Recent zbMATH articles in MSC 13https://zbmath.org/atom/cc/132024-03-13T18:33:02.981707ZWerkzeugRich groups, weak second-order logic, and applicationshttps://zbmath.org/1528.031652024-03-13T18:33:02.981707Z"Kharlampovich, Olga"https://zbmath.org/authors/?q=ai:kharlampovich.olga-g"Myasnikov, Alexei"https://zbmath.org/authors/?q=ai:myasnikov.alexei-g"Sohrabi, Mahmood"https://zbmath.org/authors/?q=ai:sohrabi.mahmoodSummary: In this chapter we initiate a study of first-order rich groups, i. e., groups where the first-order logic has the same power as the weak second-order logic. Surprisingly, there are quite a few finitely generated rich groups, they are somewhere inbetween hyperbolic and nilpotent groups (these are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev's problems in various groups
For the entire collection see [Zbl 1465.20001].Describing models of \(\mathrm{Th}(\mathbb{Z})\) in adelic termshttps://zbmath.org/1528.031662024-03-13T18:33:02.981707Z"Macintyre, Angus"https://zbmath.org/authors/?q=ai:macintyre.angus-jSummary: We consider abelian groups \(\Gamma\) elementarily equivalent to \(\mathbb{Z}\), the additive group of integers. We give an exact classification of the isomorphism types of those \(\Gamma\) which admit a pure embedding of \(\mathbb{Z}\), by using homological algebra and adelic ideas to be found in Tate's thesis (and given an elementary presentation in an expository paper of \textit{K. Conrad} [``The character group of \(\mathbb{Q}\)'', Preprint, \url{https://kconrad.math.uconn.edu/blurbs/gradnumthy/characterQ.pdf}]).
For the entire collection see [Zbl 1419.03005].Decidability and modules over Bézout domainshttps://zbmath.org/1528.031692024-03-13T18:33:02.981707Z"Toffalori, Carlo"https://zbmath.org/authors/?q=ai:toffalori.carloSummary: I survey recent results on the decidability of first order theories of modules over several noteworthy Bézout domains.
For the entire collection see [Zbl 1419.03005].Complex psd-minimal polytopes in dimensions two and threehttps://zbmath.org/1528.050032024-03-13T18:33:02.981707Z"Bogart, Tristram"https://zbmath.org/authors/?q=ai:bogart.tristram"Gouveia, João"https://zbmath.org/authors/?q=ai:gouveia.joao"Torres, Juan Camilo"https://zbmath.org/authors/?q=ai:torres.juan-camiloSummary: The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last of these, for which the least is known, and in particular on understanding which polytopes are complex psd-minimal. We prove the existence of an obstruction to complex psd-minimality which is efficiently computable via lattice membership problems. Using this tool, we complete the classification of complex psd-minimal polygons (geometrically as well as combinatorially). In dimension three we exhibit several new examples of complex psd-minimal polytopes and apply our obstruction to rule out many others.Pulling and pushing certain ideals in function ringshttps://zbmath.org/1528.060182024-03-13T18:33:02.981707Z"Dube, Themba"https://zbmath.org/authors/?q=ai:dube.themba"Stephen, Dorca Nyamusi"https://zbmath.org/authors/?q=ai:stephen.dorca-nyamusiSummary: Let \(X\) be a Tychonoff space. Associated with every subset \(S\) of \textit{\( \beta\) X} are the ideals
\[
\boldsymbol{M}^S = \{f \in C(X) \mid S \subseteq \operatorname{cl}_{\beta X} Z(f) \} \text{ and } \boldsymbol{O}^S = \{f \in C(X) \mid S \subseteq \operatorname{int}_{\beta X} \operatorname{cl}_{\beta X} Z(f) \}
\]
of the ring \(C(X)\), where \(Z(f)\) denotes the zero-set of \(f\). We show that \(\langle C(f) [ \boldsymbol{O}^K] \rangle = \boldsymbol{O}^{( \beta f )^{- 1} [ K ]}\) for any continuous map \(f : X \to Y\) and every closed subset \(K\) of \textit{\( \beta Y\)}, where \(\beta f : \beta X \to \beta Y\) is the Stone extension of \(f\) and \(C(f) : C(Y) \to C(X)\) is the ring homomorphism \(g \mapsto g \circ f\). On the other hand, \(C ( f )^{- 1} [ \boldsymbol{M}^S] = \boldsymbol{M}^{( \beta f ) [ S ]}\) for every subset \(S\) of \textit{\( \beta\) X} if and only if \(f\) is a WN-map, in the sense of \textit{R. G. Woods} [J. Lond. Math. Soc., II. Ser. 7, 453--461 (1974; Zbl 0271.54005)]. These results (and others) are corollaries of more general ones obtained in pointfree function rings.Classes of good Noetherian ringshttps://zbmath.org/1528.130012024-03-13T18:33:02.981707Z"Ionescu, Cristodor"https://zbmath.org/authors/?q=ai:ionescu.cristodor``Classes of good Noetherian rings'' by Cristodor Ionescu is an excellent book published under the imprint Birkhäuser within the Frontiers in Mathematics series. The primary objective of this book is to offer readers an exhaustive treatment of various classes of Noetherian rings and morphisms, including (quasi)-excellent rings, Nagata rings, rings with good formal fibers, and more. Originating from Emmy Noether's groundbreaking work in the 1920s, the field has seen significant evolution, enriched by contributions from Wolfgang Krull, Yasuo Akizuki, Claude Chevalley, Oscar Zariski, Masayoshi Nagata, and Alexander Grothendieck, etc.. In fact, this work builds upon the foundational contributions of esteemed authors such as H. Matsumura, and A. Brezuleanu-N. Radu. The author places a strong emphasis on presenting crucial characterizations of such rings, such as those for regular morphisms, and provides a comprehensive treatment of Popescu's Theorem on Neron desingularization. The book wraps up with an extensive bibliography, covering not just references, but also encompassing monographs, textbooks, and research papers that share thematic connections with the subject matter explored in the book and offering a historical perspective on the theory's development. The book is recommended for PhD students and researchers in commutative algebra, algebraic and arithmetic geometry, and number theory. It also includes recent developments in morphism fibers. Of course, it assumes a strong background in commutative algebra, homological algebra, and related topics to follow the subjects of the book.
Reviewer: Jebrel M. Habeb (Irbid)On a theorem of Anderson and Chunhttps://zbmath.org/1528.130022024-03-13T18:33:02.981707Z"Aliabad, Ali Rezaie"https://zbmath.org/authors/?q=ai:aliabad.ali-rezaie"Farrokhpay, Farimah"https://zbmath.org/authors/?q=ai:farrokhpay.farimah"Siavoshi, Mohammad Ali"https://zbmath.org/authors/?q=ai:siavoshi.m-aAll rings are commutative unital ring. A ring \(R\) is called strongly associate if for each \(a,b\in R\) whenever \(Ra=Rb\) implies that \(a=ub\) for some unit \(u\) of \(R\). \(R\) is called strongly regular associate if whenever \(Ra=Rb\), for \(a,b\in R\), then there exist regular (non-zerodivisor) elements \(r,s\in R\) such that \(a=rb\) and \(b=ra\) (and therefore \(a\) and \(b\) are (strongly) associate in classical quotient of \(R\)). A ring \(R\) has stable range \(1\) if whenever \(a, b\in R\) and \(Ra+Rb=R\), then there exists \(x\in R\), such that \(a+bx\) is a unit in \(R\). Finally a ring \(R\) has regular range \(1\), if whenever \(a, b\in R\) and \(Ra+Rb=R\), then there exists \(x\in R\) such that \(a+bx\) is regular in \(R\) (and there for is a unit in classical quotient ring of \(R\)). The authors proved that the polynomial ring always has regular range \(1\) and each regular range \(1\) ring is a strongly regular associate. Finally the characterized when the ring \(C(X)\) is strongly regular associate or has stable range \(1\).
