The addition of binary cubic forms

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

P5-epsilon much less than(epsilon) N(P) much less than(epsilon) P5+epsilon.

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.