Recent zbMATH articles in MSC 12https://zbmath.org/atom/cc/122024-04-02T17:33:48.828767ZWerkzeugAbstract algebra. An inquiry-based approachhttps://zbmath.org/1529.000012024-04-02T17:33:48.828767Z"Hodge, Jonathan K."https://zbmath.org/authors/?q=ai:hodge.jonathan-k"Schlicker, Steven"https://zbmath.org/authors/?q=ai:schlicker.steven-j"Sundstrom, Ted"https://zbmath.org/authors/?q=ai:sundstrom.tedFrom the publisher's description: Changes in the Second Edition
\begin{itemize}
\item Streamlining of introductory material with a quicker transition to the material on rings and groups.
\item New investigations on extensions of fields and Galois theory.
\item New exercises added and some sections reworked for clarity.
\item More online Special Topics investigations and additional Appendices, including new appendices on other methods of proof and complex roots of unity.
\end{itemize}
See the review of the first edition in [Zbl 1295.00007].Notes on the decidability of addition and the Frobenius map for polynomials and rational functionshttps://zbmath.org/1529.031192024-04-02T17:33:48.828767Z"Chompitaki, Dimitra"https://zbmath.org/authors/?q=ai:chompitaki.dimitra"Kamarianakis, Manos"https://zbmath.org/authors/?q=ai:kamarianakis.manos"Pheidas, Thanases"https://zbmath.org/authors/?q=ai:pheidas.thanasesSummary: Let \(p\) be a prime number, \(\mathbb{F}_p\) a finite field with \(p\) elements, \(F\) an algebraic extension of \(\mathbb{F}_p\) and \(z\) a variable. We consider the structure of addition and the Frobenius map (i.e., \(x\mapsto x^p)\) in the polynomial rings \(F[z]\) and in fields \(F(z)\) of rational functions. We prove that any question about \(F[z]\) in the structure of addition and Frobenius map may be effectively reduced to questions about the similar structure of the field \(F\). Furthermore, we provide an example which shows that a fact which is true for addition and the Frobenius map in the polynomial rings \(F[z]\) fails to be true in \(F(z)\). As a consequence, certain methods used to prove model completeness for polynomials do not suffice to prove model completeness for similar structures for fields of rational functions \(F(z)\), a problem that remains open even for \(F=\mathbb{F}_p\).A weak version of the strong exponential closurehttps://zbmath.org/1529.032092024-04-02T17:33:48.828767Z"D'Aquino, Paola"https://zbmath.org/authors/?q=ai:daquino.paola"Fornasiero, Antongiulio"https://zbmath.org/authors/?q=ai:fornasiero.antongiulio"Terzo, G."https://zbmath.org/authors/?q=ai:terzo.giuseppinaSummary: Assuming Schanuel's Conjecture we prove that for any irreducible variety \(V \subseteq \mathbb{C}^n \times (\mathbb{C}^\ast)^n\) over \(\mathbb{Q}^{\mathrm{alg}} \), of dimension \(n\), and with dominant projections on both the first \(n\) coordinates and the last \(n\) coordinates, there exists a generic point \(\left( {\overline{a} ,{e^{\overline{a} }}} \right) \in V\). We obtain in this way many instances of the Strong Exponential Closure axiom introduced by \textit{B. Zilber} [Ann. Pure Appl. Logic 132, No. 1, 67--95 (2005; Zbl 1076.03024)].Hensel minimality. Ihttps://zbmath.org/1529.032112024-04-02T17:33:48.828767Z"Cluckers, Raf"https://zbmath.org/authors/?q=ai:cluckers.raf"Halupczok, Immanuel"https://zbmath.org/authors/?q=ai:halupczok.immanuel"Rideau-Kikuchi, Silvain"https://zbmath.org/authors/?q=ai:rideau-kikuchi.silvainSummary: We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of t-stratifications in Hensel minimal structures and Taylor approximation results that are key to non-Archimedean versions of Pila-Wilkie point counting, Yomdin's parameterization results and motivic integration. In this first paper, we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.Euclidean polynomials for certain arithmetic progressions and the multiplicative group of \(\mathbb{F}_{p^2}\)https://zbmath.org/1529.110062024-04-02T17:33:48.828767Z"Berktav, Kadri İlker"https://zbmath.org/authors/?q=ai:berktav.kadri-ilker"Özbudak, Ferruh"https://zbmath.org/authors/?q=ai:ozbudak.ferruhIn the paper under review the author proves the following main result on Euclidean polynomials.
