VPSPACE and a transfer theorem over the complex field
We extend the transfer theorem of (14) to the complex field. That is, we investigate the links between the class VPSPACE of families of polynomials and the Blum-Shub-Smale model of computation over C. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main result is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently, then the class PAR(C) of decision problems that can be solved in parallel polynomial time over the complex field collapses to P(C). As a result. one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate P(C) from NP(C), or even from PAR(C). (C) 2009 Elsevier B.V. All rights reserved.
blum-shub-smale model computational complexity algebraic complexity valiant's model