \(S\) is not symmetric [\textit{J. Herzog}, Manuscr. Math. 3, 175--193 (1970; Zbl 0211.33801)], which combined with Theorem I.4.2 in [\textit{V. Barucci} et al., Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains. Providence, RI: American Mathematical Society (AMS) (1997; Zbl 0868.13003)], which states that the only Arf (or maximal embedding dimension) symmetric numerical semigroups are those with embedding dimension two and the well-known fact that every numerical semigroup with embedding dimension two is generic, shows that every numerical semigroup with maximal embedding dimension and multiplicity smaller than four is generic.
The same holds with Theorem 5.8: every generic numerical semigroup is uniquely presented (see for instance [\textit{V. Blanco} et al., Ill. J. Math. 55, No. 4, 1385--1414 (2011; Zbl 1279.20072)]) and it is shown in [\textit{P. A. García-Sánchez} and \textit{I. Ojeda}, Pac. J. Math. 248, No. 1, 91--105 (2010; Zbl 1208.20052)] that numerical semigroups with maximal embedding dimension with multiplicity larger than three can not be uniquely presented and consequently can not be generic. Therefore, Theorems 5.7 and 5.8 in the manuscript under review hold for any numerical semigroup with maximal embedding dimension.
Reviewer: Pedro A. García Sánchez (Granada)Computing the Conley index: a cautionary talehttps://zbmath.org/1540.370332024-09-13T18:40:28.020319Z"Mischaikow, Konstantin"https://zbmath.org/authors/?q=ai:mischaikow.konstantin"Weibel, Charles"https://zbmath.org/authors/?q=ai:weibel.charles-aThe paper explores the Conley index from an algebraic perspective. Given an isolated (also called locally maximal) invariant set \(S\) of a discrete dynamical system \(f \colon X \to X\), the homological Conley index of \(S\) is the shift equivalence class of the induced map
\[
f_* \colon H_*(P_1, P_0; k) \to H_*(P_1, P_0; k),
\]
where \((P_1, P_0)\) is any index pair that isolates \(S\). In the case \(k\) is a field, the shift equivalence class reduces to the rational canonical form and can be computed efficiently. However, Example 1.1 in the paper shows that we need more general coefficients (e.g., \(\mathbb Z\)) to distinguish between homological Conley indices of very simple invariant sets: a period-2 orbit and two hyperbolic fixed points. The authors choose \(k = \mathbb Z\), a ring for which homology computations are possible but shift equivalence classification is highly nontrivial even in very small cases.
Proposition 1.2 translates the problem of shift equivalence in the category of finitely generated abelian groups to the classification of certain \(\mathbb Z[t, t^{-1}]\)-modules under isomorphism. Most of the paper is focused on discussions related to the determination of these isomorphism classes. The authors work out the details for \(2 \times 2\) matrices over \(\mathbb Z\). The study is divided in separate sections according to the characteristic polynomial and uses strongly the theory of commutative rings. The article also contains further comments on higher-dimensional cases and a discussion concerning finite abelian groups.
Reviewer: Luis Hernández Corbato (Madrid)On Yuzvinsky's lattice sheaf cohomology for hyperplane arrangementshttps://zbmath.org/1540.520272024-09-13T18:40:28.020319Z"Mücksch, Paul"https://zbmath.org/authors/?q=ai:mucksch.paulSummary: We establish the relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space, respectively projective space associated to the module of logarithmic vector fields along the arrangement. Our main result gives a Künneth formula connecting the cohomology theories, answering a question by Yoshinaga. This, in turn, provides a characterization of the projective dimension of the module of logarithmic vector fields and yields a new proof of Yuzvinsky's freeness criterion. Furthermore, our approach affords a new formulation of Terao's freeness conjecture and a more general problem.How is a graph not like a manifold?https://zbmath.org/1540.570492024-09-13T18:40:28.020319Z"Ayzenberg, A. A."https://zbmath.org/authors/?q=ai:ayzenberg.anton-a"Masuda, M."https://zbmath.org/authors/?q=ai:masuda.mikiya"Solomadin, G. D."https://zbmath.org/authors/?q=ai:solomadin.grigory-dThe paper studies $2n$-dimensional smooth manifolds $X$ together with an equivariantly formal action of a $k$-torus. Given this data the poset $S(X)$ is defined whose elements are the face submanifolds of $X$ ordered by inclusion. It is easily seen that the poset is graded. The first author together with \textit{V. Cherepanov} [``Matroids in toric topology'', Preprint, \url{arXiv:2203.06282}] has shown that upper intervals in this poset are geometric lattices. Hence results by \textit{A. Björner} [Encycl. Math. Appl. 40, 226--283 (1992; Zbl 0772.05027)] from geometric combinatorics show that the order complex of the proper part of these posets is homotopy equivalent to a wedge of spheres of maximal possible dimension. Now the paper under review considers lower intervals and ``skeleta'' in $S(X)$. By skeleta the authors mean a rank selected subposet of $S(X)$ where all ranks below a certain number are selected. For both situations the papers exhibits high homological connectivity depending also on a number $j$ defined by the $j$-indpendence of the torus action. In a second result it is shown that for simply connected manifolds of dimension $2n$ for \(n \geq 5\) and bipartite GKM-graph there is a simplicial poset for which the equivariant cohomology of the manifold is isomorphic to a quotient of the Stanley-Reisner ring of the order complex of the poset by a linear form.