\textbf{Theorem.} Let \(n\) be a positive integer. Let \(k=8n+4\), \(\ell_1=1\), \(\ell_2=k-1\). Let \(c_0=1\) and \(c_1,\ldots,c_n\) are defined recursively by
\[
c_i=1+\sum_{j=1}^i\left[(-1)^{j+1}\binom{2(n-i+j)}{j}c_{i-j}\right], i=1,2,\ldots,n.
\]
Then \(f(x)=\sum_{i=0}^n (-1)^i c_i x^{2i}\in \mathbb{Z}[x]\setminus \mathbb{Z}\) is an Euclidean polynomial for both of the arithmetic progressions \(\ell_1\mod k\) and \(\ell_2\mod k\).
Reviewer: Jitender Singh (Amritsar)Constructing unramified extensions over quadratic fieldshttps://zbmath.org/1529.111102024-04-02T17:33:48.828767Z"Aoki, Misato"https://zbmath.org/authors/?q=ai:aoki.misato"Kida, Masanari"https://zbmath.org/authors/?q=ai:kida.masanariAuthors' abstract: Using some regular polynomials with prescribed Galois groups, we construct explicit infinite families of unramified extensions over quadratic fields by specializations satisfying certain congruence conditions.
Reviewer: Bouchaïb Sodaïgui (Valenciennes)The absolute Galois group of a \(p\)-adic fieldhttps://zbmath.org/1529.111152024-04-02T17:33:48.828767Z"Jarden, Moshe"https://zbmath.org/authors/?q=ai:jarden.moshe"Shusterman, Mark"https://zbmath.org/authors/?q=ai:shusterman.markIn this paper, the authors aim to compute the number of generators and the number of ``relations'' needed to characterize the ``presentation'' the absolute Galois group of a given finite extension of \(\mathbb{Q}_p\).
For \(p\) a prime number and \(K\) a finite extension of \(\mathbb{Q}_p\),
\textit{U. Jannsen} [Invent. Math. 70, 53--69 (1982; Zbl 0534.12009)] proved that the absolute Galois group Gal(\(K\)) of \(K\) is finitely generated, as a profinite group. The authors' goal in this article is to prove that Gal(\(K\)) is even ``finitely presented'' and to compute the number of ``generators'' and ``relations'' needed for the ``presentation''. The definition used is the following: an arbitrary profinite group \(G\) is finitely presented if there exist a short exact sequence \(1\to N \to\widehat{F}_e\to G\to 1\) for some positive integer \(e\) and elements \(y_1,\dots, y_d\in \widehat{F}_e\) such that \(N\) is the smallest closed normal subgroup of the free profinite group \(\widehat{F}_e\) on \(e\) generators that contains \(y_1,\dots,y_d\). Theses elements \(y_1,\dots, y_d\) are said to be relations of \(G\).
The other sections of the article present theorems proving the number of relations needed by using Hensel's Lemma and other results.
