Advertisement
Published in:
2006 | OriginalPaper | Chapter
Valiant’s Model: From Exponential Sums to Exponential Products
Authors : Pascal Koiran, Sylvain Perifel
Published in: Mathematical Foundations of Computer Science 2006
Publisher: Springer Berlin Heidelberg
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
We study the power of big products for computing multivariate polynomials in a Valiant-like framework. More precisely, we define a new class
V
ΠP
0
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 P
K
from NP
K
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.