Valiant's Model: From Exponential Sums to Exponential Products

We study the power of big products for computing multivariate polynomials in a Valiant-like framework. More precisely, we define a new class VΠP0 as the set of families of polynomials that are exponential products of easily computable polynomials. We investigate the consequences of the hypothesis that these big products are themselves easily computable. For instance, this hypothesis would imply that the nonuniform versions of P and NP coincide. Our main result relates this hypothesis to Blum, Shub and Smale's algebraic version of P versus NP. Let K be a field of characteristic 0. Roughly speaking, we show that in order to separate PK from NPK using a problem from a fairly large class of “simple” problems, one should first be able to show that exponential products are not easily computable. The class of “simple” problems under consideration is the class of NP problems in the structure (K,+,–,=), in which multiplication is not allowed.