Reviewer: Mouad Moutaoukil (Fès)A note on depth-b normal elementshttps://zbmath.org/1529.111252024-04-02T17:33:48.828767Z"Sheekey, John"https://zbmath.org/authors/?q=ai:sheekey.john"Thomson, David"https://zbmath.org/authors/?q=ai:thomson.david-georgeGiven a power \(q\) of a prime \(p\), a finite field \(\mathbb F_q\) with \(q\) elements and a degree \(n\) extension field \(\mathbb F_{q^n}\), an element \(\beta \in \mathbb F_{q^n}\) is said to be a \emph{normal element} over \(\mathbb F_q\) if \((\beta,\beta^q,\ldots,\beta^{q^{n-1}})\) is an \(\mathbb F_q\)-basis of \(\mathbb F_{q^n}\). The focus of this paper is on the following generalization of normal elements. For a fixed \(\alpha\in \mathbb F_{q^n}\) and a positive integer \(b\le p\), an element \(\beta\in \mathbb F_{q^n}\) has \emph{normal \(\alpha\)-depth \(b\)} if \(\beta, \beta-\alpha,\ldots,\beta-(b-1)\alpha\) are all normal elements over \(\mathbb F_q\). More precisely, the authors investigate when an element \(\beta\) and all its conjugates \(\beta^q,\beta^{q^2},\ldots,\beta^{q^{n-1}}\) have simultaneously normal \(\alpha\)-depth \(b\). Such elements are called \emph{\((\alpha,b)\)-sociable} and in this paper the authors give upper bounds and exact computations of the number of \((\alpha,b)\)-sociable elements under several different assumptions.
For the entire collection see [Zbl 1455.11009].
Reviewer: Alessandro Neri (Ghent)Polynomial constructions of Chudnovsky-type algorithms for multiplication in finite fields with linear bilinear complexityhttps://zbmath.org/1529.111282024-04-02T17:33:48.828767Z"Ballet, Stéphane"https://zbmath.org/authors/?q=ai:ballet.stephane"Bonnecaze, Alexis"https://zbmath.org/authors/?q=ai:bonnecaze.alexis"Pacifico, Bastien"https://zbmath.org/authors/?q=ai:pacifico.bastienSummary: Chudnovsky-type algorithms for the multiplication in finite extensions of finite fields are well-known for having a good bilinear complexity, both asymptotically and at finite distance. More precisely, for every degree \(n\) of the extension, the existence of a family of algorithms with linear bilinear complexity in \(n\) has been proved using the original method applied to an explicit recursive tower of function fields. However, there is currently no method to build these algorithms in polynomial time. Nevertheless, one can construct in polynomial time a Chudnovsky-type algorithm over the projective line for the multiplication in any extension degree, with a quasi-linear bilinear complexity. In this paper, we prove that we can obtain algorithms both constructible in polynomial time and having a linear bilinear complexity by mixing up these two strategies.
For the entire collection see [Zbl 1516.11002].Solution to algebraic equations of degree 4 and the fundamental theorem of algebra by Carl Friedrich Gausshttps://zbmath.org/1529.120012024-04-02T17:33:48.828767Z"Südland, Norbert"https://zbmath.org/authors/?q=ai:sudland.norbert"Volkmann, Jörg"https://zbmath.org/authors/?q=ai:volkmann.jorg"Kumar, Dinesh"https://zbmath.org/authors/?q=ai:kumar.dineshSummary: Since \textit{Geronimo Cardano}, algebraic equations of degree 4 have been solved analytically. Frequently, the solution algorithm is given in its entirety. We discovered two algorithms that lead to the same \textit{resolvente}, each with two solutions; therefore, six formal solutions appear to solve an algebraic equation of degree four. Given that a square was utilized to