Recent zbMATH articles in MSC 12 https://zbmath.org/atom/cc/12 2021-11-25T18:46:10.358925Z Werkzeug Equational theories of fields https://zbmath.org/1472.03031 2021-11-25T18:46:10.358925Z "Martin-Pizarro, Amador" https://zbmath.org/authors/?q=ai:martin-pizarro.amador "Ziegler, Martin" https://zbmath.org/authors/?q=ai:ziegler.martin.1 Let $$T$$ be a first-order theory. A formula $$\varphi(x;y)$$, with a given partition of free variables into $$x$$ and $$y$$, is called an \textit{equation} if, in every model $$M$$ of $$T$$, the collection of finite intersections of instances $$\varphi(x;a)$$ with $$a\in M$$ has the descending chain condition. The theory $$T$$ is said to be \textit{equational} if every formula is equivalent to a Boolean combination of equations modulo $$T$$. Equationality is a strengthening of stability. For example, the theory of an equivalence relation with infinitely many infinite classes, the theory of algebraically closed fields, and the theory of separably closed fields of characteristic $$p>0$$ and finite imperfection degree are equational. In the paper under review, it is proven that the theory of separably closed fields of arbitrary imperfection degree and the theory of proper pairs of algebraically closed fields are equational. Thus, the authors are able to deal with theories where the models (as fields) have infinite linear dimension over a definable subfield. Indeed, if $$K$$ is a separably closed field of characteristic $$p$$ and infinite imperfection degree, then $$K$$ has infinite linear dimension over the definable subfield $$K^p$$. Similarly, if $$(K; +, \cdot, C)$$ is a proper pair of algebraically closed fields then the linear dimension of $$K$$ over $$C$$ is infinite. Fields with automorphism and valuation https://zbmath.org/1472.03033 2021-11-25T18:46:10.358925Z "Beyarslan, Özlem" https://zbmath.org/authors/?q=ai:beyarslan.ozlem "Hoffmann, Daniel Max" https://zbmath.org/authors/?q=ai:hoffmann.daniel-max "Onay, Gönenç" https://zbmath.org/authors/?q=ai:onay.gonenc "Pierce, David" https://zbmath.org/authors/?q=ai:pierce.david-a|pierce.david-m In this short paper, the authors study problems around companionability of the theories involving a valuation and an automorphism on a field. They obtain two results which clarify the situation under the assumption of freeness'', that is the authors assume that there is no interaction between the valuation and the automorphism in question. These results are as follows. 1. The model companion of the theory of the aforementioned fields with valuation and automorphism exists. 2. A natural candidate, which is the theory of models of ACFA equipped with a valuation, is NOT this model companion. A simple criterion https://zbmath.org/1472.03034 2021-11-25T18:46:10.358925Z "Blossier, Thomas" https://zbmath.org/authors/?q=ai:blossier.thomas "Martin-Pizarro, Amador" https://zbmath.org/authors/?q=ai:martin-pizarro.amador Summary: In this article, we mimic the proof of the simplicity of the theory ACFA of generic difference fields in order to provide a criterion, valid for certain theories of pure fields and fields equipped with operators, which shows that a complete theory is simple whenever its definable and algebraic closures are controlled by an underlying stable theory. Definable $$V$$-topologies, Henselianity and NIP https://zbmath.org/1472.03035 2021-11-25T18:46:10.358925Z "Halevi, Yatir" https://zbmath.org/authors/?q=ai:halevi.yatir "Hasson, Assaf" https://zbmath.org/authors/?q=ai:hasson.assaf "Jahnke, Franziska" https://zbmath.org/authors/?q=ai:jahnke.franziska In the article under review, the authors prove that every $$t$$-Henselian NIP valued infinite field admits at most one definable V-topology, and as a consequence, they show that a positive answer to Shelah's conjecture implies the Henselianity conjecture, which states that every definable valuation on an infinite NIP field must be Henselian. Shelah's conjecture states that an infinite NIP field is either separably closed, real closed or it can be equipped with a non-trivial definable Henselian valuation (cf. [\textit{F. Jahnke} and \textit{J. Koenigsmann}, J. Symb. Log. 80, No. 1, 85--99 (2015; Zbl 1372.03078)]). Johnson has affirmatively