THE ADDITION OF BINARY CUBIC FORMS

We show that a sum of four non-degenerate binary cubic forms with inte gral coefficients necessarily possesses a non-trivial rational zero. W hen 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 s um inside a box of size P of the shape P5-epsilon much less than(epsil on) N(P) much less than(epsilon) P5+epsilon. Finally, given two binary cubic forms with non-zero discriminant, we show that almost all integ ers, lying in those congruence classes permitted by local solubility c onditions, are represented as the sum of the aforementioned forms.