# What happens to complexity classes if all $\#P$ problems have polynomial-time algorithms?

As title says what happens to other complexity classes if all $\#P$ (Sharp-P) problems have polynomial-time algorithms? What happens to PSPACE?

By definition, $\mathbb{P} = \mathbb{NP}$, since any $\mathbb{NP}$ problem could be solved by answering the question "Is the # of accepting paths non-zero", which by assumption, can be calculated in $\mathbb{P}$. As a consequence, the polynomial hierarchy collapses as well.
• I haven't read the result from Bürgisser's book, but it is probably related to writing the permanent as a determinant of some matrix, i.e. determinantal complexity, with some extra depth necessary for the reductions. Also note that $NC^{i}/poly$ are non-uniform versions (equiv. with advice), which I am not aware that are contained in $P$. Finally, the permanent is known to be $\#P$-complete for nonnegative matrices, which one could reduce the positive characteristic case to, if that is what you mean by the function/polynomial distinction. – chazisop Jul 1 '15 at 14:14