proved the conjecture when the field is dp-minimal, that is, of dp-rank $$1$$. A crucial part of his proof was to provide the field with a definable V-topology, that is, a definable topology induced by either an archimedean absolute value or by a valuation. Furthermore, Johnson showed that the V-topology is canonical and hence unique. In this paper, the authors generalise the uniqueness of the V-topology for all $$t$$-Henselian NIP valued fields, by showing (Theorem 5.3) that they admit at most one definable V-topology. Recall that a valued field is $$t$$-Henselian if for every natural number $$n\ge 2$$, the collection of tuples $$(a_0, \ldots, a_{n-1})$$ such that the polynomial $T^n +a_{n-1}T^{n-1}+\cdots+a_0$ has a simple root in $$K$$ is an open subset of $$K^n$$. As a by-product of their result, the authors show in Theorem 6.3 that a positive answer to Shelah's conjecture yields a positive answer to the Henselianity conjecture, which states that every definable valuation on an infinite NIP field must be Henselian. In [\textit{W. Johnson}, Ann. Pure Appl. Logic 172, No. 6, Article ID 102949, 33 p. (2021; Zbl 07333020)], the latter conjecture has been answered positively whenever the field has positive characteristic. Applications of constructed new families of generating-type functions interpolating new and known classes of polynomials and numbers https://zbmath.org/1472.05010 2021-11-25T18:46:10.358925Z "Simsek, Yilmaz" https://zbmath.org/authors/?q=ai:simsek.yilmaz Summary: The aim of this article is to construct some new families of generating-type functions interpolating a certain class of higher order Bernoulli-type, Euler-type, Apostol-type numbers, and polynomials. Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol-type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented. Looking for a new version of Gordon's identities https://zbmath.org/1472.05013 2021-11-25T18:46:10.358925Z "Afsharijoo, Pooneh" https://zbmath.org/authors/?q=ai:afsharijoo.pooneh Summary: We give a commutative algebra viewpoint on Andrews recursive formula for the partitions appearing in Gordon's identities, which are a generalization of Rogers-Ramanujan identities. Using this approach and differential ideals, we conjecture a family of partition identities which extend Gordon's identities. This family is indexed by $$r\geq 2$$. We prove the conjecture for $$r=2$$ and $$r=3$$. Exact $$p$$-adic computation in Magma https://zbmath.org/1472.11004 2021-11-25T18:46:10.358925Z "Doris, Christopher" https://zbmath.org/authors/?q=ai:doris.christopher Summary: We describe a new arithmetic system for the Magma computer algebra system for working with $$p$$-adic numbers exactly, in the sense that numbers are represented lazily to infinite $$p$$-adic precision. This is the first highly featured such implementation. This has the benefits of increasing user-friendliness and speeding up some computations, as well as forcibly producing provable results. We give theoretical and practical justification for its design and describe some use cases. The intention is that this article will be of benefit to anyone wanting to implement similar functionality in other languages. Surjectivity of Galois representations in rational families of abelian varieties https://zbmath.org/1472.11169 2021-11-25T18:46:10.358925Z "Landesman, Aaron" https://zbmath.org/authors/?q=ai:landesman.aaron "Swaminathan, Ashvin" https://zbmath.org/authors/?q=ai:swaminathan.ashvin-anand "Tao, James" https://zbmath.org/authors/?q=ai:tao.james "Xu, Yujie" https://zbmath.org/authors/?q=ai:xu.yujie Summary: In this article, we show that for any nonisotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-1 subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension $$g\geq 3$$, there are infinitely many abelian varieties over $$\mathbb{Q}$$ with adelic Galois representation having image equal to all of $$\mathrm{GSp}_{2g}(\widehat{\mathbb{Z}})$$. Lübeck's classification of representations of finite simple groups of Lie type and the inverse Galois problem for some orthogonal groups https://zbmath.org/1472.11171 2021-11-25T18:46:10.358925Z "Zenteno, Adrián" https://zbmath.org/authors/?q=ai:zenteno.adrian Summary: In this paper