Polynomials with factors of the form ( x q – a ) with roots modulo every integer
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 the following result:\N\NTheorem 1. \textit Let \(q\) be an odd prime and let \(\{a_{j}\}_{j=1}^{l}\subseteq \mathbb{Z}\setminus \{-1,1\}\) be a finite set of non-zero \(q\)-free integers. Let \(p_1, p_2,\cdots , p_k\), \((k \geq 2)\) be all the primes dividing \(\prod_{j=1}^{l}a_j\) and let \({p_i}^{v_{ij}}\) is the highest power of \(p_i\) dividing \(a_j\) for every \(1 \leq i \leq k\) and \(1 \leq j \leq l\). Then, the polynomial \(\prod_{j=1}^{l}(x^q-a_j)\) is intersective if and only if following conditions are satisfied:\N\begin{itemize}\N\item[1.] The set of hyperplanes \(\{(x_i)_{i=1}^{k}\in \mathbb{F}_{q}^{k} : \displaystyle{\sum_{i=1}^{k}v_{ij}x_i= 0}: 1\leq j\leq l\}\) associated with the set \(\{a_1, a_2,\cdots , a_l\}\) forms a linear covering of \(\mathbb{F}_{q}^{k}\)\N\item[2.] One of the \(a_i\) is a \(qth\) power modulo \(q^q\).\N\item[3.] For every \(1 \leq i \leq k\), there exists \(j \in \{1, 2,\cdots , l\}\) such that \(p_i \nmid a_j\) and \(a_j\) is a \(qth\) power modulo \(p_i\).\N\end{itemize}\N\NIt immediately follows that if \(l \leq q\) and the polynomial \(\prod_{j=1}^{l}(x^q-a_j)\) is intersective, then there exists \(1 \leq j_0 \leq l\) such that \(a_{j_0}\) is a perfect \(qth\) power, and so we need at least \(q+1\) many distinct factors of the form \((x^q-a_i)\) for the polynomial \(\prod(x^q-a_i)\) to be intersective without having integer roots. The author also obtains, as a special case, an equivalence between the intersectivity of the polynomial \(\prod_{j=1}^{l}(x^q-a_j)\) and that of \(\prod_{j=1}^{l}(x^q-rad_q(a_{j}^{c_j}))\), for every \({(c_j)}_{j=1}^l\in {(\mathbb{F}_q\setminus \{0\})}^l\), where \(rad_q({b})\) denotes the \(q\)-free part of the integer \(b\). As applications, illustrative examples of families of intersective polynomials for \(q=3\) and \(q=5\) are provided. Analogous examples for \(q=2\) are given in [\textit{B. Mishra}, Am. Math. Mon. 129, No. 2, 178--182 (2022; Zbl 1486.11043); Integers 22, Paper A15, 11 p. (2022; Zbl 1503.11065)].
scientific article; zbMATH DE number 7909534