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

TL;DR

This paper investigates the asymptotic count of $k$-free integers represented by a binary form with integer coefficients, establishing precise growth rates under certain conditions on $k$ and the form's factors.

Contribution

It provides new asymptotic formulas for the number of $k$-free integers represented by binary forms, extending previous results to broader cases with specific bounds on $k$ and the form's irreducible factors.

Findings

Asymptotic formula for $R_{F,k}(Z)$ with explicit constant $C_{F,k}$

Conditions on $k$ ensuring the asymptotic holds, including $k > 7r/18$

Identification of special cases $(k,r) = (2,6)$ or $(3,8)$ where results apply

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 $k \geq 2$ 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 $C_{F,k} Z^{\frac{2}{d}}$ provided that $k$ exceeds $ \frac{7r}{18}$ or $(k,r)$ is $(2,6)$ or $(3,8)$.