derive the solution in both instances, it is critical to verify each solution. This check reveals that the four Cardanic solutions are the only four solutions to an algebraic equation of degree four. This demonstrates that \textit{Carl Friedrich Gauss}' (1799) \textit{fundamental theorem of algebra} is not simple, despite the fact that it is a fundamental theorem. This seems to be a novel insight.On products of polynomials. IIhttps://zbmath.org/1529.120022024-04-02T17:33:48.828767Z"Masser, David"https://zbmath.org/authors/?q=ai:masser.david-william"Wise, Andrew"https://zbmath.org/authors/?q=ai:wise.andrewFor \(P\in{\mathbb{C}}[X]\), let \(H(P)\) denote the usual height of \(P\), viz. the maximum modulus of the coefficients of \(P\). For \(n_1\) and \(n_2\) non negative integers, let \(\mu(n_1,n_2)\) denote the infimum of \(H(P_1P_2)/H(P_1)H(P_2)\) over all nonzero polynomials \(P_1\) and \(P_2\) of degrees \(n_1\) and \(n_2\) respectively. Lower bounds for \(\mu(n_1,n_2)\) are due to \textit{A. Schinzel} [Polynomials with special regard to reducibility. Cambridge: Cambridge University Press (2000; Zbl 0956.12001)], \textit{D. W. Boyd} [Mathematika 39, No. 2, 341--349 (1992; Zbl 0758.30003)] as well as the first author and \textit{J. Wolbert} [Proc. Am. Math. Soc. 117, No. 3, 593--599 (1993; Zbl 0769.12002)] for \(\min\{n_1,n_2\}=1\). The main goal of this paper is to settle a question posed by the first author in his book [\textit{D. Masser}, Auxiliary polynomials in number theory. Cambridge: Cambridge University Press (2016; Zbl 1354.11002)] (exercise 14.40 p. 87) by proving that \(\mu(2,2)=(\sqrt{13}-3)/2\), which is attained with the polynomial \(P_1P_2=X^4+X^3-X^2-X+1\). \par
The paper ends with some remarks, including the question whether for each \(n_1\) and \(n_2\), the minimum \(\mu(n_1,n_2)\) is reached with a product \(P_1P_2\) which is a Littlewood polynomial, namely a polynomial with coefficients \(+1\) or \(-1\).
Reviewer: Michel Waldschmidt (Paris)Hilbert irreducibility, the Malle conjecture and the Grunwald problemhttps://zbmath.org/1529.120032024-04-02T17:33:48.828767Z"Motte, François"https://zbmath.org/authors/?q=ai:motte.francoisThis article has a nice 7-page introduction that I will try to summarize here. It deals with two interesting topics in Inverse Galois Theory.
The first topic is the Malle Conjecture, which establishes an estimate of the number \(N(K,G,y)\) of different Galois extensions \(L/K\) of a fixed number field \(K\) with fixed Galois group \(G\) and bounded discriminant \(N_{K/\mathbb{Q}}(d_{L/K})\leq y\). More precisely, it states that there exists a constant \(c_1\) and, for every \(\varepsilon >0\), there exist constants \(c_2,y_0\) such that
\[
c_1y^{a(G)}\leq N(K,G,y)< c_2y^{a(G)+\varepsilon},\quad\text{for all }y\geq y_0,
\]
where \(a(G):=(|G|(1-1/\ell))^{-1}\) and \(\ell\) is the smallest prime divisor of \(|G|\).
The second topic is the Grunwald Problem, which concerns the realization of finitely many fixed local Galois extensions \(L^{\mathfrak{p}}/K_{\mathfrak{p}}\), with Galois group embedding into a fixed group \(G\), by a single global \(G\)-extension \(L/K\). Here \(K_{\mathfrak{p}}\) denotes of course the completion of \(K\) at the prime \(\mathfrak{p}\).
Both topics are related in the sense that a control on the amount of solutions to a given Grunwald Problem immediately gives lower bounds on the number \(N(K,G,y)\) from the Malle Conjecture.
