Recent zbMATH articles in MSC 11Shttps://www.zbmath.org/atom/cc/11S2021-02-27T13:50:00+00:00WerkzeugIteration and the minimal resultant.https://www.zbmath.org/1453.111572021-02-27T13:50:00+00:00"Jacobs, Kenneth"https://www.zbmath.org/authors/?q=ai:jacobs.kenneth-s"Williams, Phillip"https://www.zbmath.org/authors/?q=ai:williams.phillipSummary: Let \(K\) be an algebraically closed field that is complete with respect to a non-Archimedean absolute value, and let \(\varphi\in K(z)\) have degree \(d\geq 2\). We characterize maps for which the minimal resultant of an iterate \(\varphi^n\) is given by a simple formula in terms of \(d\), \(n\), and the minimal resultant of \(\varphi\). Three characterizations of such maps are given, one measure-theoretic and two in terms of the indeterminacy locus \(I(d)\) in the parameter space \(\mathbb{P}^{2d+1}\) of (possibly degenerate) rational maps.
As an application, we are able to give a new explicit formula involving the Arakelov-Green's function attached to \(\varphi\). We end by illustrating our results with some explicit examples.Branches on division algebras.https://www.zbmath.org/1453.160152021-02-27T13:50:00+00:00"Arenas, Manuel"https://www.zbmath.org/authors/?q=ai:arenas.manuel"Arenas-Carmona, Luis"https://www.zbmath.org/authors/?q=ai:arenas-carmona.luisSummary: We describe the set of maximal orders in a 2-by-2 matrix algebra over a non-commutative local division algebra \(B\) containing a given suborder, for certain important families of such suborders, including rings of integers of division subalgebras of \(B\) or most maximal semisimple commutative subalgebras.A note on a structure theorem for prehomogeneous vector spaces.https://www.zbmath.org/1453.111592021-02-27T13:50:00+00:00"Ouchi, Masaya"https://www.zbmath.org/authors/?q=ai:ouchi.masayaSummary: In this note, we give a structure theorem for all prehomogeneous vector spaces defined over the complex number field \(\mathbb{C}\). Also it means a necessary and sufficient condition for a triplet \((G, \rho, V )\) defined over \(\mathbb{C}\) to be a prehomogeneous vector space. For this purpose, we give a general structural correspondence between isotropy subgroups and fixed point sets when a group acts on a non-empty set.Arboreal Cantor actions.https://www.zbmath.org/1453.370132021-02-27T13:50:00+00:00"Lukina, Olga"https://www.zbmath.org/authors/?q=ai:lukina.olgaSummary: In this paper, we consider minimal equicontinuous actions of discrete countably generated groups on Cantor sets, obtained from the arboreal representations of absolute Galois groups of fields. In particular, we study the asymptotic discriminant of these actions. The asymptotic discriminant is an invariant obtained by restricting the action to a sequence of nested clopen sets, and studying the isotropies of the enveloping group actions in such restricted systems. An enveloping (Ellis) group of such an action is a profinite group. A large class of actions of profinite groups on Cantor sets is given by arboreal representations of absolute Galois groups of fields. We show how to associate to an arboreal representation an action of a discrete group, and give examples of arboreal representations with stable and wild asymptotic discriminant.Convergence of \(p\)-adic sequences.https://www.zbmath.org/1453.111552021-02-27T13:50:00+00:00"De Carvalho, Maria Pires"https://www.zbmath.org/authors/?q=ai:de-carvalho.maria-pires"Lourenço, João Nuno P."https://www.zbmath.org/authors/?q=ai:lourenco.joao-nuno-pSummary: Accustomed to working with the real numbers and the usual notion of distance, we wonder at the impact that a change in the metric can cause. Given a prime \(p\), the \(p\)-adic norm of a rational number measures its size with respect to the powers of \(p\), declaring that an irreducible fraction is small if its numerator is divisible by a high positive power of \(p\). This arithmetic method to estimate distances is quite different from the common absolute value and, in particular, the family of convergent sequences is distinct from the real one. We will concern ourselves with the existence of series with rational terms whose convergence holds no matter the chosen norm and such that, though their limits vary with the metric, we know how to master this dependence.\(p\)-adic \(L\)-functions and classical congruences.https://www.zbmath.org/1453.110012021-02-27T13:50:00+00:00"Lin, Xianzu"https://www.zbmath.org/authors/?q=ai:lin.xianzuThis paper is structured as follows: after an introduction, Section 2 gives preliminaries that will be used throughout the article. In Section 3, the author reviews Barsky and Washington's \(p\)-adic expansion of power sums and its applications. In Section 4, the paper gives a similar \(p\)-adic expansion for sums of Lehmer's type and derives many corollaries. Sections 5 and 6 are devoted to extensions of Gauss's and Jacobi's congruences, and of Wilson's theorem, respectively.