we prove that for each integer of the form $$n = 4 \varpi$$(where $$\varpi$$ is a prime between 17 and 73) at least one of the following groups: $$\mathrm{P} {\Omega}_n^\pm(\mathbb{F}_{\ell^s}), \mathrm{PSO}_n^\pm(\mathbb{F}_{\ell^s}), \mathrm{PO}_n^\pm(\mathbb{F}_{\ell^s})$$ or $$\mathrm{PGO}_n^\pm(\mathbb{F}_{\ell^s})$$ is a Galois group of $$\mathbb{Q}$$ for almost all primes $$\ell$$ and infinitely many integers $$s > 0$$. This is achieved by making use of the classification of small degree representations of finite simple groups of Lie type in defining characteristic due to Lübeck and a previous result of the author on the image of the Galois representations attached to RAESDC automorphic representations of $$\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$$. Torsion points of Drinfeld modules over large algebraic extensions of finitely generated function fields https://zbmath.org/1472.11184 2021-11-25T18:46:10.358925Z "Asayama, Takuya" https://zbmath.org/authors/?q=ai:asayama.takuya \textit{W.-D. Geyer} and \textit{M. Jarden} considered in [Isr. J. Math. 31, 257--297 (1978; Zbl 0406.14025)] torsion points of elliptic curves defined over some class of infinitely generated fields over their prime fields. Furthermore, they conjectured that the same results obtained over elliptic curves hold for abelian varieties. The aim of the paper under review is to prove some results for Drinfeld modules analogous to the conjecture of Geyer and Jarden. Let $$K$$ be an algebraic function field of one variable over the finite field of $$q$$ elements $${\mathbb F}_q$$. Let $$\tilde K$$ be a fixed algebraic closure of $$K$$ and let $$K^{\mathrm{sep}}$$ be the separable closure of $$K$$ in $$\tilde K$$. For a positive integer $$e$$, let $$\sigma =(\sigma_1,\ldots,\sigma_e)\in\mathrm{Gal}(K^{\mathrm{sep}}/K)^e$$ be an $$e$$-tuple. Set $$\tilde K(\sigma)$$ and $$K^{\mathrm{sep}}(\sigma)$$ be the fixed fields in $$\tilde K$$ and $$K^{\mathrm{sep}}$$ by $$\sigma$$, respectively. Let $$\infty$$ be a fixed place of $$K$$, and let $$A$$ be the ring of elements of $$K$$ which are regular outside $$\infty$$. Let $$L$$ be a field containing $${\mathbb F}_q$$ and let $$\iota\colon A\to L$$ be a fixed injective ring homomorphism. Assume that $$L$$ is finitely generated over $$K$$. The main result is the following. For almost all $$\sigma\in\mathrm{Gal} (L^{\mathrm{sep}}/L)^e$$ (in the sense of the Haar measure) and for every Drinfeld $$A$$-module $$\varphi$$ defined over $$\tilde L(\sigma)$$ with $$\mathrm{End}_{\tilde L}(\varphi)=A$$, we have: (1) If $$e=1$$, the group $$_{\varphi}\tilde L(\sigma)_{\mathrm{tor}}$$ is infinite and there exist infinitely many non-zero prime ideals $$P$$ of $$A$$ such that $$_{\varphi}\tilde L(\sigma)[P]\neq 0$$. (2) If $$e\geq 2$$, the group $$_{\varphi}\tilde L(\sigma)_{\mathrm{tor}}$$ is finite. (3) The group $$_{\varphi}\tilde L(\sigma)[P^{\infty}]:=\bigcup_{n=1}^{\infty} {_{\varphi}\tilde L(\sigma)} [P^n]$$ is finite for every non-zero prime ideal $$P$$ of $$A$$. Here $$_{\varphi}\tilde L(\sigma)$$ denotes the structure as an $$A$$-module and $$_{\varphi}\tilde L(\sigma)[P]$$ denotes the $$P$$-torsion. The difference between the cases $$e=1$$ and $$e\geq 2$$ arises from the fact that $$\sum_P N(P)^{-e}$$ diverges for $$e=1$$ and converges for $$e\geq 2$$, where the sum runs over all non-zero prime ideals $$P$$ of $$A$$ and where $$N(P):=|A/P|$$. In order to prove the main result, the authors prove that it suffices to show some statements on each Drinfeld module over $$L$$ and then they derive these statements in the last section. A generalized Iwasawa's theorem and its application https://zbmath.org/1472.11283 2021-11-25T18:46:10.358925Z "Lai, King Fai" https://zbmath.org/authors/?q=ai:lai.king-fai "Tan, Ki-Seng" https://zbmath.org/authors/?q=ai:tan.ki-seng Let $$K/k$$ be a $$\mathbb{Z}_p^d$$-extension of a number field $$k$$ ($$p\in \mathbb{N}$$ a prime) with Galois group $$\Gamma$$ and associated Iwasawa algebra $$\Lambda:=\mathbb{Z}_p[[\Gamma]]$$. For any intermediate number field $$k\subseteq E\subsetneq K$$, let $$A_E$$ be the $$p$$-part of the class group of $$E$$ and $$A'_E$$ the quotient of $$A_E$$ modulo the classes generated by primes