DIFFERENTIAL IDENTITIES

We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this repr esentation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product . As a corollary, we obtain Bombieri's inequality (originally proved i n [4]), with significant improvements. The hypercube representation of a polynomial was elaborated in order to meet the requests of massivel y parallel computation on the ''Connection Machine'' at Etablissement Technique Central de l'Armement; we see here once again (after [3] and [5]) the theoretical power of the model.