Reviewer: Mouad Moutaoukil (Fès)Peters type polynomials and numbers and their generating functions: approach with \(p\)-adic integral method.https://www.zbmath.org/1453.110332021-02-27T13:50:00+00:00"Simsek, Yilmaz"https://www.zbmath.org/authors/?q=ai:simsek.yilmazSummary: The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and \(p\)-adic integrals method. (Peters polynomials \(s_n(x;\lambda,\mu)\) are defined by
\[
\frac{(1+t)^x}{(1+(1+t)^\lambda)^\mu}= \sum_{n=0}^\infty s_n(x;\lambda,\mu)\frac{t^n}{n!}.
\]
Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well-known special numbers and polynomials are presented. By using \(p\)-adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol-type Peters numbers and polynomials). By using these functions with their partial derivative equations and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well-known formulas. Finally, two open problems for interpolation functions for Apostol-type Peters numbers and polynomials are revealed.On symmetric identities of Carlitz's type \(q\)-Daehee polynomials.https://www.zbmath.org/1453.110362021-02-27T13:50:00+00:00"Kim, Won Joo"https://www.zbmath.org/authors/?q=ai:kim.wonjoo"Jang, Lee-Chae"https://www.zbmath.org/authors/?q=ai:jang.lee-chae"Kim, Byung Moon"https://www.zbmath.org/authors/?q=ai:kim.byungmoonSummary: In this paper, we study Carlitz's type \(q\)-Daehee polynomials and investigate the symmetric identities for them by using the \(p\)-adic \(q\)-integral on \(\mathbb{Z}_p\) under the symmetry group of degree \(n\).A survey on \(p\)-adic integrals.https://www.zbmath.org/1453.260392021-02-27T13:50:00+00:00"Duran, Ugur"https://www.zbmath.org/authors/?q=ai:duran.ugur"Dutta, Hemen"https://www.zbmath.org/authors/?q=ai:dutta.hemenThe survey is devoted to the Volkenborn integral of a \(\mathbb Q_p\)-valued function on \(\mathbb Z_p\) (see [\textit{W. H. Schikhof}, Ultrametric calculus. An introduction to \(p\)-adic analysis. Cambridge University Press (1984; Zbl 0553.26006)]) and its generalizations, such as the so-called fermionic \(p\)-adic integral, their \(q\)-analogs and corresponding weighted integrals. The authors consider various special polynomials related to the above integration methods.
The authors do not even mention many other types of integrals of non-Archimedean analysis and mathematical physics (analogs of classical integrals of holomorphic and meromorphic functions, integrals of complex-valued functions on \(p\)-adic spaces etc).
For the entire collection see [Zbl 1425.35003].
