Recent zbMATH articles in MSC 12https://zbmath.org/atom/cc/122021-01-08T12:24:00+00:00WerkzeugReducibility of polynomials after a polynomial substitution.https://zbmath.org/1449.120012021-01-08T12:24:00+00:00"Drungilas, Paulius"https://zbmath.org/authors/?q=ai:drungilas.paulius"Dubickas, Arturas"https://zbmath.org/authors/?q=ai:dubickas.arturasLet \(K\) be a field, and let \(f \in K[x]\) be a polynomial of degree \(d \geq 3\) which is irreducible over \(K.\) \textit{M. Ulas} [J. Number Theory 202, 37--59 (2019; Zbl 1435.11068)] raised the problem of the existence of a polynomial of degree \(\leq d-1\) such that the composition polynomial \(f(g(x))\) is reducible in \(K.\) He proved the existence of such polynomial in case of \(d\leq 4.\) Here the authors solve the above problem.
There is an integer \(\ell\) in the range \(2 \leq \ell\leq d-1\) and a polynomial \(h\in K[x]\) of degree \(d\ell\) such that \(f(h(x))\) of degree \(d\ell\) is reducible in \( K[x].\) In particular, for any \(K\) and \(f\) as above, there is \(h\in K[x]\) of degree \(\ell= d-1\) such that \(f(h(x))\) of degree \(d(d-1)\) has an irreducible factor \(f^{*}(x) := x^df(x^{-1}) \in K[x]\) of degree \(d.\)
It is also shown that for any non-constant polynomial \(g \in K[x],\) the polynomial \(f(g(x))\) is irreducible over \( K\) if and only if for some root \(\alpha\) of \(f\) the polynomial \(g(x)-\alpha\) is irreducible over \(K(\alpha).\)
The authors also characterized all quartic polynomials \(f \in K[x],\) where \(K\) is a field of characteristic zero, for which \(f(g(x))\) remains irreducible over \(K\) under any quadratic substitution \(g \in K[x].\) This characterization is given in terms of K-rational points on an elliptic curve of genus 1.
As a corollary, they prove that the polynomial \(g(x)^4 + 1\) is irreducible over \(\mathbb{Q}\) for any quartic polynomial \( g \in \mathbb{Q}[x].\)
Reviewer: Piroska Lakatos (Debrecen)Elliptic threshold secret division scheme.https://zbmath.org/1449.140082021-01-08T12:24:00+00:00"Spel'nikov, A. B."https://zbmath.org/authors/?q=ai:spelnikov.a-bSummary: The new aspect of proactive systems realization was developed based on elliptic curves arithmetic. Neural network model of secret division scheme at elliptic curve is introduced. Prolongation mechanism of the scheme was developed. Neural network model of secret regeneration is introduced. Method of construction and functioning report of the scheme were developed. Calculation of safe time period of secret generator settings existence is introduced. Calculation of complexity and time needed for prolongation of this model is introduced.Survey on the Kakutani problem in \(p\)-adic analysis. I.https://zbmath.org/1449.120022021-01-08T12:24:00+00:00"Escassut, Alain"https://zbmath.org/authors/?q=ai:escassut.alainSummary: Let \(\mathbb{K}\) be a complete ultrametric algebraically closed field and let \(A\) be the Banach \(\mathbb{K}\)-algebra of bounded analytic functions in the ``open'' unit disk \(D\) of \(\mathbb{K}\) provided with the Gauss norm. Let \(\operatorname{Mult}(A,\|.\|)\) be the set of continuous multiplicative semi-norms of \(A\) provided with the topology of pointwise convergence, let \(\operatorname{Mult}_m(A,\|.\|)\) be the subset of the \(\Phi \in \operatorname{Mult}(A,\|.\|)\) whose kernel is a maximal ideal and let \(\operatorname{Mult}_1(A,\|.\|)\) be the subset of the \(\Phi \in \operatorname{Mult}(A,\|.\|)\) whose kernel is a maximal ideal of the form \((x - a)A\) with \(a \in D\). By analogy with the Archimedean context, one usually calls ultrametric corona problem, or ultrametric Kakutani problem the question whether \(\operatorname{Mult}_1(A,\|.\|)\) is dense in \(\operatorname{Mult}_m(A,\|.