Number of integers represented by families of binary forms

TL;DR

This paper extends previous work on counting integers represented by cyclotomic forms to more general binary forms, providing asymptotic bounds and applying results to specific families.

Contribution

It introduces new asymptotic upper bounds for the common values of non-isomorphic binary forms of degree greater than 3, broadening the scope of integer representation analysis.

Findings

Established asymptotic upper bounds for common values of binary forms

Extended previous results to more general families of forms

Applied bounds to specific examples of binary form families

Abstract

We extend our previous results on the number of integers which are values of some cyclotomic form of degree larger than a given value (see \cite{FW1}), to more general families of binary forms with integer coefficients. Our main ingredient is an asymptotic upper bound for the cardinality of the set of values which are common to two non isomorphic binary forms of degree greater than $3$. We apply our results to some typical examples of families of binary forms.