S-parts of values of univariate polynomials

Abstract

Let $S=\{p_1,\dots,p_s\}$ be a finite non-empty set of distinct prime numbers, let $f\in \mathbb{Z}[X]$ be a polynomial of degree $n\ge 1$, and let $S'\subseteq S$ be the subset of all $p\in S$ such that $f$ has a root in $\mathbb{Z}_p$. For any non-zero integer $y$, write $y=p_1^{k_1}\dots p_s^{k_s}y_0$, where $k_1,\dots,k_s$ are non-negative integers and $y_0$ is an integer coprime to $p_1,\dots,p_s$. We define the $f$-normalized $S$-part of $y$ by $[y]_{f,S}:=p_1^{k_1 r_{p_1,S}(f)}\dots p_s^{k_s r_{p_s,S}(f)}$, with $r_{p,S}(f)=1$ if $p\in S\setminus S'$ and $r_{p,S}(f)=R_{S'}(f)/R_{p}(f)$ if $p\in S'$, where $R_p(f)$ denotes the largest multiplicity of a root of $f$ in $\mathbb{Z}_p$ and $R_{S'}(f):=\max_{p\in S'} R_p(f)$. For positive real numbers $\varepsilon, B$ with $\varepsilon<R_{S'}(f)/n$, we consider the number $\widetilde{N}(f,S,\varepsilon,B)$ of integers $x$ such that $|x|\le B$ and $0<|f(x)|^{\varepsilon}\le [f(x)]_{f,S}$. We prove that if $s':=\#S'\ge 1$, then $\widetilde{N}(f,S,\varepsilon,B)\asymp_{f,S,\varepsilon} B^{1-(n\varepsilon)/R_{S'}(f)}(\log B)^{s'-1}$ as $B\to \infty$. Moreover, if $f$ has no multiple roots in $\mathbb{Z}_p$ for any $p\in S'$ and $s':=\#S'\ge 2$, then there exists a constant $C(f,S,\varepsilon)>0$ such that $\widetilde{N}(f,S,\varepsilon,B)\sim C(f,S,\varepsilon)\,B^{1-n\varepsilon}(\log B)^{s'-1}$ as $B\to \infty$.