lying above $$p$$. Consider the direct (resp. inverse) limits $$A_K$$ and $$A'_K$$ (resp. $$X_K$$ and $$X'_K$$) of the $$A_E$$ and $$A'_E$$ with respect to the natural inclusions (resp. norm) maps. Those are modules over the Iwasawa algebra $$\Lambda$$ and their structure is the main topic of various conjectures: most notably the Greenberg Generalized Conjecture, which predicts the pseudo-nullity of $$X_{\tilde{k}}$$, with $$\tilde{k}$$ the compositum of all the $$\mathbb{Z}_p$$-extensions of $$k$$ (we recall that a finitely generated $$\Lambda$$-module $$M$$ is pseudo-null, i.e., $$M\sim_\Lambda 0$$, if it has at least two relatively prime annihilators). The paper mainly deals with the subquotient of $$X_K$$ (resp. $$X'_K$$) given by the inverse limit of the kernels of the capitulation maps $$c_{K/E}: A_E \longrightarrow A_K$$ (resp. $$c'_{K/E}: A'_E \longrightarrow A'_K$$). Let $\dot{X}_K := \lim_{\begin{subarray}{c} \longleftarrow \\ E \end{subarray}} \mathrm{Ker}\,c_{K/E}\quad \text{and}\quad \dot{X}'_K := \lim_{\begin{subarray}{c} \longleftarrow \\ E \end{subarray}}\mathrm{Ker}\,c'_{K/E} \,,$ the authors prove that $$\dot{X}\sim_\Lambda 0$$ and $$\dot{X}'\sim_\Lambda 0$$, thus generalizing a result of \textit{I. Iwasawa} [Ann. Math. (2) 98, 246--326 (1973: Zbl 0285.12008)] which was proved only for $$d=1$$. The main technical point is a careful description of the topology of $$\Gamma$$ (hence of $$\Lambda$$-modules) which is carried out by providing explicit generators for augmentation ideals of the group rings associated to the finite layers of $$K/k$$. Once this is done, with the crucial ingredient of the $$\mathbb{Z}_p$$-flat sets described by \textit{P. Monsky} [Math. Ann. 255, 217--227 (1981: Zbl 0437.12016)], the proof closely follows the classical one by Iwasawa. In the final section the authors use the main theorem to obtain duality pseudo-isomorphisms $X_K^\# \sim_\Lambda \lim_{\begin{subarray}{c} \longleftarrow \\ E \end{subarray}} A_E^\vee \quad\text{and}\quad (X'_K)^\# \sim_\Lambda \lim_{\begin{subarray}{c} \longleftarrow \\ E \end{subarray}} (A'_E)^\vee,$ (where $$\,^\vee$$ denotes the Pontrjagin dual and $$\,^\#$$ the $$\Lambda$$-module with the twisted action induced by $$\gamma\longrightarrow \gamma^{-1}$$) via the theory of $$\Gamma$$-systems presented in [\textit{K.F. Lan} et al., Trans. Am. Math. Soc. 370, No. 3, 1925--1958 (2018: Zbl 1444.11236)]. A family of permutation trinomials over $$\mathbb{F}_{q^2}$$ https://zbmath.org/1472.11295 2021-11-25T18:46:10.358925Z "Bartoli, Daniele" https://zbmath.org/authors/?q=ai:bartoli.daniele "Timpanella, Marco" https://zbmath.org/authors/?q=ai:timpanella.marco Summary: Let $$p>3$$ and consider a prime power $$q=p^h$$. We completely characterize permutation polynomials of $$\mathbb{F}_{q^2}$$ of the type $$f_{a,b}(X)=X(1+aX^{q(q-1)}+bX^{2(q-1)})\in\mathbb{F}_{q^2}[X]$$. In particular, using connections with algebraic curves over finite fields, we show that the already known sufficient conditions are also necessary. Möbius-Frobenius maps on irreducible polynomials https://zbmath.org/1472.11298 2021-11-25T18:46:10.358925Z "Brochero Martínez, F. E." https://zbmath.org/authors/?q=ai:brochero-martinez.fabio-enrique "Oliveira, Daniela" https://zbmath.org/authors/?q=ai:oliveira.daniela-s "Reis, Lucas" https://zbmath.org/authors/?q=ai:reis.lucas Summary: Let $$n$$ be a positive integer and let $$\mathbb{F}_{q^n}$$ be the finite field with $$q^n$$ elements, where $$q$$ is a prime power. We introduce a natural action of the \textit{projective semilinear group} $${\mathrm{P}\Gamma\mathrm{L}} (2, q^n)={\mathrm{PGL}} (2, q^n)\rtimes{\mathrm{Gal}} ({\mathbb{F}_{q^n}} /\mathbb{F}_q)$$ on the set of monic irreducible polynomials over the finite field $$\mathbb{F}_{q^n}$$. Our main results provide information on the characterisation and number of fixed points. Recursive constructions of $$k$$-normal polynomials using some rational transformations over finite fields https://zbmath.org/1472.11299 2021-11-25T18:46:10.358925Z "Kim, Ryul" https://zbmath.org/authors/?q=ai:kim.ryul "Son, Hyang-Sim" https://zbmath.org/authors/?q=ai:son.hyang-sim Arithmetic constraints of polynomial maps through discrete logarithms