S-parts of values of univariate polynomials, binary forms and decomposable forms at integral points

TL;DR

This paper investigates bounds on the $S$-part of polynomial and form values at integers, extending previous results with new inequalities, density estimates, and generalizations to binary and decomposable forms using advanced number theory tools.

Contribution

It extends bounds on the $S$-part of polynomial values, generalizes results to binary and decomposable forms, and provides density estimates using modern Diophantine approximation techniques.

Findings

Inequality $[f(x)]_S \

c \

d<1$ holds for polynomials with roots, with $d>1/n$ under certain conditions.

Abstract

Let $S$ be a finite set of primes. The $S$-part $[m]_S$ of a non-zero integer $m$ is the largest positive divisor of $m$ that is composed of primes from $S$. In 2013, Gross and Vincent proved that if $f(X)$ is a polynomial with integer coefficients and with at least two roots in the complex numbers, then for every integer $x$ at which $f(x)$ is non-zero, we have (*) $[f(x)]_S\leq c\cdot |f(x)|^d$, where $c$ and $d$ are effectively computable and $d<1$. Their proof uses Baker-type estimates for linear forms in complex logarithms of algebraic numbers. As an easy application of the $p$-adic Thue-Siegel-Roth theorem we show that if $f(X)$ has degree $n\geq 2$ and no multiple roots, then an inequality such as (*) holds for all $d>1/n$, provided we do not require effectivity of $c$. Further, we show that such an inequality does not hold anymore with $d=1/n$ and sufficiently small $c$. In addition we prove a density result, giving for every $\epsilon>0$ an asymptotic estimate with the right order of magnitude for the number of integers $x$ with absolute value at most $B$ such that $f(x)$ has $S$-part at least $|f(x)|^{\epsilon}$. The result of Gross and Vincent, as well as the other results mentioned above, are generalized to values of binary forms and decomposable forms at integral points. Our main tools are Baker type estimates for linear forms in complex and $p$-adic logarithms, the $p$-adic Subspace Theorem of Schmidt and Schlickewei, and a recent general lattice point counting result of Barroero and Widmer.