The main result of this article (Theorem AB) is a generalization to an arbitrary number field \(K\) of a result by \textit{P. Dèbes} [Isr. J. Math. 218, 101--131 (2017; Zbl 1425.11179)], which deals with the case \(K=\mathbb{Q}\). They both count \(G\)-extensions obtained by \textit{specialization} from a regular \(G\)-extension \(F/K(T)\), giving lower bounds for such a number. More precisely, given a regular \(G\)-extension \(F/K(T)\), the result establishes a lower bound of the form \(y^{(1-1/|G|)/\delta}\) for the number \(N(F/K(T),y,\mathcal{F}_y)\) of \(G\)-extensions obtained from \(F/K(T)\), where \(y\) is big enough and bounds the discriminant as before, and \(\mathcal{F}_y\) denotes fixed local conditions on a finite set of primes depending on \(y\) and \(\delta\). This immediately yields a ``Malle type'' lower bound for \(G\) whenever it is a regular Galois group over \(K\) (Theorem A). Under the same hypothesis, it also yields a quantitative solution to some Grunwald problems (Theorem B), improving results from \textit{P. Dèbes} and \textit{N. Ghazi} [Ann. Inst. Fourier 62, No. 3, 989--1013 (2012; Zbl 1255.14022)], where they proved the existence of solutions without regard to their quantity.
\textit{P. Dèbes}' result in [Isr. J. Math. 218, 101--131 (2017; Zbl 1425.11179)] used an explicit version of Hilbert's Irreducibility Theorem due to \textit{Y. Walkowiak} [Acta Arith. 116, No. 4, 343--362 (2005; Zbl 1071.12002)]. Thus, a generalization as the one given here needs a generalization of Walkowiak's result. And this is the other main result of this article (Theorem C). Given an irreducible polynomial \(P(X_1,X_2)\in\mathcal O_K[X_1,X_2]\), monic on \(X_2\), it bounds by above the number \(N(P,B)\) of roots \((x_1,x_2)\in \mathcal O_K^2\) of \(P\) that have bounded height \(H(x_i)\leq B\) for some suitably defined height function \(H\). Along with some other tools, this allows to bound by below the number of specializations on one variable that give an irreducible polynomial on the other variable, which yields the intended application. The proof of such a statement relies on yet another generalization (Theorem 3.4) to arbitrary number fields of a result by \textit{D. R. Heath-Brown} [Ann. Math. (2) 155, No. 2, 553--598 (2002; Zbl 1039.11044)].
Reviewer: Giancarlo Lucchini Arteche (Santiago)Modular forms and some cases of the inverse Galois problemhttps://zbmath.org/1529.120042024-04-02T17:33:48.828767Z"Zywina, David"https://zbmath.org/authors/?q=ai:zywina.davidThis paper solves the inverse Galois problem over \(\mathbb Q\) for the groups \(\text{PSL}_2(\mathbb F_p)\) for all primes \(p\), and \(\text{PSL}_2(\mathbb F_{p^3})\) for all odd primes \(p \equiv \pm 2, \pm 3, \pm 4, \pm 6 \pmod{13}\), that is, it is shown that these groups arise as the Galois groups of finite Galois extensions of \(\mathbb Q\).
The method considers the Galois representations \[\bar \rho_{f,\mathfrak p} : G_{\mathbb Q} \rightarrow \text{GL}_2(\mathbb F_{\mathfrak p})\] attached to a specific classical modular form \(f\) of known level, weight, and character, where \(\mathfrak p\) is a prime of the coefficient field of \(f\) above \(p\) and \(\mathbb F_{\mathfrak p}\) is the residue field of \(\mathfrak p\). It is known if \(f\) does not have complex multiplication, then for \(p\) sufficiently large, the Galois representation \(\bar \rho_{f,\mathfrak p}\) has projective image \(\text{PSL}_2(\mathbb F_{\mathfrak p})\) or \(\text{PGL}_2(\mathbb F_{\mathfrak p})\) by work of Ribet (Theorem 1.1). A large part of the overall method involves developing strong criteria to ensure the projective image is known for all primes, or all primes of a certain form.
