Sur le théorème du produit

Abstract

We present new sharp effective versions of Faltings’ product theorem. This result, a generalization of Roth’s lemma, shows that if the zeroes of index σ of a multihomogeneous polynomial P have a component Z in common with the zeroes of index σ+ϵ then this Z (subset of a product of projective spaces) is itself a product. Here the index is taken with respect to the degrees δ i of P as weights and the result holds whenever δ i /δ i+1 is big enough in terms of ϵ. Furthermore, effective versions by Evertse and Ferretti bound the degree and height of Z. If m is the number of factors and c the codimension of Z, our result assumes only δ i /δ i+1 ≥(m/ϵ) c . This is better than the previous bounds by a factor c! and we improve in the same way the estimates for degrees and heights. The key point is the use of Samuel multiplicity introduced in these questions by Philippon, through his zero-estimates. The main corollary of the theorem shows that, in the above setting, any component of the zeros of index ϵ are contained in a (non-trivial) product under a similar, more restrictive condition on the degrees of P. We deduce first such a corollary, in the usual manner, with the bound δ i /δ i+1 ≥(mn/ϵ) n (where n is the dimension of the multi-projective space). This condition is the one obtained by Philippon but we get slightly better estimates for degrees and heights (through a direct proof). Last, using a different approach, we prove another corollary with the better bound δ i /δ i+1 ≥(m/ϵ) n ; in this case estimates for degrees and heights are less accurate but not significantly in view of certain applications.