On the representation of $k$-free integers by binary forms

Abstract

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$ and let $r$ denote the largest degree of an irreducible factor of $F$ over the rationals. Let $k$ be an integer with $k\ge 2$ and suppose that there is no prime $p$ such that $p^k$ divides $F(a,b)$ for all pairs of integers $(a,b)$. Let $R_{F,k}(Z)$ denote the number of $k$-free integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a positive number $C_{F,k}$ such that $R_{F,k}(Z)$ is asymptotic to $C_{F,k}{Z}^{2/d}$ provided that $k$ exceeds $7r/18$ or $(k,r)$ is $(2,6)$ or $(3,8)$.