I'm trying to find if p(x,y,z,...)=0 has a Gaussian integer root (more specifically, I want to find if p has a Gaussian integer root where the imaginary components are even, but if that can't be done, don't worry about it).
I can use the continuous Newton's Method to experimentally find complex roots of the polynomial, but I'd like to determine if a root of the sort I've specified exists without having to find more than just a few roots experimentally.
Unfortunately, I know absolutely 0 Galois Theory. I'm aware that this is possible if the order of the polynomial is less than or equal to 4, but that's horribly limiting for my problem. In general, the order of my polynomial will be pretty huge.
Can anyone at least point me in the right direction for solving this problem?
Edit: Since my original problem is probably unsolvable:
My full problem is, given a root to a p(x,y,z,...) with integer coefficients, find whether or not p(x,y,z,...) = 1 (mod 2) using only reductions to be that can be done in polynomial time in the number of variables.
I suspect this problem, however, is simply too hard to solve as it is essentially SAT with a bit of extra information.
However, the reason I'm wondering if this is solvable is because it would imply that you can use differential equation solvers (such as analog electrical circuits) to solve hard computation problems.