An abcd theorem over function fields and applications

We provide a lower bound for the number of distinct zeros of a sum 1 + u + v for two rational functions u, v, in term of the degree of u, v, which is sharp whenever u, v have few distinct zeros and poles compared to their degree. This sharpens the "abcd-theorem" of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface x a + y a + z c = 1 contains only finitely many rational or elliptic curves, provided a ≥ 10 4 and c ≥ 2; this provides special cases of a known conjecture of Bogomolov. © Société Mathématique de France.