The case of \(\text{PSL}_2(\mathbb F_p)\) was settled in [\textit{D. Zywina}, Duke Math. J. 164, No. 12, 2253--2292 (2015; Zbl 1332.12007)] using the Galois representations arising from a specific elliptic surface. This paper demonstrates that the method of considering the Galois representations arising from modular forms is sufficiently strong to prove the case of \(\text{PSL}_2(\mathbb F_p)\) in a natural way, as well as other new cases.
Reviewer: Imin Chen (Burnaby)Number of zeros of exponential polynomials in zero residue characteristichttps://zbmath.org/1529.120052024-04-02T17:33:48.828767Z"Escassut, Alain"https://zbmath.org/authors/?q=ai:escassut.alainLet \({\mathbb{L}}\) be a field of zero characteristic, complete with respect to an absolute value \(|\cdot|\). Let \(\exp\) be the exponential function \(\exp(x)=\sum_{j=1}^\infty x^j/j!\). Let \(F(x)=\sum_{i=1}^k f_i(x) \exp(\zeta_i x)\) be an exponential polynomial, with \(k\ge 1\), \(\zeta_i\in {\mathbb{L}}\), \(f_i\in{\mathbb{L}}[x]\setminus\{0\}\). Upper bounds for the number of zeroes of \(F\) in a disk of \({\mathbb{L}}\) is a classical topic. When \({\mathbb{L}}={\mathbb{C}}\), it was investigated by \textit{S. Dancs} and \textit{P. Turán} [Publ. Math. 11, 257--265, 266--272 (1964; Zbl 0136.06201)], \textit{K. Mahler} [Acta Math. Acad. Sci. Hung. 18, 83--96 (1967; Zbl 0207.35602)], \textit{A. O. Gelfond} [Transcendental and algebraic numbers. New York: Dover Publications, Inc. (1960; Zbl 0090.26103)], \textit{R. Tijdeman} [Nederl. Akad. Wet., Proc., Ser. A 74, 1--7 (1971; Zbl 0211.09201)] and others, in connection with transcendental number theory. For the case where \({\mathbb{L}}\) is ultrametric with residue characteristic \(p>0\), see the papers by \textit{A. J. van der Poorten} [Nederl. Akad. Wet., Proc., Ser. A 79, 46--49 (1976; Zbl 0318.30006); J. Aust. Math. Soc., Ser. A 22, 12--26 (1976; Zbl 0348.12022)] and \textit{Ph. Robba} [in: Groupe d'Étude d'Anal. Ultramétr., 4e Année 1976/77, Exp. No. 9, 3 p. (1978; Zbl 0381.12010)].
Here the author considers the case where \({\mathbb{L}}\) is ultrametric with residue characteristic zero. Assuming \(|\zeta_i|<1\), he proves that the number of zeroes of \(F\) in the disk \(\{x\in {\mathbb{L}} \mid |x|<1\}\), taking multiplicities into account, is at most \({\operatorname{deg}}f_1+\dots+{\operatorname{deg}}f_k+k-1\), which is optimal.
Reviewer: Michel Waldschmidt (Paris)On cuts of the quotient field of a ring of formal power serieshttps://zbmath.org/1529.130182024-04-02T17:33:48.828767Z"Galanova, Natal'ya Yur'evna"https://zbmath.org/authors/?q=ai:galanova.natalya-yurevna(no abstract)Abstract algebra. Translated from the Chinesehttps://zbmath.org/1529.200012024-04-02T17:33:48.828767Z"Deng, Shaoqiang"https://zbmath.org/authors/?q=ai:deng.shaoqiang"Zhu, Fuhai"https://zbmath.org/authors/?q=ai:zhu.fuhaiPublisher's description: This book is translated from the Chinese version published by Science Press, Beijing, China, in 2017. It was written for the Chern class in mathematics of Nankai University and has been used as the textbook for the course Abstract Algebra for this class for more than five years. It has also been adapted in abstract algebra courses in several other distinguished universities across China.