Reviewer: Alborz Azarang (Ahvāz)Rings of invariants for three dimensional modular representationshttps://zbmath.org/1528.130032024-03-13T18:33:02.981707Z"Herzog, Jürgen"https://zbmath.org/authors/?q=ai:herzog.jurgen"Trivedi, Vijaylaxmi"https://zbmath.org/authors/?q=ai:trivedi.vijaylaxmiLet \(p>3\) be a prime number, let \(\mathbb{F}\) be a field of characteristic \(p,\) let \(G\) be a finite abelian group and let \(V\) be an \(n\)-dimensional representation of \(G\) over \(\mathbb{F}.\) Let \(\mathbb{F}[V]^G\) be the ring of invariants. The aim of the paper is to study the ring of invariants in the case \(G=(\mathbb{Z}/p\mathbb{Z})^r,\) for a 3-dimensional generic representation. The ring of invariants are computed and it is shown that these rings are complete intersection rings with embedding dimension \(\lceil \frac{r}{2} \rceil +3.\)
Reviewer: Cristodor-Paul Ionescu (Bucureşti)Maximal subrings of classical integral domainshttps://zbmath.org/1528.130042024-03-13T18:33:02.981707Z"Alinaghizadeh, M."https://zbmath.org/authors/?q=ai:alinaghizadeh.mohammadreza"Azarang, A."https://zbmath.org/authors/?q=ai:azarang.alborzSummary: It is shown that if \(R\) is an integral domain with \(|R| = 2^{\mathfrak{c}}\), then either \(R\) has a maximal subring or \(R\) has a prime ideal \(P\) which is not a maximal ideal of \(R, Char (R/P) = Char (R)\) and \(|R/P| < |R|\). We prove that if \(R = A \oplus I\) is a ring, where \(A\) is a subring of \(R\) and \(I\) is an ideal of \(R\) and \(I/I^2\) is a finitely generated nonzero \(R/I\)-module, then \(R\) has a maximal subring. We show that if \(R\) is an infinite Noetherian ring which is not integral over its prime subring and \(R\) has no maximal subring, then \(R\) has a prime ideal \(P\) (which is not a maximal ideal of \(R)\) such that \(R/P\) is not integral over its prime subring but for each ideal \(I\) of \(R\) with \(P \subsetneq I, R/I\) is integral over its prime subring, in particular \(dim (R/P) = 1\) or \(2\) and \(|R/P| = |R|\) is countable. The existence of maximal subrings in Prüfer and Bézout domains is investigated. Moreover, we prove that if \(R\) is a maximal subring of an integral domain \(T\), where \(T\) is not a field, and \(R\) is a Prüfer domain, then \(R\) is integrally closed in \(T\) and consequently \(T\) is an overring of \(R\) and \(T\) is a Prüfer domain. Conversely, if \(R\) is integrally closed in \(T\) and \(T\) is a Prüfer domain, then \(R\) is a Prüfer domain. Finally, we study the existence of maximal subrings in a ring \(T\) by the use of survival pairs and valuation pairs too.An algebraic approach to sets defining minimal dominating sets of regular graphshttps://zbmath.org/1528.130052024-03-13T18:33:02.981707Z"Nasernejad, Mehrdad"https://zbmath.org/authors/?q=ai:nasernejad.mehrdadSummary: Suppose that \(V = \{1, \ldots, n\}\) is a non-empty set of \(n\) elements, \(\mathcal{S} = \{ S_1, \ldots, S_m\}\) a non-empty set of \(m\) non-empty subsets of \(V\). In this paper, by using some algebraic notions in commutative algebra, we investigate the question arises whether there exists an undirected finite simple graph \(G\) with \(V(G)=V\), where \(\mathcal{S}\) is the set whose elements are the minimal dominating sets of \(G\).Absorbing multiplication modules over pullback ringshttps://zbmath.org/1528.130062024-03-13T18:33:02.981707Z"Atani, S. Ebrahimi"https://zbmath.org/authors/?q=ai:ebrahimi-atani.shahabaddin"Bazari, M. Sedghi Shanbeh"https://zbmath.org/authors/?q=ai:bazari.maryam-sedghi-shanbehSummary: Following some ideas and a technique introduced in [\textit{R. Ebrahimi Atani} and the first author, Commun. Algebra 41, No. 2, 776--791 (2013; Zbl 1280.13003)] we give a complete classification, up to isomorphism, of all indecomposable 2-absorbing multiplication modules with finitedimensional top over pullback of two discrete valuation domains with the same residue field.Applications of Swan's Bertini theorem to unimodular rowshttps://zbmath.org/1528.130072024-03-13T18:33:02.981707Z"Keshari, Manoj K."https://zbmath.org/authors/?q=ai:keshari.manoj-kumar"Sharma, Sampat"https://zbmath.org/authors/?q=ai:sharma.sampatThis paper under review deals with the nice group structure on the elementary orbit space of unimodular rows. Let R be a commutative Noetherian \(d\)-dimensional ring with \(1\).
Let \(\mathrm{Um}_n(R)\) denote the set of unimodular rows of length \(n\) in a ring \(R\)., and \(\mathrm{E}_n(R)\) denote the normal subgroup of \(\mathrm{GL}_n(R)\) generated by elementary matrices. In 1983, W. van der Kallen introduced a group structure to the orbit space \(\mathrm{Um}_{d+1}(R)/\mathrm{E}_{d+1}(R)\) for \(n\ge 2\). The product formula is said to be \textit{nice} if it satisfies
\[
[x_1,v_2,\ldots,v_n]*[v_1,v_2,\ldots,v_n]=[x_1v_1,v_2,\ldots,v_n].
\]
In this paper, the authors considered affine algebra \(R\) of dimension \(d\ge 4\) over a perfect field \(k\) of characteristic different from \(2\). Let \(MS_n(R)\) denote the universal Mennicke \(n\)-symbol. Using Swan's Bertini theorem, the authors proved : \(MS_{d+1}(R)\) is uniquely divisible prime to char \(k\) if \(R\) is reduced and \(k\) is infinite with c.d. \((k)\le 1\). This is a generalization of Fasel's result.
For an ideal \(I\subset R\) in \(R\), they have proved the following results for relative groups:
(1) \(\mathrm{Um}_{d+1}(R,I)/\mathrm{E}_{d+1}(R,I)\) has nice group structure if c.d. \((k)\le 2\).
(2) \(\mathrm{Um}_{d}(R)/\mathrm{E}_{d}(R)\) has nice group structure if k is algebraically closed of char \(k\ne 2, 3\) and either \(k=\overline{\mathbb{F}}_p\) or \(R\) is normal.
