Massively parallel computations on many-variable polynomials

Abstract

We show that a multivariate homogeneous polynomial can be represented on a hypercube in such a way that sums, products and partial derivatives can be performed by massively parallel computers. This representation is derived from the theoretical results of Beauzamy-Bombieri-Enflo-Montgomery [1]. The norm associated with it, denoted by [·], is itself a very efficient tool: when products of polynomials are performed, the best constant in inequalities of the form [PQ]≥C[P][Q] are provided, and the extremal pairs (that is, the pairs of polynomials for which the product is as small as possible) can be identified.