\|)\). In order to recall the study of this problem that was made in several successive steps, here we first recall how to characterize the various continuous multiplicative semi-norms of \(A\), with particularly the nice construction of certain multiplicative semi-norms of \(A\) whose kernel is neither a null ideal nor a maximal ideal, due to J. Araujo. Here we prove that multbijectivity implies density. The problem of multbijectivity will be described in a further paper [Part II, Sarajevo J. Math. 16(29), No. 1, 55--70 (2020; Zbl 07258265)].A generalization of the theorem of Von Staudt-Hua-Buekenhout-Cojan in the real \(\overset{=}{\partial}-\mathcal{F}\mathbb{R}^k_{td}\), \(1\le k\le 2n+1\), space on real geometric projective \(P_k\), \(1\le k\le 2n+1\), finite dimensional space. II.https://zbmath.org/1449.510012021-01-08T12:24:00+00:00"Cojan, Stelian Paul"https://zbmath.org/authors/?q=ai:cojan.stelian-paulSummary: Quite often it is possible to discover an alternative way to define a geometric locus which is totally different from the original one. When this is possible we obtain new interesting insight on the geometric object analogous at the improvement achieved when different ways to prove a given theorem are discovered. The purpose of our article is to describe some well-known loci using an alternative approach.
For Part I see [the author, Int. J. Geom. 7, No. 2, 50--58 (2018; Zbl 1412.58002)].Representation of the elements of the finite field \(\mathbb{F}_p\) by fractions.https://zbmath.org/1449.110062021-01-08T12:24:00+00:00"Louboutin, S."https://zbmath.org/authors/?q=ai:louboutin.stephane-r"Murchio, A."https://zbmath.org/authors/?q=ai:murchio.aThe authors reword Thue's Lemma, see e.g. \textit{V. Shoup} [A computational introduction to number theory and algebra. 2nd ed. Cambridge: Cambridge University Press (2009; Zbl 1196.11002)] to state that for any odd prime \(p\), \[ (\mathbb{Z}/p \mathbb{Z})^* = \{\pm a/b \mid 1 \le a, b < \sqrt{p}\}.\] However, when \(p\) is replaced by \(n= 2p\) (and \(p>3\)), by considering the element \(p-2\) in \((\mathbb{Z}/p \mathbb{Z})^*\) they show that the bound of \(\sqrt{n}\) must here be replaced by at least the ceiling of \(\dfrac{p+3-\frac{p}{3}}{3}\), where \( \frac{p}{3}\) is the Legendre symbol, and give a conjecture for the exact bound in general.
Reviewer: Thomas A. Schmidt (Corvallis)On constructing two classes of permutation polynomials over finite fields.https://zbmath.org/1449.111062021-01-08T12:24:00+00:00"Cheng, Kaimin"https://zbmath.org/authors/?q=ai:cheng.kaiminSummary: In this paper, we construct two classes of permutation polynomials over finite fields. First, by one well-known lemma proposed by previous researchers, we characterize one class of permutation polynomials of the finite field. Second, by using the onto property of functions related to the elementary symmetric polynomial in multivariable and the general trace function, we construct another class of permutation polynomials of the finite field. This extends the results of previous researchers to the more general cases.Extensions of Vieira's theorems on the zeros of self-inversive polynomials.https://zbmath.org/1449.300102021-01-08T12:24:00+00:00"Losonczi, László"https://zbmath.org/authors/?q=ai:losonczi.laszloSummary: Recently \textit{R. S. Vieira} [Ramanujan J. 42, No. 2, 363--369 (2017; Zbl 1422.30013) ] found sufficient conditions for self-inversive polynomials to have some of their zeros on the unit circle. We extend his results by giving the location of those zeros. In case of fourth degree real reciprocal polynomials we compare Vieira's sufficient conditions with the necessary and sufficient conditions obtained by help of Chebyshev transformation.