https://zbmath.org/1472.11305 2021-11-25T18:46:10.358925Z "Reis, Lucas" https://zbmath.org/authors/?q=ai:reis.lucas Summary: Let $$q$$ be a prime power, let $$\mathbb{F}_q$$ be the finite field with $$q$$ elements and let $$\theta$$ be a generator of the cyclic group $$\mathbb{F}_q^\ast$$. For each $$a \in \mathbb{F}_q^\ast$$, let $$\log_\theta a$$ be the unique integer $$i \in \{1, \ldots, q - 1 \}$$ such that $$a = \theta^i$$. Given polynomials $$P_1, \ldots, P_k \in \mathbb{F}_q [x]$$ and divisors $$1 < d_1, \ldots, d_k$$ of $$q - 1$$, we discuss the distribution of the functions $F_i : y \mapsto \log_\theta P_i(y) \pmod d_i,$ over the set $$\mathbb{F}_q \setminus\bigcup_{i = 1}^k \{y \in \mathbb{F}_q \mid P_i(y) = 0 \}$$. Our main result entails that, under a natural multiplicative condition on the pairs $$( d_i, P_i)$$, the functions $$F_i$$ are asymptotically independent. We also provide some applications that, in particular, relates to past work. Non-existence of Hopf-Galois structures and bijective crossed homomorphisms https://zbmath.org/1472.12001 2021-11-25T18:46:10.358925Z "Tsang, Cindy" https://zbmath.org/authors/?q=ai:tsang.cindy-sin-yi Work of \textit{C. Greither} and \textit{B. Pareigis} [J. Algebra 106, 239--258 (1987; Zbl 0615.12026)] and of \textit{N. P. Byott} [Commun. Algebra 24, No. 10, 3217--3228 (1996; Zbl 0878.12001)] yields that the enumeration of Hopf-Galois structures on a Galois extension of fields with Galois group $$G$$ is equivalent to the enumeration of all the regular subgroups of all the holomorphs $$\mathrm{Hol}(N)$$ of the groups $$N$$ of the same order as $$G$$. Various methods have been introduced to study these regular subgroups. In the paper under review, the author shows how to describe them in terms of bijective crossed homomorphisms from $$G$$ to $$N$$, and proves the usefulness of this approach by addressing two questions. \textit{S. Carnahan} and \textit{L. Childs} had shown [J. Algebra 218, No. 1, 81--92 (1999; Zbl 0988.12003)] that if $$G$$ is a finite simple group, then the images of the right and left regular representations of $$G$$ are the only subgroups of $$\mathrm{Hol}(G)$$ which are isomorphic to $$G$$. In their first result, the author extends this to finite quasisimple groups, that is, those perfect groups $$G$$ such that $$G/Z(G)$$ is simple. The second group of results addresses the following question. Suppose $$G$$ and $$N$$ are non-isomorphic groups of the same order. Is there a regular subgroup of $$\mathrm{Hol}(N)$$ isomorphic to $$G$$? \textit{N. P. Byott} [Bull. Lond. Math. Soc. 36, No. 1, 23--29 (2004; Zbl 1038.12002)] had given a negative answer when $$G$$ is non-abelian simple. The author extends this result to some classes of quasisimple groups. Hopf-Galois structures on a Galois $$S_n$$-extension https://zbmath.org/1472.12002 2021-11-25T18:46:10.358925Z "Tsang, Cindy" https://zbmath.org/authors/?q=ai:tsang.cindy-sin-yi A finite Galois extension $$L\supseteq K$$ can be a Hopf-Galois extension for other finite-dimensional $$K$$-Hopf algebras than just the group algebra $$K[G]$$ of the Galois group $$G$$ of the field extension. In a groundbreaking paper [J. Algebra 106, 239--258 (1987; Zbl 0615.12026)] \textit{C. Greither} and \textit{B. Pareigis} established a bijection between the Hopf-Galois structures on $$L\supseteq K$$ and the set $$\mathcal{E}(G)$$ of those regular subgroups of the symmetric group of $$G$$ which are normalized by the image of the left regular representation of $$G$$. Moreover, the set $$\mathcal{E}(G)$$ is the disjoint union of the subsets $$\mathcal{E}(G,N)$$ consisting of the regular subgroups in $$\mathcal{E}(G)$$ that are isomorphic to $$N$$ and where $$N$$ runs over the isomorphism classes of all groups having the same order as $$G$$. \textit{N. P. Byott} [Commun. Algebra 24, No. 10, 3217--3228 (1996; Zbl 0878.12001)] showed that the cardinality of $$\mathcal{E}(G,N)$$ essentially can be computed from the number of those regular subgroups of the holomorph of $$N$$ that are isomorphic to $$G$$. The goal of the paper under review is to enumerate the Hopf-Galois structures for a Galois extension $$L\supseteq K$$ with the symmetric group $$S_n$$ on $$n$$ letters as a Galois