The aim of this book is to introduce the fundamental theories of groups, rings, modules, and fields, and help readers set up a solid foundation for algebra theory. The topics of this book are carefully selected and clearly presented. This is an excellent mathematical exposition, well-suited as an advanced undergraduate textbook or for independent study. The book includes many new and concise proofs of classical theorems, along with plenty of basic as well as challenging exercises.Locally solvable and solvable-by-finite maximal subgroups of \(\mathrm{GL}_n (D)\)https://zbmath.org/1529.200492024-04-02T17:33:48.828767Z"Khanh, Huynh Viet"https://zbmath.org/authors/?q=ai:huynh-viet-khanh."Hai, Bui Xuan"https://zbmath.org/authors/?q=ai:bui-xuan-hai.Summary: This paper aims to study solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup of the general skew linear group \(\mathrm{GL}_n (D)\) over a division ring \(D\). It turns out that in the case where \(D\) is non-commutative, if such maximal subgroups exist, then either it is abelian or \([D\colon F]<\infty\). Also, if \(F\) is an infinite field and \(n\geq 5\), then every locally solvable maximal subgroup of a normal subgroup of \(\mathrm{GL}_n (F)\) is abelian.Skew bracoidshttps://zbmath.org/1529.201062024-04-02T17:33:48.828767Z"Martin-Lyons, Isabel"https://zbmath.org/authors/?q=ai:martin-lyons.isabel"Truman, Paul J."https://zbmath.org/authors/?q=ai:truman.paul-jSkew brace is an algebraic structure introduced in [\textit{L. Guarnieri} and \textit{L. Vendramin}, Math. Comput. 86, 2519--2534 (2017; Zbl 1371.16037)] as a tool to study the non-degenerate set-theoretic solutions to the Yang-Baxter equation. Recall that a (left) \textit{skew brace} is a triplet \((B,\star,\cdot)\) for which \((B,\star)\) and \((B,\cdot)\) are groups satisfying
\[
a\cdot (b\star c) = (a\cdot b)\star a^{-1} \star (a\cdot c)
\]
for all \(a,b,c\in B\), where \(a^{-1}\) denotes the inverse of \(a\) with respect to \(\star\). It is known that for any group \(N = (N,\star)\), there is a one-to-one correspondence between the binary operations \(\cdot\) on \(N\) such that \((N,\star,\cdot)\) is a skew brace and the regular subgroups lying in the holomorph \(\mathrm{Hol}(N)\) of \(N\).
On the other hand, by work of \textit{C. Greither} and \textit{B. Pareigis} [J. Algebra 106, 239--258 (1987; Zbl 0615.12026)] and \textit{N. P. Byott} [Commun. Algebra 24, 3217--3228 (1996; Zbl 0878.12001)], for any finite Galois extension \(L/K\) with Galois group \(G = (G,\cdot)\), there is a (not necessarily one-to-one) correspondence between Hopf-Galois structures on \(L/K\) and the regular subgroups isomorphic to \(G\) lying in \(\mathrm{Hol}(N)\) for \(N\) ranging over the groups of the same order as \(G\).
Since both objects are related to regular subgroups lying in the holomorph, one sees that there is a connection between skew braces and Hopf-Galois structures on finite Galois extensions. Their correspondence has recently been made explicit by [\textit{L. Stefanello} and \textit{S. Trappeniers}, Bull. Lond. Math. Soc. 55, 1726--1748 (2023; Zbl 07738097)], where it was shown that for any finite Galois extension \(L/K\) with Galois group \(G = (G,\cdot)\), there is a bijection between the binary operations \(\star\) on \(G\) such that \((G,\star,\cdot)\) is a skew brace and the Hopf-Galois structures on \(L/K\).
