VPSPACE and a transfer theorem over the complex field

TL;DR

This paper extends a transfer theorem to the complex field, linking polynomial space computability of polynomial families to complexity class collapses in the Blum-Shub-Smale model, highlighting the importance of evaluating VPSPACE families.

Contribution

It generalizes the transfer theorem to the complex field and establishes a connection between VPSPACE evaluation complexity and class collapses in the BSS model.

Findings

Efficient evaluation of VPSPACE families implies PAR collapses to P over C.

Separating P from NP over C requires VPSPACE families to be hard to evaluate.

The result emphasizes the role of VPSPACE evaluation complexity in complexity class separations.

Abstract

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.