Polynomials with factors of the form ( x q – a ) with roots modulo every integer
type: Item
instance of: scholarly article
title: Polynomials with factors of the form ( x q – a ) with roots modulo every integer
DOI: 10.1080/00927872.2024.2362332
author: Bhawesh Mishra
published in: Communications in Algebra
publication date: 2024-09-10T00:00:00Z
review text: A set \(S\subseteq \mathbb{Z}\) is called intersective if given any set \(T \subseteq \mathbb{Z}\) with positive upper density, \(\overline{d}(T) := \displaystyle{\lim_{n\rightarrow \infty}} \sup \frac{|T\cap \{-n,\cdots, -2,-1,0,1,2,\cdots, n|}{2n+1}\), one has \(S \cap\) \((T-T)\not=0,\) where \((T-T):=\{t_1-t_2: t_1, t_2 \in T \}\). A polynomial \(f\) of integer coefficients is called intersective if the set of its values \(\{f(x) : x \in \mathbb{Z}\}\) is an intersective. Intersective polynomials are of interest in additive combinatorics and number theory. It was shown in [\textit{T. Kamae} and \textit{M. Mendès France}, Isr. J. Math. 31, 335--342 (1978; Zbl 0396.10040)], that a polynomial \(f\) is intersective if and only if \(f(x)\equiv 0\) (mod \(m\)) is solvable for every positive integer \(m>1\). The most meaningful case is when a polynomial \(f(x)= 0\) has roots modulo every positive integer, but does not have a linear monic factor \(x-a\). It is not hard to see that if a polynomial \(f\) has an integer root, then \(f\) clearly has roots modulo every positive integer. However, there exist an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots (For more details, see [\textit{B. Mishra}, Am. Math. Mon. 129, No. 2, 178--182 (2022; Zbl 1486.11043)]). Furthermore, by virtue of the Chinese remainder theorem, the congruence \(f(x)\equiv 0\) (mod \(m\)) is solvable for every positive integer \(m>1\) if and only if it is also solvable for all the powers of primes. \textit{D. Berend} and \textit{Y. Bilu} [Proc. Am. Math. Soc. 124, No. 6, 1663--1671 (1996; Zbl 1055.11523)] provided a simple criterion to decide whether or not a polynomial with integer coefficients has a root modulo every non-zero integer.\N\NThe goal in this paper is to establish necessary and sufficient conditions for the polynomial \(\prod_{j=1}^{l}(x^q-a_j)\) to have roots modulo any positive integer. More precisely, the author mainly obtains…
reviewed by: El Hassane Fliouet
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 7909534
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7909534