Reviewer: Rabeya Basu (Pune)Residual intersections and linear powershttps://zbmath.org/1528.130082024-03-13T18:33:02.981707Z"Eisenbud, David"https://zbmath.org/authors/?q=ai:eisenbud.david"Huneke, Craig"https://zbmath.org/authors/?q=ai:huneke.craig-l"Ulrich, Bernd"https://zbmath.org/authors/?q=ai:ulrich.berndLet \(S\) be a Noetherian ring and let \(I \subset S\) be an ideal of codimension \(g\). The notion of residual intersections generalizes the notion of linkage. Let \(J = (a_1,\dots,a_s)\) be an ideal contained in \(I\) and define \(K = J:I\). If \(\hbox{codim } K \geq s\) then \(K\) is called the \(s\)-residual intersection of \(I\) with respect to \(J\). If in addition \(\hbox{codim } (I+K) > s\) then the residual intersection if said to be geometric. Under strong hypotheses, residual intersections have nice properties and these have been well studied. The main result of this paper weakens these hypotheses and gives a natural rank 1, self-dual, maximal Cohen-Macaulay (MCM) module over certain residual intersections of certain ideals. More precisely, now we assume \(S\) to be a standard graded polynomial ring over an infinite field \(k\). Let \(I \subset S\) be a non-zero ideal generated by forms of the same degree \(\delta\). Let \(\ell = \ell(I)\) be the analytic spread of \(I\). Let \(J \subset I\) be an ideal generated by \(\ell -1\) general forms in \(I\) of degree \(\delta\). Set \(R = S/(J:I)\), \(\bar R = S/(J:I^\infty)\), \(\bar I = I \bar R\), \(M = M(IR)\) (the latter is a module defined earlier in the paper). If \(R\) is reduced away from \(V(I)\) and all sufficiently high powers of \(I\) are linearly presented then the authors give strong conclusions about \(R, \bar R\) and \(M\), including the rank 1, self-dual, MCM properties mentioned above. They also give several consequences of this result, including a theorem about the ideal of \(2 \times 2\) minors of a \(2 \times n\) generic matrix. Many examples are given to show the breadth of this result.
Reviewer: Juan C. Migliore (Notre Dame)On the first nontrivial strand of syzygies of projective schemes and condition \(\mathrm{ND}(\ell)\)https://zbmath.org/1528.130092024-03-13T18:33:02.981707Z"Ahn, Jeaman"https://zbmath.org/authors/?q=ai:ahn.jeaman"Han, Kangjin"https://zbmath.org/authors/?q=ai:han.kangjin"Kwak, Sijong"https://zbmath.org/authors/?q=ai:kwak.sijongLet $X \subseteq \mathbb{P}^{n+e}$ be a non-degenerate closed subscheme of dimesnion $n$ and codimension $e$ defined over an algebraically closed field $\mathbf{k}$. Although some results of the paper under review hold in this generality, we assume that $X$ is a projective variety and the characteristic of the base field $\mathbf{k}$ is zero for convenience. After Green's pioneering work on syzygies, there has been a great deal of interest in understanding the Betti tables of projective varieties. The Betti table of $X \subseteq \mathbb{P}^{n+e}$ consists of the graded Betti numbers
\[
\beta_{i,j}(X):= \dim_{\mathbf{k}} \operatorname{Tor}_i^R(R/I_{X|\mathbb{P}^r}, \mathbf{k})_{i+j},
\]
where $R:=\mathbf{k}[x_0, \ldots, x_{n+e}]$ is the homogeneous coordinate ring of $\mathbb{P}^{n+e}$. Previously, Han-Kwak proved that
\[
\beta_{i,1}(X) \leq i \binom{e+1}{ i+1}\text{ for }i \geq 1
\]
and the equality holds for some (or each) $1 \leq i \leq e$ if and only if $X \subseteq \mathbb{P}^{n+e}$ is arithmetically Cohen-Macaulay with $2$-linear resolution. One may expect to generalize this result to the first nontrivial strand of the Betti table -- assuming $H^0(\mathbb{P}^{n+e}, \mathscr{I}_{X|\mathbb{P}^{n+e}}(\ell)) = 0$ for $\ell \geq 1$, we seek to find a reasonable upper bound for $\beta_{i, \ell}(X)$. For this purpose, we need an additional condition that is $\operatorname{ND}(\ell)$ condition introduced in this paper. We say that $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{ND}(\ell)$ condition if $H^0(\Lambda, \mathscr{I}_{X \cap \Lambda|\Lambda}(\ell)) = 0$ for every general linear subspace $\Lambda \subseteq \mathbb{P}^{n+e}$ with $\dim \Lambda \geq e$. It is equivalent to $\operatorname{Gin}(I_{X|\mathbb{P}^{n+e}}) \subseteq (x_0, \ldots, x_{e-1})^{\ell+1}$, where $\operatorname{Gin}(I_{X|\mathbb{P}^{n+e}})$ is the generic initial ideal with respect to the degree reverse lexicographic order (Proposition 2.3). Note that $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{ND}(1)$ condition if and only if $X \subseteq \mathbb{P}^{n+e}$ is non-degenerate. This means that $\operatorname{ND}(1)$ condition is automatic while $\operatorname{ND}(\ell)$ condition for $\ell \geq 2$ is nontrivial. In Section 4, relevant examples and some questions on $\operatorname{ND}(\ell)$ condition are presented. The first main result of the paper is Theorem 1.1: If $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{ND}(\ell)$ condition, then
\[
\beta_{i, \ell}(X) \leq \binom{i+\ell-1}{ \ell} \binom{e + \ell}{i + \ell}\text{ for }i \geq 1
\]
and the equality holds for some (or each) $1 \leq i \leq e$ if and only if $X \subseteq \mathbb{P}^{n+e}$ is arithmetically Cohen-Macaulay with $(\ell+1)$-linear resolution.
Next, recall that $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{N}_{d,p}$ condition if $\beta_{i,j}(X) = 0$ for $i \leq p$ and $j \geq d$. A well-known result of Eisenbud-Green-Hulek-Popescu says that if $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{N}_{2,e}$, then $X$ is arithmetically Cohen-Macaulay with $2$-linear resolution. To generalized this result, we also need $\operatorname{ND}(\ell)$ condition. More precisely, the second main result of the paper (Theorem 1.2) states ``if $X \subseteq \mathbb{P}^{n+e}$ satisfies $\operatorname{N}_{d,e}$ condition and $\operatorname{ND}(d-1)$ condition, then $X$ is arithmetically Cohen-Macaulay with $d$-linear resolution.''
Reviewer: Jinhyung Park (Daejeon)The facet ideals of chessboard complexeshttps://zbmath.org/1528.130102024-03-13T18:33:02.981707Z"Jiang, Chengyao"https://zbmath.org/authors/?q=ai:jiang.chengyao"Zhao, Yakun"https://zbmath.org/authors/?q=ai:zhao.yakun"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.11|wang.hong.12"Zhu, Guangjun"https://zbmath.org/authors/?q=ai:zhu.guangjunGraph complexes have provided an important link between combinatorics and algebra, topology, and geometry. The two most important graph complexes are the matching complex and independence complex. The chessboard complex is the matching complex of a complete bipartite graph $K_{m,n}$. In this paper the authors describe the irreducible decomposition of the facet ideal of the chessboard complex, with $n\geq m$. They also provide some lower bounds for the depth and regularity of the facet ideal.