Reviewer: Volkmar Welker (Marburg)The geometry of supersymmetry: a concise introductionhttps://zbmath.org/1540.580062024-09-13T18:40:28.020319Z"Norbert, Poncin Sarah Schouten"https://zbmath.org/authors/?q=ai:norbert.poncin-sarah-schoutenThis paper reviews the theoretical development of supergeometry and colored supergeometry, tracing their evolution from the basic principles to the latest research findings available at the time of writing.
It starts with a precise explanation of supersymmetry. Then it collect definitions and notions of supermanifolds, also known as \(\mathbb{Z}_2\)-graded manifolds, explores their morphisms, and defines the concept of smoothness in this context. Then it reminds calculus on supermanifolds using tangent sheaves and their sections, which are tools to define super differential forms.
The next part asserts the motivation and need for defining colored supergeometry in the contexts of physics, algebra, and geometry. Then, it reviews the definition of a smooth \(\mathbb{Z}_2^n\)-graded manifold and explores some results in colored supergeometry such as invertability of \(\mathbb{Z}_2^n\) functions and the theorem of colored morphisms.
The final section focuses on the integration theory. It begins with a review of linear \(\mathbb{Z}_2\) algebras, emphasizing \(\mathbb{Z}_2\)-modules, linear maps, and \(\mathbb{Z}_2\)-Berezinian. Then it generalizes these concepts to the \(\mathbb{Z}_2^n\)-graded case. To describe the integration on \(\mathbb{Z}_2^n\)-graded manifolds, it first reviews integration on the ordinary smooth manifolds. Finally, using the \(\mathbb{Z}_2^n\)-Berezinian sheaf of a \(\mathbb{Z}_2^n\)-manifold, the section defines integration for both \(\mathbb{Z}_2\) and \(\mathbb{Z}_2^n\)-manifolds.
Reviewer: Fereshteh Bahadorykhalily (Shiraz)Embedding principle for rings and abelian groupshttps://zbmath.org/1540.682982024-09-13T18:40:28.020319Z"Watase, Yasushige"https://zbmath.org/authors/?q=ai:watase.yasushigeSummary: The article concerns about formalizing a certain lemma on embedding of algebraic structures in the Mizar system, claiming that if a ring \(A\) is embedded in a ring \(B\) then there exists a ring \(C\) which is isomorphic to \(B\) and includes \(A\) as a subring. This construction applies to algebraic structures such as abelian groups and rings.Some issues on the automatic computation of plane envelopes in interactive environmentshttps://zbmath.org/1540.683282024-09-13T18:40:28.020319Z"Botana, Francisco"https://zbmath.org/authors/?q=ai:botana.francisco"Recio, Tomas"https://zbmath.org/authors/?q=ai:recio.tomasSummary: This paper addresses some concerns, and describes some proposals, on the ellusive concept of envelope of an algebraic family of varieties, and on its automatic computation. We describe how to use the recently developed Gröbner cover algorithm to study envelopes of families of algebraic curves, and we give a protocol towards its implementation in dynamic geometry environments. The proposal is illustrated through some examples. A beta version of GeoGebra is used to highlight new envelope abilities in interactive environments, and limitations of our approach are discussed, since the computations are performed in an algebraically closed field.Short proofs of ideal membershiphttps://zbmath.org/1540.683302024-09-13T18:40:28.020319Z"Hofstadler, Clemens"https://zbmath.org/authors/?q=ai:hofstadler.clemens"Verron, Thibaut"https://zbmath.org/authors/?q=ai:verron.thibautSummary: A cofactor representation of an ideal element, that is, a representation in terms of the generators, can be considered as a certificate for ideal membership. Such a representation is typically not unique, and some can be a lot more complicated than others. In this work, we consider the problem of computing sparsest cofactor representations, i.e., representations with a minimal number of terms, of a given element in a polynomial ideal. While we focus on the more general case of noncommutative polynomials, all results also apply to the commutative setting.