Reviewer: Anatoly N. Kochubei (Kyïv)Rational orbits of prehomogeneous vector spaces.https://www.zbmath.org/1453.111602021-02-27T13:50:00+00:00"Yukie, Akihiko"https://www.zbmath.org/authors/?q=ai:yukie.akihikoFrom the text: This paper is an announcement of recent results of the author ([``On equivariant maps related to the space of pairs of exceptiona lJordan algebras'', Comment. Math. Univ. St. Pauli. (to appear), \url{arXiv:1603.0075}; ``Rational orbits of the space of pairs of exceptional Jordan algebras'', Preprint, \url{arXiv:1603.00739}] with \textit{R. Kato} and [the author, Tohoku Math. J. (2) 71, No. 1, 35--52 (2019; Zbl 1443.11252)]).Characteristic cycle of a rank one sheaf and ramification theory.https://www.zbmath.org/1453.140612021-02-27T13:50:00+00:00"Yatagawa, Yuri"https://www.zbmath.org/authors/?q=ai:yatagawa.yuriSummary: We compute the characteristic cycle of a rank one sheaf on a smooth surface over a perfect field of positive characteristic. We construct a canonical lifting on the cotangent bundle of Kato's logarithmic characteristic cycle using ramification theory and prove the equality of the characteristic cycle and the canonical lifting. As corollaries, we obtain a computation of the singular support in terms of ramification theory and the Milnor formula for the canonical lifting.Characteristic class and the \(\varepsilon \)-factor of an étale sheaf.https://www.zbmath.org/1453.140602021-02-27T13:50:00+00:00"Umezaki, Naoya"https://www.zbmath.org/authors/?q=ai:umezaki.naoya"Yang, Enlin"https://www.zbmath.org/authors/?q=ai:yang.enlin"Zhao, Yigeng"https://www.zbmath.org/authors/?q=ai:zhao.yigengThe authors of this article prove a twist formula for the \(\varepsilon\)-factor of a constructible sheaf on a projective smooth variety over a finite field in terms of characteristic class of the sheaf. This formula is a modified version of the formula conjectured by \textit{K. Kato} and \textit{T. Saito} [Ann. Math. (2) 168, No. 1, 33--96 (2008; Zbl 1172.14011)]. They give two applications of the twist formula. First, they prove that the characteristic classes of constructible etale sheaves on projective smooth varieties over a finite field are compatible with proper push-forward. Secondly, they show that the two Swan classes in the literature are the same on proper smooth surfaces over a finite field.
Reviewer: Mouad Moutaoukil (Fès)On Brown polynomials. II.https://www.zbmath.org/1453.120092021-02-27T13:50:00+00:00"Ershov, Yu. L."https://www.zbmath.org/authors/?q=ai:ershov.yurij-leonidovichLet \(v\) be a valuation of a field \(K\) with value group \(G\) and valuation ring \(R\) having maximal ideal \(\mathcal M\). Let \(v^x\) denote the Gaussian prolongation of \(v\) to the rational function field \(K(x)\) defined by \(v^x(\sum_{i}a_ix^i)= \min_i\{v_p(a_i)\}, a_i\in K\). Let \(\phi(x)\in R[x]\) be a monic polynomial which is irreducible module \(\mathcal M\). Given a positive element \(\lambda\) of the divisible closure of \(G\) and some integer \(e>0\), a polynomial \(f(x)\in R[x]\) called a Brown polynomial of the type \((\phi,~e,~\lambda)\) if the \(\phi\)-expansion of \(f\) given by \(f=\sum_{i}f_i\phi^i\), \(f_i\in R[x]\) with deg \(f_i<\) deg \(\phi\) satisfies the following two conditions:
\begin{itemize}
\item[(i)] \(v^x(f_e)=0,~v^x(f_0)=e\lambda\),
\item[(ii)] \(v^x(f_i)\geq (e-i)\lambda\) for \(0< i < e\).
\end{itemize}
In this paper, it is proved that the product of two Brown polynomials and a factor of a Brown polynomial is again a Brown polynomial. It is also shown that when \((K,~v)\) is Henselian, then any monic irreducible polynomial belonging to \(R[x]\) is a Brown polynomial. The proofs are well explained; the paper extends some already known results in this direction.