Reviewer: Monica La Barbiera (Messina)Regularity of powers of edge ideals of Cohen-Macaulay weighted oriented forestshttps://zbmath.org/1528.130112024-03-13T18:33:02.981707Z"Kumar, Manohar"https://zbmath.org/authors/?q=ai:kumar.manohar"Nanduri, Ramakrishna"https://zbmath.org/authors/?q=ai:nanduri.ramakrishnaA weighted oriented graph is a graph \(D = (V(D), E(D),w)\), where \(V(D)\) is its vertex set, \(E(D) = \{(x, y) \ | \ \text{there is an edge from } x \text{ to } y\}\) is its edge set and \(w\) is its weight function which assigns a weight \(w(x)\) to each vertex \(x\) of \(D\). The underlying graph \(G\) of \(D\) is a simple graph with \(V(G) = V(D)\) and \(E(G) = \{\{x, y\} \ |\ (x, y)\in E(D)\}\). A vertex \(x\) of \(D\) is called a sink if there is no vertex \(y\) with \((x,y)\in E(D)\). Let \(V(D) = \{x_1, \dots , x_n\}\) and \(R = K[x_1, \dots , x_n]\), a polynomial ring over a field \(K\). Then the edge ideal of \(D\) is defined as the ideal
\[
I (D) = \langle x_i x_j^{w(x_j)} \ | \ (x_i , x_j) \in E(D)\rangle.
\]
In the paper under review, the regularity of powers of \(I(D)\) is computed where \(D\) is a Cohen-Macaulay weighted oriented forest. In fact firstly the result is gained for the first power and then inductively it is shown:
Let \(D\) be a weighted oriented unmixed forest (equivalently Cohen-Macaulay forest). Let \(\{\{x_1, y_1\}, \dots , \{x_r , y_r \}\}\) be a perfect matching in the underlying graph \(G\) of \(D\). Suppose \(y_1, \dots, y_r\) are sinks. Then for any \(k \geq 1\),
\begin{align*}
\mathrm{reg}(I (D)^k ) = \max \{ &(k - 1)(\max \{ w(y_{i_j})\} + 1) + \sum_{j=1}^s w(y_{i_j}) + 1 :\\
& \text{none of the edges } \{x_{i_j} , y_{i_j} \} \text{ being adjacent}\}.
\end{align*}
Recall that two edges are said to be adjacent if there exists an edge between them.
Reviewer: Fahimeh Khosh-Ahang Ghasr (Ilam)Elementary construction of minimal free resolutions of the Specht ideals of shapes \((n-2,2)\) and \((d,d,1)\)https://zbmath.org/1528.130122024-03-13T18:33:02.981707Z"Shibata, Kosuke"https://zbmath.org/authors/?q=ai:shibata.kosuke"Yanagawa, Kohji"https://zbmath.org/authors/?q=ai:yanagawa.kohjiLet \(\lambda\) be a partition of \(n\), \(T\) a Young tableau of shape \(\lambda\) filled in bijectively with the integers \(1,2,\dots,n\), and \(R=K[x_1,\dots,x_n]\) the \(n\)-variable polynomial algebra over a field \(K\) of characteristic zero. Denote by \(f_T\in R\) the product of the linear forms \(x_i-x_j\) where \(i,j\) are contained in the same column of \(T\) and \(i\) is above \(j\). Note that the symmetric group \(S_n\) acts on the algebra \(R\) by permutation of the variables, and the subspace of \(R\) spanned by the \(f_T\) as \(T\) ranges over the Young tableaux of shape \(\lambda\) is isomorphic to the Specht module labelled by \(\lambda\). The ideal \(I_{\lambda}\) of \(R\) spanned by this subspace is called a \textit{Specht ideal}. The authors construct a minimal free resolution of \(R/I_{\lambda}\) in the cases when \(\lambda=(n-2,2)\) and \(\lambda=(d,d,1)\). The construction is explicit, and the paper uses only the basic theory of Specht modules.
Reviewer: Matyas Domokos (Budapest)On \(w_\infty\)-Warfield cotorsion modules and Krull domainshttps://zbmath.org/1528.130132024-03-13T18:33:02.981707Z"Pu, Yongyan"https://zbmath.org/authors/?q=ai:pu.yongyan"Zhao, Wei"https://zbmath.org/authors/?q=ai:zhao.wei.8"Tang, Gaohua"https://zbmath.org/authors/?q=ai:tang.gaohua"Wang, Fanggui"https://zbmath.org/authors/?q=ai:wang.fanggui"Xiao, Xuelian"https://zbmath.org/authors/?q=ai:xiao.xuelianAuthors' abstract: Let \(R\) be a commutative domain with \(1\) and \(Q\)(\(\ne R\)) its field of quotients. In this note an \(R\)-module \(M\) is called \(w_{\infty}\)-Warfield cotorsion if \(M\in\mathcal{WC}\cap\mathcal{P}^{\perp}_{w_{\infty}}\), where \(\mathcal{WC}\) denotes the class of all Warfield cotorsion \(R\)-modules and \(\mathcal{P}_{w_{\infty}}\) the class of all \(w_{\infty}\)-projective \(R\)-modules. It is shown that \(R\) is a PVMD if and only if all \(w\)-cotorsion \(R\)-modules are \(w_{\infty}\)-Warfield cotorsion, and that \(R\) is a Krull domain if and only if every \(w\)-Matlis cotorsion strong \(w\)-module over \(R\) is a \(w_{\infty}\)-Warfield cotorsion \(w\)-module.
Reviewer: François Couchot (Caen)Dominant local rings and subcategory classificationhttps://zbmath.org/1528.130142024-03-13T18:33:02.981707Z"Takahashi, Ryo"https://zbmath.org/authors/?q=ai:takahashi.ryoSummary: We introduce a new notion of commutative noetherian local rings, which we call dominant. We explore fundamental properties of dominant local rings and compare them with other local rings. We also provide several methods to get a new dominant local ring from a given one. Finally, we classify resolving subcategories of the module category \(\mathsf{mod}\,R\) and thick subcategories of the derived category \(\mathsf{D}^b(R)\) and the singularity category \(\mathsf{D}^{sg}(R)\) for a local ring \(R\) whose certain localizations are dominant local rings. Our results recover and refine all the known classification theorems described in this context.Some results on top local cohomology moduleshttps://zbmath.org/1528.130152024-03-13T18:33:02.981707Z"Aghapournahr, Moharram"https://zbmath.org/authors/?q=ai:aghapournahr.moharram"Bahmanpour, Kamal"https://zbmath.org/authors/?q=ai:bahmanpour.kamalLet \(R\) 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
\[
H^{i}_{\mathfrak{a}}(M) \cong \underset{n\in \mathbb{N}}\varinjlim \mathrm{Ext}^{i}_{R}(R/ \mathfrak{a}^{n},M).
\]
Moreover, the cohomological dimension of \(M\) with respect to \(\mathfrak{a}\) is defined by \[
\mathrm{cd}(\mathfrak{a},M)= \sup \{i\geq 0\text{ such that }H^{i}_{\mathfrak{a}}(M)\neq 0\}.