We show that the problem of computing cofactor representations with a bounded number of terms is decidable and \textsf{NP}-complete. Moreover, we provide a practical algorithm for computing sparse (not necessarily optimal) representations by translating the problem into a linear optimization problem and by exploiting properties of signature-based Gröbner basis algorithms. We show that, for a certain class of ideals, representations computed by this method are actually optimal, and we present experimental data illustrating that it can lead to noticeably sparser cofactor representations.\(\mathfrak{gl}(3)\) polynomial integrable system: different faces of the 3-body/\(\mathcal{A}_2\) elliptic Calogero modelhttps://zbmath.org/1540.810712024-09-13T18:40:28.020319Z"Turbiner, Alexander V."https://zbmath.org/authors/?q=ai:turbiner.alexander"Lopez Vieyra, Juan Carlos"https://zbmath.org/authors/?q=ai:lopez-vieyra.juan-carlos"Guadarrama-Ayala, Miguel A."https://zbmath.org/authors/?q=ai:guadarrama-ayala.miguel-aSummary: It is shown that the \(\mathfrak{gl}(3)\) polynomial integrable system, introduced by \textit{V. V. Sokolov} and \textit{A. V. Turbiner} in [J. Phys. A, Math. Theor. 48, No. 15, Article ID 155201, 15 p. (2015; Zbl 1329.81434)], is equivalent to the \(\mathfrak{gl}(3)\) quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian as well as their third-order integral can be rewritten in terms of \(\mathfrak{gl}(3)\) algebra generators. In turn, all these \(\mathfrak{gl}(3)\) generators can be represented by the non-linear elements of the universal enveloping algebra of the 5-dimensional Heisenberg algebra \(\mathfrak{h}_5(\hat{p}_{1,2},\hat{q}_{1,2}, I)\), thus, the Hamiltonian and integral are two elements of the universal enveloping algebra \(U_{\mathfrak{h}_5}\). In this paper, four different representations of the \(\mathfrak{h}_5\) Heisenberg algebra are used: (I) by differential operators in two real (complex) variables, (II) by finite-difference operators on uniform or exponential lattices. We discovered the existence of two 2-parametric bilinear and trilinear elements (denoted \(H\) and \(I\), respectively) of the universal enveloping algebra \(U(\mathfrak{gl}(3))\) such that their Lie bracket (commutator) can be written as a linear superposition of nine so-called artifacts - the special bilinear elements of \(U(\mathfrak{gl}(3))\), which vanish once the representation of the \(\mathfrak{gl}(3)\)-algebra generators is written in terms of the \(\mathfrak{h}_5(\hat{p}_{1,2},\hat{q}_{1,2}, I)\)-algebra generators. In this representation all nine artifacts vanish, two of the above-mentioned elements of \(U(\mathfrak{gl}(3))\) (called the Hamiltonian \(H\) and the integral \(I)\) commute(!); in particular, they become the Hamiltonian and the integral of the 3-body elliptic Calogero model, if \((\hat{p},\hat{q})\) are written in the standard coordinate-momentum representation. If \((\hat{p},\hat{q})\) are represented by finite-difference/discrete operators on uniform or exponential lattice, the Hamiltonian and the integral of the 3-body elliptic Calogero model become the isospectral, finite-difference operators on uniform-uniform or exponential-exponential lattices (or mixed) with polynomial coefficients. If \((\hat{p},\hat{q})\) are written in complex \((z, \bar{z})\) variables the Hamiltonian corresponds to a complexification of the 3-body elliptic Calogero model on \(\mathbb{C}^2\).Dynamics of localized waves in quasi-one-dimensional imbalanced binary Bose-Einstein condensateshttps://zbmath.org/1540.811932024-09-13T18:40:28.020319Z"Ismailov, K. K."https://zbmath.org/authors/?q=ai:ismailov.k-k"Baizakov, B. B."https://zbmath.org/authors/?q=ai:baizakov.bakhtiyor-b"Abdullaev, F. Kh."https://zbmath.org/authors/?q=ai:abdullaev.fatkhulla-kh"Salerno, M."https://zbmath.org/authors/?q=ai:salerno.marioSummary: In the framework of coupled Gross-Pitaevskii equations, we explore the dynamics of localized waves in quasi-1D imbalanced binary Bose-Einstein condensates where the intra-component interaction is repulsive, while the inter-component one is attractive. The existence regimes of stable self-trapped localized states in the form of symbiotic solitons have been analyzed. Imbalanced mixtures, where the number of atoms in one component exceeds the number of atoms in the other component, are considered in parabolic potential and box-like trap. A variational approach has been developed which allows us to find the stationary state of the system and frequency of small amplitude oscillations near the equilibrium. It is shown that the strength of inter-component coupling can be retrieved from the frequency of the localized state's vibrations. When all the intra-species and inter-species interactions are repulsive, we numerically find a new type of symbiotic solitons resembling dark-bright solitons. The motion of the minority component in the surrounding gas of the majority component with different velocities reveals its superfluid properties.Counting and enumerating feasible rotating schedules by means of Gröbner baseshttps://zbmath.org/1540.901422024-09-13T18:40:28.020319Z"Falcón, Raúl"https://zbmath.org/authors/?q=ai:falcon.raul-m"Barrena, Eva"https://zbmath.org/authors/?q=ai:barrena.eva"Canca, David"https://zbmath.org/authors/?q=ai:canca.david"Laporte, Gilbert"https://zbmath.org/authors/?q=ai:laporte.gilbertSummary: This paper deals with the problem of designing and analyzing rotating schedules with an algebraic computational approach. Specifically, we determine a set of Boolean polynomials whose zeros can be uniquely identified with the set of rotating schedules related to a given workload matrix subject to standard constraints. These polynomials constitute zero-dimensional radical ideals, whose reduced Gröbner bases can be computed to count and even enumerate the set of rotating schedules that satisfy the desired set of constraints. Thereby, it enables to analyze the influence of each constraint in the same.