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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 integercoefficients and with at least two roots in the complex numbers, then for everyinteger $x$ 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 proofuses Baker-type estimates for linear forms in complex logarithms of algebraicnumbers. As an easy application of the $p$-adic Thue-Siegel-Roth theorem weshow that if $f(X)$ has degree $n\geq 2$ and no multiple roots, then aninequality such as (*) holds for all $d>1/n$, provided we do not requireeffectivity of $c$. Further, we show that such an inequality does not holdanymore with $d=1/n$ and sufficiently small $c$. In addition we prove a densityresult, giving for every $\epsilon>0$ an asymptotic estimate with the rightorder 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 ofGross and Vincent, as well as the other results mentioned above, aregeneralized to values of binary forms and decomposable forms at integralpoints. 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.

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