\]
The authors determine the set of all attached prime ideals of the top local cohomology module \(H^{\mathrm{cd}(\mathfrak{a},M)}_{\mathfrak{a}}(M)\) of any finitely generated \(R\)-module \(M\) with respect to the ideal \(\mathfrak{a}\) of \(R\) in terms of certain elements of \(\mathrm{Supp}_{R}(M)\). As a consequence, they show that for any pair of finitely generated \(R\)-modules \(M\) and \(N\) with \(\mathrm{Supp}_{R}(N) \subseteq\mathrm{Supp}_{R}(M)\), if \(\mathrm{cd}(\mathfrak{a},M)= \mathrm{cd}(\mathfrak{a},N)=c \geq 0\), then \(\mathrm{Att}_{R}\left(H^{c}_{\mathfrak{a}}(N)\right)\subseteq \mathrm{Att}_{R}\left(H^{c}_{\mathfrak{a}}(M)\right)\) and \(\sqrt{\mathrm{ann}_{R}\left(H^{c}_{\mathfrak{a}}(M)\right)}\subseteq \sqrt{\mathrm{ann}_{R}\left(H^{c}_{\mathfrak{a}}(N)\right)}\). Furthermore, in the special case that \(R\) is a local ring, they prove some similar results concerning the associated prime ideals of Matlis dual functors of top local cohomology modules.
Reviewer: Hossein Faridian (Clemson)An extension of \(S\)-Noetherian rings and moduleshttps://zbmath.org/1528.130162024-03-13T18:33:02.981707Z"Jara, Pascual"https://zbmath.org/authors/?q=ai:jara.pascualLet \(A\) be a commutative unitary ring and \(S\) a multiplicatively closed subset of \(A\). The ideas in [\textit{E. Hamann} et al., Pac. J. Math. 135, No. 1, 65--79 (1988; Zbl 0627.13007)] were extended by \textit{D. D. Anderson} and \textit{T. Dumitrescu} [Commun. Algebra 30, No. 9, 4407--4416 (2002; Zbl 1060.13007)] who defined and studied the so-called \(S\)-Noetherian rings: \(A\) is \(S\)-Noetherian if for every ideal \(I\) of \(A\), there exists a finitely generated subideal \(J\) of \(I\) such that \(I/J\) is annihilated by some \(s\in S\). A module variant of this definition was also given. Subsequently, there have been a lot of activity concerning \(S\)-Noetherianity (see the bibliography of the present paper).
In the present paper, the author extends the S-Noetherian concept using the more abstract notion of hereditary torsion theory instead of \(S\). Let \(\sigma\) be a hereditary torsion theory in Mod-\(A\) given by a Gabriel filter of ideals \(L(\sigma)\), that is, a non-empty filter of ideals such that an ideal \(J\) belongs to \(L(\sigma)\) provided there exists \(I\in L(\sigma)\) such that \((J : x) \in L(\sigma)\) for every \(x \in I\). The author defines \(A\) to be a totally \(\sigma\)-Noetherian ring if for every ideal \(I\) of \(A\), there exists a finitely generated subideal \(J\) of \(I\) such that \(I/J\) is annihilated by some \(H\in L(\sigma)\). A module variant of this definition is also given. Besides proving most of the results in [loc. cit.] in this new setup, the paper contains many new results. As an example we mention the Hilbert-like basis theorem. Let \(\sigma\) be a finite type hereditary torsion theory in Mod-\(A\) (that is, every \(I\in L(\sigma)\) contains some finitely generated \(J\in L(\sigma)\)) such that \(\cap_{n\geq 1}H^n\in L(\sigma)\) for each \(H\in L(\sigma)\). If \(A\) is totally \(\sigma \)-Noetherian, then the polynomial ring \(A[X]\) is totally \(\sigma'\)-Noetherian, where \(\sigma'\) is the induced torsion theory on \(A[X]\) (i.e. \(L(\sigma')=\{ I[X]\ |\ I\in L(\sigma) \}\)).
Reviewer: Tiberiu Dumitrescu (Bucureşti)Unique decompositions into \(w\)-ideals for strong Mori domainshttps://zbmath.org/1528.130172024-03-13T18:33:02.981707Z"Ay Saylam, Başak"https://zbmath.org/authors/?q=ai:ay-saylam.basak"Gürbüz, Ezgi"https://zbmath.org/authors/?q=ai:gurbuz.ezgi"Hamdi, Haleh"https://zbmath.org/authors/?q=ai:hamdi.halehA commutative ring \(R\) has the unique decomposition into ideals (UDI) property if, for any \(R\)-module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In [\textit{P. Goeters} and \textit{B. Olberding}, J. Pure Appl. Algebra 165, No. 2, 169--182 (2001; Zbl 1094.13526)], the UDI property was characterized for Noetherian integral domains. The aim of this paper is to study the \(w\)-version of the UDI property for strong Mori domains, i.e. domains satisfying the ascending chain condition on \(w\)-ideals. Note that the \(w\)-PID in the paper is nothing but the UFD (see [\textit{H. Kim} and \textit{F. Wang}, Bull. Korean Math. Soc. 50, No. 2, 475--483 (2013; Zbl 1282.13008), p. 477]).
Reviewer: Hwankoo Kim (Asan)A characterization of almost Dedekind domains in view of strong representabilityhttps://zbmath.org/1528.130182024-03-13T18:33:02.981707Z"Pakyari, Neda"https://zbmath.org/authors/?q=ai:pakyari.neda"Rostami, Esmaeil"https://zbmath.org/authors/?q=ai:rostami.esmaeil"Nekooei, Reza"https://zbmath.org/authors/?q=ai:nekooei.rezaIn the previous work [\textit{N. Pakyari} et al., Commun. Algebra 48, No. 9, 3891--3903 (2020; Zbl 1446.13007)] of the authors of this paper, it was shown that if \(R\) is an almost Dedekind domain, then every representable \(R\)-module is strongly representable. The present paper aims to prove the converse, i.e., for an integral domain \(R\), \(R\) is an almost Dedekind domain if and only if \(R\) is one-dimensional and every representable \(R\)-module is strongly representable.
Reviewer: Hwankoo Kim (Asan)Nil\({}_*\)-Artinian ringshttps://zbmath.org/1528.130192024-03-13T18:33:02.981707Z"Zhang, Xiaolei"https://zbmath.org/authors/?q=ai:zhang.xiaolei"Qi, Wei"https://zbmath.org/authors/?q=ai:qi.weiAn ideal \(I\) of a commutative unitary ring \(R \) is said to be a nil ideal provided that any element in \(I\) is nilpotent. The main motivation of this paper is to introduce and study Nil\(_\ast\)-Artinian rings, where a ring \(R\) is called Nil\(_\ast\)-Artinian if any descending chain of nil ideals stabilizes.
The authors study Nil\(_\ast\)-Artinian property in terms of quotients, localizations, polynomial ring extensions and Nagata idealizations, and then they investigate the transfer of Nil\(_\ast\)-Artinian property to amalgamated algebras (after D'Anna-Finocchiaro-Fontana). Besides, some examples are given to distinguish Nil\(_\ast\)-Artinian rings, Nil\(_\ast\)-Noetherian rings (after Zhang) and Nil\(_\ast\)-coherent rings (after Xiang).