group by enumerating the set $$\mathcal{E}(S_n)$$. For $$n=1$$ and $$n=2$$ there is only the usual Hopf-Galois structure $$K[S_n]$$, but for $$n=3$$ there are already $$5$$ Hopf-Galois structures. The latter is a special case of a result of \textit{N. P. Byott} [J. Pure Appl. Algebra 188, No. 1--3, 45--57 (2004; Zbl 1047.16022)]. The cases $$n=4$$, where $$S_n$$ is still solvable, and $$n=6$$, where $$S_n$$ has a non-trivial outer automorphism, are settled by using MAGMA. It should also be noted that several intermediate steps were previously obtained in the paper [J. Algebra 218, No. 1, 81--92 (1999; Zbl 0988.12003)] of \textit{S. Carnahan} and \textit{L. N. Childs} as well as in the author's paper [J. Pure Appl. Algebra 223, No. 7, 2804--2821 (2019; Zbl 1472.12001)]. (Remark of the reviewer: As $$\#\mathcal{E}(S_6,\mathrm{PGL}(2,9))=0$$, the condition for $$N$$ in Theorem 1.3 should not contain the group $$\mathrm{PGL}(2,9)$$.) On parametric and generic polynomials with one parameter https://zbmath.org/1472.12003 2021-11-25T18:46:10.358925Z "Dèbes, Pierre" https://zbmath.org/authors/?q=ai:debes.pierre "König, Joachim" https://zbmath.org/authors/?q=ai:konig.joachim "Legrand, François" https://zbmath.org/authors/?q=ai:legrand.francois "Neftin, Danny" https://zbmath.org/authors/?q=ai:neftin.danny Summary: Given fields $$k\subseteq L$$, our results concern one parameter \textit{L-parametric} polynomials over $$k$$, and their relation to generic polynomials. The former are polynomials $$P(T, Y) \in k [T] [Y]$$ of group $$G$$ which parametrize all Galois extensions of $$L$$ of group $$G$$ via specialization of $$T$$ in $$L$$, and the latter are those which are $$L$$-parametric for every field $$L \supseteq k$$. We show, for example, that being $$L$$-parametric with $$L$$ taken to be the single field $$\mathbb{C}((V))(U)$$ is in fact sufficient for a polynomial $$P(T,Y)\in\mathbb{C}[T][Y]$$ to be generic. As a corollary, we obtain a complete list of one parameter generic polynomials over a given field of characteristic 0, complementing the classical literature on the topic. Our approach also applies to an old problem of Schinzel: subject to the Birch and Swinnerton-Dyer conjecture, we provide one parameter families of affine curves over number fields, all with a rational point, but with no rational generic point. On the invariants of inseparable field extensions https://zbmath.org/1472.12004 2021-11-25T18:46:10.358925Z "Fliouet, El Hassane" https://zbmath.org/authors/?q=ai:fliouet.el-hassane Summary: Let $$K$$ be a finitely generated extension of a field $$k$$ of characteristic $$p\neq 0$$. In 1947, Dieudonné initiated the study of maximal separable intermediate fields. He gave in particular the form of an important subclass of maximal separable intermediate fields $$D$$ characterized by the property $$K\subseteq k({D}^{p^{-\infty }})$$, and which are called the distinguished subfields of $$K/k$$. In 1970, Kraft showed that the distinguished maximal separable subfields are precisely those over which $$K$$ is of minimal degree. This paper grew out of an attempt to find a new characterization of distinguished subfields of $$K/k$$ by means of new inseparability invariants. New bounds and an efficient algorithm for sparse difference resultants https://zbmath.org/1472.12005 2021-11-25T18:46:10.358925Z "Yuan, Chun-Ming" https://zbmath.org/authors/?q=ai:yuan.chunming "Zhang, Zhi-Yong" https://zbmath.org/authors/?q=ai:zhang.zhiyong Summary: The sparse difference resultant introduced in [\textit{Wei Li} et al., J. Symb. Comput. 68, Part 1, 169--203 (2015; Zbl 1328.65266)] is a basic concept in difference elimination theory. In this paper, we show that the sparse difference resultant of a generic Laurent transformally essential system can be computed via the sparse resultant of a simple algebraic system arising from the difference system. Moreover, new order bounds of sparse difference resultant are found. Then we propose an efficient algorithm to compute sparse difference resultant which is the quotient of two determinants whose elements are the coefficients of the polynomials in the algebraic system. The complexity of the algorithm is analyzed and experimental results show