VPSPACE and a transfer theorem over the complex field

Pascal Koiran (1) , Sylvain Perifel (1)

1 LIP - Laboratoire de l'Informatique du Parallélisme

We extend the transfer theorem of [KP2007] 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 of decision problems that can be solved in parallel polynomial time over the complex field collapses to P. 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 from NP over C, or even from PAR.

Mots clés: Computational complexity; Algebraic complexity; Blum-Shub-Smale model; Valiant's model

Domaines: Complexité [cs.CC]

Pascal Koiran, Sylvain Perifel. VPSPACE and a transfer theorem over the complex field. 2007. ⟨ensl-00153701⟩