Reviewer: Marco Fontana (Roma)\(S\)-prime ideals in principal domainhttps://zbmath.org/1528.130202024-03-13T18:33:02.981707Z"Aqalmoun, Mohamed"https://zbmath.org/authors/?q=ai:aqalmoun.mohamedSummary: Let \(R\) be a commutative ring and \(S\) be a multiplicative subset of \(R\). The \(S\)-prime ideal is a generalization of the concept of prime ideal. In this paper, we completely determine all \(S\)-prime and \(S\)-maximal ideals of a principal domain. It is shown that the intersection of any descending chain of \(S\)-prime ideals in a principal domain is an \(S\)-prime ideal, also the \(S\)-radical is investigated.Cotangent bundles of minuscule \(G/P\)https://zbmath.org/1528.130212024-03-13T18:33:02.981707Z"Fries, Marcus"https://zbmath.org/authors/?q=ai:fries.marcusSummary: We extend the results of the standard monomial theory to the cotangent variety of the Grassmannian, \(T^\ast\,\text{Grass}(k,n)\). We exhibit a standard basis in terms of triples of tableaux. We are also able to show that \(T^\ast\,\text{Grass}(k,n)\) is arithmetically Cohen-Macaulay and normal in all characteristics. Results on Cohen-Macaulayness and normality are shown for other algebraic groups and cominuscule parabolic subgroups, but only in characteristic zero.Combinatorics of \(F\)-polynomialshttps://zbmath.org/1528.130222024-03-13T18:33:02.981707Z"Fei, Jiarui"https://zbmath.org/authors/?q=ai:fei.jiaruiSummary: We use the stabilization functors to study the combinatorial aspects of the \(F\)-polynomial of a representation of any finite-dimensional basic algebra. We characterize the vertices of their Newton polytopes. We give an explicit formula for the \(F\)-polynomial restricting to any face of its Newton polytope. For acyclic quivers, we give a complete description of all facets of the Newton polytope when the representation is general. We also prove that the support of the \(F\)-polynomial is saturated for any rigid representation. We provide many examples and counterexamples and pose several conjectures.Associahedra for finite-type cluster algebras and minimal relations between \(g\)-vectorshttps://zbmath.org/1528.130232024-03-13T18:33:02.981707Z"Padrol, Arnau"https://zbmath.org/authors/?q=ai:padrol.arnau"Palu, Yann"https://zbmath.org/authors/?q=ai:palu.yann"Pilaud, Vincent"https://zbmath.org/authors/?q=ai:pilaud.vincent"Plamondon, Pierre-Guy"https://zbmath.org/authors/?q=ai:plamondon.pierre-guyThe authors prove that the mesh mutations are the minimal relations among the g-vectors with respect to any initial seed in any finite-type cluster algebra and used this algebraic result to derive geometric properties of the g-vector fan: They show that the space of all its polytopal realizations is a simplicial cone, and then observe that this property implies that all its realizations can be described as the intersection of a high-dimensional positive orthant with well-chosen affine spaces.
Moreover, the authors use a similar approach to study the space of polytopal realizations of the g-vector fans of another generalization of the associahedron: nonkissing complexes (also known as support \(\tau\)-tilting complexes) of gentle algebras. Further, they prove that the space of realizations of the nonkissing fan is simplicial when the gentle bound quiver is brick and 2-acyclic, and described in this case its facetdefining inequalities in terms of mesh mutations. Furthermore, the authors prove algebraic results on 2-Calabi-Yau triangulated categories, and on extriangulated categories that are of independent interest.
Reviewer: Udhayakumar Ramalingam (Vellore)A Nobile-like theorem for jet schemes of hypersurfaceshttps://zbmath.org/1528.140182024-03-13T18:33:02.981707Z"Barajas, Paul"https://zbmath.org/authors/?q=ai:barajas.paul"Duarte, Daniel"https://zbmath.org/authors/?q=ai:duarte.daniel-c-sSummary: We prove that the blowup of the jet scheme of a singular hypersurface along a certain jet-related module is not an isomorphism. In conjunction with recent developments in the theory of Nash blowups, our result holds over fields of arbitrary characteristic. Our approach is based on explicit presentations given by a higher-order Jacobian matrix combined with a certain jet-related matrix.On the local étale fundamental group of KLT threefold singularities. With an appendix by János Kollárhttps://zbmath.org/1528.140192024-03-13T18:33:02.981707Z"Carvajal-Rojas, Javier"https://zbmath.org/authors/?q=ai:carvajal-rojas.javier"Stäbler, Axel"https://zbmath.org/authors/?q=ai:stabler.axelThe article explores the finiteness of local fundamental groups of klt singularities over algebraically closed fields of positive characteristic. Over the complex numbers, it is established that the local fundamental group of a klt singularity is finite [\textit{C. Xu}, Compos. Math. 150, No. 3, 409--414 (2014; Zbl 1291.14057)] and [\textit{L. Braun}, Invent. Math. 226, No. 3, 845--896 (2021; Zbl 1479.14029)]. It is conjectured that the same holds true for the local étale fundamental group in characteristic \(p>0\).
This article prove the conjecture for klt 3-fold singularities in characteristic \(p>5\). The proof relies on recent advancements in the MMP (existence of plt blow-ups) and techniques from \(F\)-splitting theory.
Other noteworthy results in the article include the extension of finite unipotent torsors from a large open set on klt 3-folds and the finiteness of prime-to-\(p\) torsion of local class groups of klt 3-folds with characteristic \(p>5\). The latter result, without the characteristic restriction, is demonstrated in Appendix B by J. Kollar.
Reviewer: Fabio Bernasconi (Neuchâtel)Corrigendum to: ``The variety of polar simplices''https://zbmath.org/1528.140542024-03-13T18:33:02.981707Z"Ranestad, Kristian"https://zbmath.org/authors/?q=ai:ranestad.kristian"Schreyer, Frank-Olaf"https://zbmath.org/authors/?q=ai:schreyer.frank-olafSummary: We point out an important error in our paper [ibid. 18, 469--505 (2013; Zbl 1281.14035)] and provide the necessary corrections.Existence of the \(det^{S^2}\) maphttps://zbmath.org/1528.150052024-03-13T18:33:02.981707Z"Staic, Mihai D."https://zbmath.org/authors/?q=ai:staic.mihai-dLet \(V_d\) be a \(d\)-dimensional vector space and let \(V_d^{\otimes m}\) denote the \(m\)-th tensor power of \(V_d\). This paper is concerned with the existence (up to a scalar) of a nontrivial linear transformation \(\det^{S^2}: V_d^{\otimes d(2d-1)} \to k\), where \(k\) is a field, satisfying the property that
\[
{\det}^{S^2}(\otimes_{1 \leq i < j \leq 2d} (v_{i,j})) = 0
\]
whenever there exist \(1 \leq x < y < z \leq 2d\) such that \(v_{x,y} = v_{x,z} = v_{y,z}\). Notice that the usual determinant \(\det\) has the property \(\det(\otimes_{1 \leq i \leq d}(v_i)) = 0\) if there exist \(1 \leq x < y \leq d\) such that \(v_x = v_y\).
