The addition of binary cubic forms

Abstract

We show that a sum of four non-degenerate binary cubic forms with integral coefficients necessarily possesses a non-trivial rational zero. When each of these binary cubic forms has non-zero discriminant, we are able to obtain bounds on the number, N(P), of integral zeros of the sum inside a box of size P of the shape P 5− ϵ ≪ ϵ N(P) ≪ ϵ P 5+ ϵ . Finally, given two binary cubic forms with non-zero discriminant, we show that almost all integers, lying in those congruence classes permitted by local solubility conditions, are represented as the sum of the aforementioned forms.