the efficiency of the algorithm. Spectrum of a linear differential equation with constant coefficients https://zbmath.org/1472.12006 2021-11-25T18:46:10.358925Z "Azzouz, Tinhinane A." https://zbmath.org/authors/?q=ai:azzouz.tinhinane-a The author computes the spectrum, in the sense of [\textit{V. G. Berkovich}, Spectral theory and analytic geometry over non-Archimedean fields. Providence, RI: American Mathematical Society (1990; Zbl 0715.14013), Chapter 7], of an ultrametric linear differential operator with constant coefficients, defined over an affinoid domain of the analytic affine line. Its spectrum is a finite union of either closed disks or topological closures of open disks. It is shown to satisfy a continuity property. Of limit key polynomials https://zbmath.org/1472.13007 2021-11-25T18:46:10.358925Z "Alberich-Carramiñana, Maria" https://zbmath.org/authors/?q=ai:alberich-carraminana.maria "Boix, Alberto F. F." https://zbmath.org/authors/?q=ai:boix.alberto-f "Fernández, Julio" https://zbmath.org/authors/?q=ai:fernandez.julio "Guàrdia, Jordi" https://zbmath.org/authors/?q=ai:guardia.jordi "Nart, Enric" https://zbmath.org/authors/?q=ai:nart.enric "Roé, Joaquim" https://zbmath.org/authors/?q=ai:roe.joaquim Let $$K$$ be a field and $$v$$ a valuation on the polynomial ring $$K[x]$$, with value group $$\Gamma_v$$. For each $$\gamma\in \Gamma_v$$, we have the following abelian groups $$\mathcal{P}_{\gamma}^+=\{ g\in K[x]; \mu(g)>\gamma\}\subset\mathcal{P}_{\gamma}=\{ g\in K[x]; \mu(g)\geq\gamma\}$$. The graded algebra $$gr_v(K[x])=\oplus_{\gamma\in\Gamma_v}\mathcal{P}_{\gamma}/ \mathcal{P}_{\gamma}^+$$ is an integral domain. A MacLane-Vaquie (MLV) key polynomial for $$v$$ is a monic polynomial $$\phi\in K[X]$$ whose initial term generates a prime ideal in $$gr_v(K[x])$$, which cannot be generated by the initial term of a polynomial of smaller degree. The abstract key polynomials for $$v$$ are defined in a technical way. In the paper under review, the authors try to find relations between the MLV key polynomials for valuations $$\mu\leq v$$ and the abstract key polynomials for $$v$$. Erratum to: Slope filtrations'' https://zbmath.org/1472.14022 2021-11-25T18:46:10.358925Z "André, Yves" https://zbmath.org/authors/?q=ai:andre.yves Corrects Example 1.2.2.(2) and Lemma 1.2.18 in [the author, ibid. 1, No. 1, 1--85 (2009; Zbl 1213.14039)] and notes that Lemma 1.2.8 should be discarded (Proposition 1.4.18, the only place where this lemma is used, is modified accordingly). On the monodromy theorem for the family of $$p$$-adic differential equations https://zbmath.org/1472.14025 2021-11-25T18:46:10.358925Z "Mebkhout, Zoghman" https://zbmath.org/authors/?q=ai:mebkhout.zoghman The author proves a semi-global monodromy theorem for a $$p$$-adic de Rham bundle in the neighbourhood of a generic point of a hypersurface of a smooth scheme over a perfect field with characteristic $$p>0$$ in higher dimensions. The result is formulated using the notion of a Frobenius endomorphism $$\sigma$$ of a $$p$$-adic field $$L$$, i.e. an endomorphism whose restriction to $$\mathbb{Q}_p$$ is the identity, which is continuous and such that for each $$a\in \mathcal{O}_L$$ (the ring of integers) one has $$|\sigma (a)-a^p|<1$$. Computing the equisingularity type of a pseudo-irreducible polynomial https://zbmath.org/1472.14069 2021-11-25T18:46:10.358925Z "Poteaux, Adrien" https://zbmath.org/authors/?q=ai:poteaux.adrien "Weimann, Martin" https://zbmath.org/authors/?q=ai:weimann.martin In the paper under review, the authors characterize a class of germs of plane curve singularities, containing irreducible ones, whose equisingularity type can be computed in an expected quasi-linear time with respect to the discriminant valuation of a Weierstrass equation. Rings in which idempotents generate maximal or minimal ideals https://zbmath.org/1472.16036 2021-11-25T18:46:10.358925Z "Dube, Themba" https://zbmath.org/authors/?q=ai:dube.themba "Ghirati, Mojtaba" https://zbmath.org/authors/?q=ai:ghirati.mojtaba "Nazari, Sajad" https://zbmath.org/authors/?q=ai:nazari.sajad "Taherifar, Ali" https://zbmath.org/authors/?q=ai:taherifar.a The authors characterize rings with identity in which every left ideal generated