Theorem 3.7 in the paper proves its existence, but the author leaves the uniqueness of \({\det}^{S^2}\) as an open question. A geometric interpretation of the condition \({\det}^{S^2}\) is given in terms of dependence relations on the elements \(v_{s,k}\), much like how the usual \(\det\) function detects linear dependence. A specific application is also discussed: the \({\det}^{S^2}\) map can detect if a certain partition (see Definition 2.2) of the complete graph \(K_{2d}\) has cycles. At the end of the paper, the author wonders if the construction of \({\det}^{S^2}\) is related to the resultant of two polynomials (the determinant of their Sylvester matrix).
Reviewer: Hayden Julius (Niagara Falls)Zavadskij modules over cluster-tilted algebras of type \(\mathbb{A}\)https://zbmath.org/1528.160132024-03-13T18:33:02.981707Z"Moreno Cañadas, Agustín"https://zbmath.org/authors/?q=ai:moreno-canadas.agustin"Serna, Robinson-Julian"https://zbmath.org/authors/?q=ai:serna.robinson-julian"Marín Gaviria, Isaías David"https://zbmath.org/authors/?q=ai:marin-gaviria.isaias-davidArising in the context of differentiation algorithms in poset representation theory and the study of general orders, \textit{W. Rump} [J. Pure Appl. Algebra 153, No. 2, 171--190 (2000; Zbl 0964.16016)] introduced Zavadskij modules. Over a finite dimensional \(k\)-algebra, indecomposable Zavadskij modules are characterised by being tame uniserial.
In the present paper, given any finite dimensional \(k\)-algebra \(A\), the authors prove that the indecomposable Zavadskij \(A\)-modules are precisely those uniserial \(A\)-modules with a mast of multiplicity one in each vertex (in the quiver \(Q\) associated to \(A\)) (Theorem 7). In case \(A\) is hereditary, the latter condition can be dropped, while for tree quivers \(Q\), there is a bijection between the set of paths of \(Q\) and the indecomposable Zavadskij modules (Corollary 8). The authors go on to give a criterion for a direct sum of indecomposable Zavadskij modules to be Zavadskij (Theorem 9).
Further, they study the number of indecomposable Zavadskij modules over cluster-tilted algebras of type \(\mathbb{A}_n\), using the geometric model introduced by \textit{P. Caldero} et al. [Trans. Am. Math. Soc. 358, No. 3, 1347--1364 (2006; Zbl 1137.16020)]. The formula given relates the number of indecomposable Zavadskij modules to a sum of triangular numbers associated to the fans in the triangulation giving the cluster-tilted algebra (Theorem 14). They note that the formula gives the dimension of the cluster-tilted algebra. In the case of a Dynkin algebra of type \(\mathbb{A}_n\), the number can be calculated more explicitly (Corollaries 16 and 17). As an application and following \textit{S. Nowak} and \textit{D. Simson} [Commun. Algebra 30, No. 1, 455--476 (2002; Zbl 1005.16037)], the authors give categorifications of integer sequences (in the sense of \textit{P. Fahr} and \textit{C. M. Ringel} [J. Integer Seq. 15, No. 2, Article 12.2.1, 12 p. (2012; Zbl 1291.11037)]) of the sequences A000217, A005563, A002370, and A152948 encoded in Sloane's On-Line Encyclopedia of Integer Sequences.
The paper is divided into six sections. Following an introduction, they recall important aspects on modules over finite-dimensional \(k\)-algebras, cluster-tilted algebras of type \(\mathbb{A}\), Zavadskij modules and Rump's characterisation thereof. Section~3 contains Theorems 7 and 9, Corollary 8 and their proofs along with examples. Section 4 establishes the formulas, again accompanied by an example. Section 5 then connects the Zavadskij modules with integer sequences while concluding remarks constitute Section 6.
Reviewer: Sebastian Eckert (Bielefeld)SITT-ring properties in bi-amalgamated rings along idealshttps://zbmath.org/1528.160182024-03-13T18:33:02.981707Z"Aruldoss, A."https://zbmath.org/authors/?q=ai:aruldoss.a"Selvaraj, Chelliah"https://zbmath.org/authors/?q=ai:selvaraj.chelliahSummary: Let \(f:A\to B\) and \(g:A\to C\) be two ring homomorphisms and let \(J\) and \(J'\) be two ideals of \(B\) and \(C\), respectively, such that \(f^{-1}(J)=g^{-1}(J')\). In this paper, we give a characterization for the amalgamation of \(A\) with \(B\) along \(J\) with respect to \(f\) (denoted by \(A\bowtie^fJ)\) to be a SITT-ring and also we give a characterization for the bi-amalgamation of \(A\) with \((B,C)\) along \((J,J')\) with respect to \((f,g)\) (denoted by \(A\bowtie^{f,g}(J,J'))\) to be a SITT-ring. We also give some characterizations for strong weakly SIT-rings.Umbral calculus in Ore extensionshttps://zbmath.org/1528.160262024-03-13T18:33:02.981707Z"Benouaret, Chahrazed"https://zbmath.org/authors/?q=ai:benouaret.chahrazed"Salinier, Alain"https://zbmath.org/authors/?q=ai:salinier.alainSummary: The aim of the paper is to show the existence of some ingredients for an umbral calculus on some Ore extensions, in a manner analogous to Rota's classical umbral calculus which deals with a univariate polynomial ring on a field of characteristic zero. For that, we introduce the notion of a quasi-derivation in order to specify Ore extensions on which building up this umbral calculus is possible. This allows in particular to define an action of the Ore extension on tensor products of modules. We develop also a Pincherle calculus for operators and we define a coalgebra structure on the Ore extension.Rings with \(x^n +x\) or \(x^n -x\) nilpotenthttps://zbmath.org/1528.160352024-03-13T18:33:02.981707Z"Abyzov, Adel N."https://zbmath.org/authors/?q=ai:abyzov.adel-nailevich"Danchev, Peter V."https://zbmath.org/authors/?q=ai:danchev.peter-vassilev"Tapkin, Daniel T."https://zbmath.org/authors/?q=ai:tapkin.daniel-tThe main aim of the paper is to describe the rings \(R\) such that all \(r\in R\), at least one of the elements \(r^n-r\) or \(r^n + r\) is nilpotent, where \(n\geq 2\) is a fixed integer. This is realized in Theorem 2.15, where many details about these rings are presented. The paper also contains interesting results about matrix rings over finite fields.
Reviewer: Simion Sorin Breaz (Cluj-Napoca)Separability properties of nilpotent \(\mathbb{Q} [x]\)-powered groupshttps://zbmath.org/1528.200512024-03-13T18:33:02.981707Z"Majewicz, Stephen"https://zbmath.org/authors/?q=ai:majewicz.stephen"Zyman, Marcos"https://zbmath.org/authors/?q=ai:zyman.marcosSummary: In this paper, we study conjugacy and subgroup separability properties in the class of nilpotent \(\mathbb{Q} [x]\)-powered groups. Many of the techniques used to study these properties in the context of ordinary nilpotent groups carry over naturally to this more general class. Among other results, we offer a generalization of a theorem due to G. Baumslag. The generalized version states that if \(G\) is a finitely \(\mathbb{Q} [x]\)-generated \(\mathbb{Q} [x]\)-torsion-free nilpotent \(\mathbb{Q} [x]\)-powered group and \(H\) is a \(\mathbb{Q} [x]\)-isolated subgroup of \(G\), then for any prime \(\pi \in \mathbb{Q} [x]\), \( \bigcap^\infty_{i=1} G \pi i H = H\).
