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Linear equations and multiplicative polynomial equations in infinitely many variables

December 2024
Communications in Algebra

DOI:10.1080/00927872.2024.2438729

Authors:

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Finitely many implies infinitely many

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Melvyn Nathanson

Many mathematical statements have the following form. If something is true for all finite subsets of an infinite set I, then it is true for all of I. This paper describes some old and new results on infinite sets of linear and polynomial equations with the property that solutions for all finite subsets of the set of equations imply the existence of a solution for the infinite set of equations.

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F. Riesz

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Alexander Abian

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Dec 1921

Eduard Helly

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S Banach
Banach S.

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International audience We give a multivariate analog of the type B Eulerian polynomial introduced by Brenti. We prove that this multivariate polynomial is stable generalizing Brenti's result that every root of the type B Eulerian polynomial is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability—a generalization of ... [Show full abstract] real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Nous prèsentons un raffinement multivariè d'un polynôme eulèrien de type B dèfini par Brenti. En prouvant que ce polynôme est stable nous gènèralisons un rèsultat de Brenti selon laquel chaque racine du polynôme eulèrien de type B est rèelle. Notre preuve combine un raffinement de la statistique des descentes pour les permutations signèes avec la stabilitè—une gènèralisation de la propriètè d'avoir uniquement des racines rèelles aux polynômes en plusieurs variables. La connexion est que nos polynômes eulèriens raffinès satisfont une rècurrence donnèe par un opèrateur linèaire qui prèserve la stabilitè.

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May 2023

We consider sparse polynomials in N variables over a finite field, and ask whether they vanish on a set

S^N

, where S is a set of nonzero elements of the field. We see that if for a polynomial f, there is

\mathbf{c}\in S^N

with

f (\mathbf{c}) \neq 0

, then there is such a

\mathbf{c}

in every sphere inside

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Given a finite set W in

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\prod_{i=1}^n V_i

where

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under a linear transformation, that is,

W\stackrel{\lambda}{\to} \prod_{i=1}^n V_i

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\lambda\in Gl_n (\bar{k})

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\end{document}, with respect to a partition of variables \begin{document}

\textbf{x}

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Subramanian Visweswaran

Rings considered in this article are commutative with identity. A subring of a ring is assumed to contain the identity element of the ring. Let S be a multiplicatively closed subset of a ring R satisfying the following property (P): whenever

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Frédéric Ayant

In the present paper we evaluate a multiple integral transformation involving the product of a extension of the Hurwitz-lerch Zeta-function, the multivariable Aleph-function, the multivariable I-function defined by Prasad and general class of polynomials of several variables. The importance of the result established in this paper lies in the fact they involve the Aleph-function of several ... [Show full abstract] variables which is sufficiently general in nature and capable to yielding a large of results merely by specializating the parameters their in.

Last Updated: 02 Jan 2025