by an idempotent different from $$0$$ and $$1$$ is either a maximal left ideal or a minimal left ideal. These rings are called IMm-rings. Several special classes of IMm-rings are considered. In particular, if $$R$$ is a semiprimitive commutative ring that has infinitely many maximal ideals, $$R$$ being an IMm-ring is characterized by means of the Zariski topology of the maximal spectrum $$\text{Max}(R):=\{M\subseteq R: M \text{ is a maximal ideal of } R\}$$. Finally the authors study rings with a weaker form of the IMm-property''. On the Dixmier-Moeglin equivalence for Poisson-Hopf algebras https://zbmath.org/1472.17080 2021-11-25T18:46:10.358925Z "Launois, Stéphane" https://zbmath.org/authors/?q=ai:launois.stephane "León Sánchez, Omar" https://zbmath.org/authors/?q=ai:leon-sanchez.omar Summary: We prove that the Poisson version of the Dixmier-Moeglin equivalence holds for cocommutative affine Poisson-Hopf algebras. This is a first step towards understanding the symplectic foliation and the representation theory of (cocommutative) affine Poisson-Hopf algebras. Our proof makes substantial use of the model theory of fields equipped with finitely many possibly noncommuting derivations. As an application, we show that the symmetric algebra of a finite dimensional Lie algebra, equipped with its natural Poisson structure, satisfies the Poisson Dixmier-Moeglin equivalence. Degree three invariants for semisimple groups of types $$B$$, $$C$$, and $$D$$ https://zbmath.org/1472.20102 2021-11-25T18:46:10.358925Z "Baek, Sanghoon" https://zbmath.org/authors/?q=ai:baek.sanghoon Summary: We determine the group of reductive cohomological degree 3 invariants of all split semisimple groups of types $$B$$, $$C$$, and $$D$$. We also present a complete description of such cohomological invariants. As an application, we show that the group of degree 3 unramified cohomology of the classifying space $$BG$$ is trivial for all split semisimple groups $$G$$ of types $$B$$, $$C$$, and $$D$$. Hilbert transforms and the equidistribution of zeros of polynomials https://zbmath.org/1472.42002 2021-11-25T18:46:10.358925Z "Carneiro, Emanuel" https://zbmath.org/authors/?q=ai:carneiro.emanuel "Das, Mithun Kumar" https://zbmath.org/authors/?q=ai:das.mithun-kumar "Florea, Alexandra" https://zbmath.org/authors/?q=ai:florea.alexandra-m "Kumchev, Angel V." https://zbmath.org/authors/?q=ai:kumchev.angel-v "Malik, Amita" https://zbmath.org/authors/?q=ai:malik.amita "Milinovich, Micah B." https://zbmath.org/authors/?q=ai:milinovich.micah-b "Turnage-Butterbaugh, Caroline" https://zbmath.org/authors/?q=ai:turnage-butterbaugh.caroline-l "Wang, Jiuya" https://zbmath.org/authors/?q=ai:wang.jiuya Summary: We improve the current bounds for an inequality of \textit{P. Erdős} and \textit{P. Turán} [Ann. Math. (2) 51, 105--119 (1950; Zbl 0036.01501)] related to the discrepancy of angular equidistribution of the zeros of a given polynomial. Building upon a recent work of \textit{K. Soundararajan} [Am. Math. Mon. 126, No. 3, 226--236 (2019; Zbl 1409.42003)], we establish a novel connection between this inequality and an extremal problem in Fourier analysis involving the maxima of Hilbert transforms, for which we provide a complete solution. Prior to Soundararajan [loc. cit.], refinements of the discrepancy inequality of Erdős and Turán had been obtained by \textit{T. Ganelius} [Ark. Mat. 3, 1--50 (1954; Zbl 0055.06905)] and \textit{M. Mignotte} [C. R. Acad. Sci., Paris, Sér. I 315, No. 8, 907--911 (1992; Zbl 0773.31002)]. Quantum state transfer on a class of circulant graphs https://zbmath.org/1472.81043 2021-11-25T18:46:10.358925Z "Pal, Hiranmoy" https://zbmath.org/authors/?q=ai:pal.hiranmoy Summary: We study the existence of quantum state transfer on non-integral circulant graphs. We find that continuous-time quantum walks on quantum networks based on certain circulant graphs with $$2^k (k \in \mathbb{Z})$$ vertices exhibit pretty good state transfer when there is no external dynamic control over the system. We generalize a few previously known results on pretty good state transfer on circulant graphs, and this way we re-discover all integral circulant graphs on $$2^k$$ vertices exhibiting perfect state transfer.