For the entire collection see [Zbl 1435.20002].Even-primitive vectors in induced supermodules for general linear supergroups and in costandard supermodules for Schur superalgebrashttps://zbmath.org/1528.200842024-03-13T18:33:02.981707Z"Marko, František"https://zbmath.org/authors/?q=ai:marko.frantisekSummary: Let \(G=\mathrm{GL}(m|n)\) be the general linear supergroup over an algebraically closed field \(K\) of characteristic zero, and let \(G_{ev}=\mathrm{GL}(m)\times \mathrm{GL}(n)\) be its even subsupergroup. The induced supermodule \(H^0_G(\lambda )\), corresponding to a dominant weight \(\lambda\) of \(G\), can be represented as \(H^0_{G_{ev}}(\lambda )\otimes \Lambda (Y)\), where \(Y=V_m^*\otimes V_n\) is a tensor product of the dual of the natural \(\mathrm{GL} (m)\)-module \(V_m\) and the natural \(\mathrm{GL} (n)\)-module \(V_n\), and \(\Lambda (Y)\) is the exterior algebra of \(Y\). For a dominant weight \(\lambda\) of \(G\), we construct explicit \(G_{ev} \)-primitive vectors in \(H^0_G(\lambda )\). Related to this, we give explicit formulas for \(G_{ev} \)-primitive vectors of the supermodules \(H^0_{G_{ev}}(\lambda )\otimes \otimes^k Y\). Finally, we describe a basis of \(G_{ev} \)-primitive vectors in the largest polynomial subsupermodule \(\nabla (\lambda )\) of \(H^0_G(\lambda )\) (and therefore in the costandard supermodule of the corresponding Schur superalgebra \(S(m|n))\). This yields a description of a basis of \(G_{ev} \)-primitive vectors in arbitrary induced supermodule \(H^0_G(\lambda )\).Univariate ideal membership parameterized by rank, degree, and number of generatorshttps://zbmath.org/1528.681472024-03-13T18:33:02.981707Z"Arvind, V."https://zbmath.org/authors/?q=ai:arvind.vikraman"Chatterjee, Abhranil"https://zbmath.org/authors/?q=ai:chatterjee.abhranil"Datta, Rajit"https://zbmath.org/authors/?q=ai:datta.rajit"Mukhopadhyay, Partha"https://zbmath.org/authors/?q=ai:mukhopadhyay.partha.1Summary: Let \(F[X]\) be the polynomial ring over the variables \(X=\{x_1,x_2,\dots,x_n\}\). An ideal \(I= \langle p_1(x_1),\dots,p_n(x_n)\rangle\) generated by univariate polynomials \(\{p_i(x_i)\}_{i=1}^n\) is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results.
\begin{itemize}\item[--] Let \(f(X)\in \mathbb{F}[\ell_1,\dots,\ell_r]\) be a (low rank) polynomial given by an arithmetic circuit where \(\ell_i: 1\le i\le r \) are linear forms, and \(I= \langle p_1(x_1),\dots,p_n(x_n)\rangle\) be a univariate ideal. Given \(\vec{\alpha}\in\mathbb{F}^n\), the (unique) remainder \(f(X)\pmod I\) can be evaluated at \(\vec{\alpha}\) in deterministic time \(d^{O(r)}\cdot\operatorname{poly}(n)\), where \(d=\max\{\deg(f),\deg(p_ 1),\dots,\deg(p_n)\}\). This yields a randomized \(n^{O(r)}\) algorithm for minimum vertex cover in graphs with rank-\(r\) adjacency matrices. It also yields an \(n^{O(r)}\) algorithm for evaluating the permanent of a \(n\times n\) matrix of rank \(r\), over any field \(\mathbb{F}\). Over \(\mathbb{Q}\), an algorithm of similar run time for low rank permanent is due to \textit{A. I. Barvinok} [Math. Oper. Res. 21, No. 1, 65--84 (1996; Zbl 0846.90115)] via a different technique.
\item[--] Let \(f(X)\in\mathbb{F}[X]\) be given by an arithmetic circuit of degree \(k\) (\(k\) treated as fixed parameter) and \(I=\langle p_1(x_1),\dots,p_n(x_n)\rangle\). We show that in the special case when \(I=\langle x_1^{e_1},\dots,x_n^{e_n}\rangle\), we obtain a randomized \(O^*(4.08^k)\) algorithm that uses \(\operatorname{poly}(n,k)\) space.
\item[--] Given \(f(X)\in\mathbb{F}[X]\) by an arithmetic circuit and \(I=\langle p_1(x_1),\dots,p_k(x_k)\rangle\), membership testing is W[1]-hard, parameterized by \(k\). The problem is MINI[1]-hard in the special case when \(I=\langle x_1^{e_1},\dots,x_k^{e_k}\rangle\).\end{itemize}
For the entire collection see [Zbl 1407.68032].Polarized gravitational waves in the parity violating scalar-nonmetricity theoryhttps://zbmath.org/1528.830112024-03-13T18:33:02.981707Z"Chen, Zheng"https://zbmath.org/authors/?q=ai:chen.zheng|chen.zheng.1"Yu, Yang"https://zbmath.org/authors/?q=ai:yu.yang"Gao, Xian"https://zbmath.org/authors/?q=ai:gao.xian(no abstract)On polynomial functions Modulo \(p^e\) and faster bootstrapping for homomorphic encryptionhttps://zbmath.org/1528.940512024-03-13T18:33:02.981707Z"Geelen, Robin"https://zbmath.org/authors/?q=ai:geelen.robin"Iliashenko, Ilia"https://zbmath.org/authors/?q=ai:iliashenko.ilia"Kang, Jiayi"https://zbmath.org/authors/?q=ai:kang.jiayi"Vercauteren, Frederik"https://zbmath.org/authors/?q=ai:vercauteren.frederikSummary: In this paper, we perform a systematic study of functions \(f: \mathbb{Z}_{p^e} \rightarrow \mathbb{Z}_{p^e}\) and categorize those functions that can be represented by a polynomial with integer coefficients. More specifically, we cover the following properties: necessary and sufficient conditions for the existence of an integer polynomial representation; computation of such a representation; and the complete set of equivalent polynomials that represent a given function.
As an application, we use the newly developed theory to speed up bootstrapping for the BGV and BFV homomorphic encryption schemes. The crucial ingredient underlying our improvements is the existence of null polynomials, i.e. non-zero polynomials that evaluate to zero in every point. We exploit the rich algebraic structure of these null polynomials to find better representations of the digit extraction function, which is the main bottleneck in bootstrapping. As such, we obtain sparse polynomials that have 50\% fewer coefficients than the original ones. In addition, we propose a new method to decompose digit extraction as a series of polynomial evaluations. This lowers the time complexity from \(\mathcal{O}(\sqrt{pe})\) to \(\mathcal{O}(\sqrt{p}\sqrt[4]{e})\) for digit extraction modulo \(p^e\), at the cost of a slight increase in multiplicative depth. Overall, our implementation in \texttt{HElib} shows a significant speedup of a factor up to 2.6 over the state-of-the-art.
For the entire collection see [Zbl 1525.94003].