Friable values of binary forms

Abstract

Let $F\in \mathbb{Z}[X,Y]$ be an integral binary form of degree $g\ge 2$, and let

where as usual $P^{+}(n)$ denotes the largest prime factor of $n$. It is proved that ${\Psi }_F(x,y)\asymp x^2$ for $y=x^{g-2+\epsilon }$ in general, and $y=x^{1/\sqrt{\mathrm{e}}+\epsilon }$ if $g=3$. Better results are obtained if $F$ is reducible.