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  of families of polynomials and the Blum–Shub–Smale model of computation over C. Roughly speaking, a family of polynomials is in  if its coefficients can be computed in polynomial space. Our main result is that if (uniform, constant-free)  families can be evaluated efficiently, then the class  of decision problems that can be solved in parallel polynomial time over the complex field collapses to . As a result, one must first be able to show that there are  families which are hard to evaluate in order to separate  from , or even from .