Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

Streszczenie

Let $S = \{p_1, \ldots , p_s\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m = p_1^{a_1} \ldots p_s^{a_s} b$, where $a_1, \ldots , a_s$ are non-negative integers and $b$ is an integer relatively prime to $p_1 \cdots p_s$. Then we define the $S$-part $[m]_S$ of $m$ by $[m]_S := p_1^{a_1} \ldots p_s^{a_s}$. In 2013, Gross and Vincent proved that if $f(X)$ is a polynomial with integral coefficients with at least two distinct roots, then there exist effectively computable positive numbers $\kappa_1$ and $\kappa_2$, depending only on $f(X)$ and $S$, such that \begin{equation}\label{ggg}\tag{$*$} [f(x)]_S \lt \kappa_2 |f(x)|^{1 - \kappa_1} \end{equation} for every integer $x$ with $f(x)\not=0$. Their proof uses a Baker-type estimate for logarithmic forms. Under the additional hypotheses that $f(X)$ has degree $n\geq 2$ and no multiple roots, we deduce an ineffective analogue of \eqref{ggg}, with instead of $1-\kappa_1$ an exponent ${1}/{n}+\epsilon$ for every $\epsilon \gt 0$ and instead of $\kappa_2$ an ineffective number depending on $f(X)$, $S$ and $\epsilon$. This is in fact an easy application of the $p$-adic Thue–Siegel–Roth Theorem. We show that the exponent ${1}/{n}$ is best possible. Lastly, we give an estimate for the density of the set of integers $x$ for which $[f(x)]_S$ is large, i.e., for every small $\epsilon \gt 0$ we estimate in terms of $B$ the number of integers $x$ with $|x|\leq B$ such that $[f(x)]_S\geq |f(x)|^{\epsilon}$.

We considerably extend both the result of Gross and Vincent, its ineffective analogue, and the density result by proving similar results for the $S$-parts of values of homogeneous binary forms and, more generally, of values of decomposable forms at integer points, under suitable assumptions.