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.

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