# S-parts of values of univariate polynomials

## TL;DR

This paper investigates the distribution of integers related to the values of univariate polynomials, focusing on the asymptotic behavior of integers satisfying specific divisibility and valuation conditions involving prime sets and polynomial roots.

## Contribution

It introduces a new measure called the $f$-normalized $S$-part and derives asymptotic formulas for the count of integers meeting certain polynomial-related inequalities, extending previous understanding of polynomial value distributions.

## Key findings

- Asymptotic estimate for the count of integers with bounded size satisfying polynomial divisibility conditions.
- Explicit asymptotic formula involving powers of B and logarithmic factors depending on the roots of the polynomial.
- Conditions under which the asymptotic behavior simplifies to a constant multiple of a main term.

## 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 $B\to \infty$.