Connections between Graph Isomorphism and Polynomial Equivalence

The paper you linked in the comments - and references therein - already seems to answer your first question.

For your second question: I have little reason to think that there is a theorem of the form "If GI is in P, then [something about derandomizing PIT]." For example, it is possible that GI is in P, but Polynomial Equivalence is not. (Note that PolyEq and RingIso are relatively close to one another, and RingIso is as hard as integer factoring...)

Note that equivalence problems in general tend to be nontrivially harder than their specific instances or testing for triviality. For example, Formula Isomorphism is believed to be intermediate between the first two levels of $\mathsf{PH}$, although testing isomorphism to the trivial formula is $\mathsf{coNP}$-complete. (This is the Boolean analogue of your suggestion of trying to use PolyEq to solve PIT. Of course, one could argue that our intuition for the intermediate status of Formula Iso is largely based on Graph Iso...) As another example, testing for Knottedness is in $\mathsf{NP} \cap \mathsf{coNP}$ (assuming GRH; albeit only recently), but the current best upper bound on testing Knot Equivalence is that it can be done in time that is a tower of 2's of height $c^n$, where $n$ is the number of crossings and $c = 10^{10^6}$ (see here). Although it is possible that both Knottedness and Knot Equivalence are in $\mathsf{P}$, the latter seems significantly harder than the former.

Solved by user129 (1 )
Dec 08, 2015 at 05:08