On the representation of integers by binary forms

TL;DR

This paper establishes an asymptotic formula for counting integers represented by a binary form with integer coefficients, degree at least 3, and non-zero discriminant, as their absolute value grows.

Contribution

It proves that the number of integers represented by such binary forms grows asymptotically like a constant times Z^{2/d}.

Findings

Asymptotic formula for R_F(Z) as Z approaches infinity.

Identification of the constant C_F depending on the form.

Extension of classical results to forms with degree at least 3.

Abstract

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$. Let $R_F(Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a positive number $C_F$ such that $R_F(Z)$ is asymptotic to $C_F Z^{\frac{2}{d}}$.