The work of Greither-Pareigis [loc. cit.] and Byott [loc. cit.] considers the more general setting where \(L/K\) is a finite separable extension which is not necessarily Galois. It is natural to ask whether one can extend the Stefanello-Trappeniers bijection to all separable extensions by generalizing the definition of skew braces.
In the paper under review, the authors propose the following definition: A (left) \textit{skew bracoid} is a 5-tuple \((G,\cdot,N,\star,\odot)\) for which \((G,\cdot)\) and \((N,\star)\) are groups and \(\odot\) is a transitive action of \((G,\cdot)\) on \(N\) such that
\[
g \odot (\eta \star \mu) = (g\odot \eta) \star (g\odot e_N)^{-1} \star (g\odot \mu)
\]
for all \(g\in G\) and \(\eta,\mu\in N\). One can recover the definition of a skew brace by taking \(G = (B,\cdot)\), \(N=(B,\star)\), and \(\odot = \cdot\). The authors build the theory of skew bracoids by studying sub-structures, homomorphisms, etc. Moreover, using skew bracoids, they are able to generalize the Stefanello-Trappeniers bijection to arbitrary finite separable extensions (see Theorem 5.1 of the paper).
Reviewer: Cindy Tsang (Tōkyō)Symplectic 4-dimensional semifields of order \(8^4\) and \(9^4\)https://zbmath.org/1529.510042024-04-02T17:33:48.828767Z"Lavrauw, Michel"https://zbmath.org/authors/?q=ai:lavrauw.michel"Sheekey, John"https://zbmath.org/authors/?q=ai:sheekey.johnThe authors classify symplectic 4-dimensional semifields over \({\mathbb F}_q\), for \(q \le 9\). They show that every symplectic 4-dimensional semifield over \({\mathbb F}_q\) for \(q\) even, \(q \le 8\) is associative and hence a field. Every symplectic 4-dimensional semifield over \({\mathbb F}_q\) for \(q\) odd, \(q \le 9\) is Knuth-equivalent to a Dickson commutative semifield. For \(q \le 7\), these results were known previously and are confirmed by the authors.
The proofs are mainly computational and the authors carefully describe their ideas and algorithms.
Reviewer: Norbert Knarr (Stuttgart)Complete solution of the LSZ model via topological recursionhttps://zbmath.org/1529.810952024-04-02T17:33:48.828767Z"Branahl, Johannes"https://zbmath.org/authors/?q=ai:branahl.johannes"Hock, Alexander"https://zbmath.org/authors/?q=ai:hock.alexanderSummary: We prove that the Langmann-Szabo-Zarembo (LSZ) model with quartic potential, a toy model for a quantum field theory on noncommutative spaces grasped as a complex matrix model, obeys topological recursion of Chekhov, Eynard and Orantin. By introducing two families of correlation functions, one corresponding to the meromorphic differentials \(\omega_{g,n}\) of topological recursion, we obtain Dyson-Schwinger equations that eventually lead to the abstract loop equations being, together with their pole structure, the necessary condition for topological recursion. This strategy to show the exact solvability of the LSZ model establishes another approach towards the exceptional property of integrability in some quantum field theories. We compare differences in the loop equations for the LSZ model (with complex fields) and the Grosse-Wulkenhaar model (with hermitian fields) and their consequences for the resulting particular type of topological recursion that governs the models.Boomerang uniformity of some classes of functions over finite fieldshttps://zbmath.org/1529.940232024-04-02T17:33:48.828767Z"Garg, Kirpa"https://zbmath.org/authors/?q=ai:garg.kirpa"Hasan, Sartaj Ul"https://zbmath.org/authors/?q=ai:hasan.sartaj-ul"Stănică, Pantelimon"https://zbmath.org/authors/?q=ai:stanica.pantelimonSummary: We give bounds for the boomerang uniformity of the perturbation of some special classes of permutation functions, namely, Gold and inverse functions via trace maps. Consequently, we obtain some classes of functions with low boomerang uniformity, as often required for practical purposes.