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 $S^{\prime }\subseteq S$ 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 $p\in S\setminus S^{\prime }$ and $r_{p,S}(f)=R_{S^{\prime }}(f)/R_p(f)$ if $p\in S^{\prime }$, where $R_p(f)$ denotes the largest multiplicity of a root of $f$ in ${\mathbb{Z}}_p$ and $R_{S^{\prime }}(f):={\max }_{p\in S^{\prime }}{R}_p(f)$. For positive real numbers $\epsilon ,B$ with $\epsilon <R_{S^{\prime }}(f)/n$, we consider the number $\tilde{N}(f,S,\epsilon ,B)$ of integers $x$ such that $|x|\le B$ and $0<|f(x){|}^{\epsilon }\le [f(x){]}_{f,S}$. We prove that if $s^{\prime }:=\#S^{\prime }\ge 1$, then $\tilde{N}(f,S,\epsilon ,B){\asymp }_{f,S,\epsilon }{B}^{1-(n\epsilon )/R_{S^{\prime }}(f)}(\log B{)}^{s^{\prime }-1}$ as $B\to \infty$. Moreover, if $f$ has no multiple roots in ${\mathbb{Z}}_p$ for any $p\in S^{\prime }$ and $s^{\prime }:=\#S^{\prime }\ge 2$, then there exists a constant $C(f,S,\epsilon )>0$ such that $\tilde{N}(f,S,\epsilon ,B)\sim C(f,S,\epsilon )B^{1-n\epsilon }(\log B{)}^{s^{\prime }-1}$ as $B\to \infty$.