VPSPACE and a Transfer Theorem over the Reals

We introduce a new class VPSPACE of families of polynomials. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main theorem is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class \sf PAR_{\mathbb R}\sf {PAR}_{\mathbb {R}} of decision problems that can be solved in parallel polynomial time over the real numbers collapses to \sfP_{\mathbb R}\sf{P}_{\mathbb {R}}. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate \sfP_{\mathbb R}\sf{P}_{\mathbb {R}} from \sfNP_{\mathbb R}\sf{NP}_{\mathbb {R}}, or even from \sfPAR_{\mathbb R}\sf{PAR}_{\mathbb {R}}.