For Part I see Sib. Math. J. 53, No. 4, 656--658 (2012); translation from Sib. Mat. Zh. 53, No. 4, 819--821 (2012; Zbl 1275.12002).
Reviewer: Sudesh Kaur Khanduja (Mohali)Twisted Alexander polynomial and matrix-weighted zeta function.https://www.zbmath.org/1453.570092021-02-27T13:50:00+00:00"Goda, Hiroshi"https://www.zbmath.org/authors/?q=ai:goda.hiroshiSummary: The twisted Alexander polynomial is an invariant of the pair of a knot and its group representation. Herein, we introduce a digraph obtained from an oriented knot diagram, which is used to study the twisted Alexander polynomial of knots. In this context, we show that the inverse of the twisted Alexander polynomial of a knot may be regarded as the matrix-weighted zeta function that is a generalization of the Ihara-Selberg zeta function of a directed weighted graph.Zeros of \(p\)-adic hypergeometric functions, \(p\)-adic analogues of Kummer's and Pfaff's identities.https://www.zbmath.org/1453.111562021-02-27T13:50:00+00:00"Saikia, Neelam"https://www.zbmath.org/authors/?q=ai:saikia.neelamThe author studies character sums, Gauss sums, Jacobi sums, \(p\)-adic gamma function, \(p\)-adic hypergeometric functions and, \(p\)-adic Kummer-type and Pfaff-type identities. The author gives all the zeros and nonzero values of a family of hypergeometric series in the \(p\)-adic setting. The author also shows that these values of hypergeometric series in the padic setting lead to transformations of hypergeometric series in the \(p\)-adic setting which can be described as \(p\)-adic analogues of Kummer's and Pfaff's linear transformations on classical hypergeometric series. The author interprets certain summation identities for hypergeometric series in the \(p\)-adic setting besides Gaussian hypergeometric series
Reviewer: Yilmaz Simsek (Antalya)The polylog quotient and the Goncharov quotient in computational Chabauty-Kim theory. I.https://www.zbmath.org/1453.110882021-02-27T13:50:00+00:00"Corwin, David"https://www.zbmath.org/authors/?q=ai:corwin.david"Dan-Cohen, Ishai"https://www.zbmath.org/authors/?q=ai:dan-cohen.ishaiArithmetic subderivatives: \(p\)-adic discontinuity and continuity.https://www.zbmath.org/1453.110042021-02-27T13:50:00+00:00"Haukkanen, Pentti"https://www.zbmath.org/authors/?q=ai:haukkanen.pentti"Merikoski, Jorma K."https://www.zbmath.org/authors/?q=ai:merikoski.jorma-kaarlo"Tossavainen, Timo"https://www.zbmath.org/authors/?q=ai:tossavainen.timoSummary: In a previous paper, we proved that the arithmetic subderivative \(D_S\) is discontinuous at any rational point with respect to the ordinary absolute value. In the present paper, we study this question with respect to the \(p\)-adic absolute value. In particular, we show that \(D_S\) is in this sense continuous at the origin if \(S\) is finite or \(p \not\in S\).Mahler coefficients of locally scaling transformations on \(\mathbb{Z}_p \).https://www.zbmath.org/1453.111582021-02-27T13:50:00+00:00"Memić, Nacima"https://www.zbmath.org/authors/?q=ai:memic.nacimaSummary: We establish a formula describing the relation between Mahler and van der Put coefficients of continuous functions on the ring \(\mathbb{Z}_p\) of \(p\)-adic integers. We show its application to a class of Lipschitz functions.Computing the Galois group of a polynomial over a \(p\)-adic field.https://www.zbmath.org/1453.111542021-02-27T13:50:00+00:00"Doris, Christopher"https://www.zbmath.org/authors/?q=ai:doris.christopher