Suppose that $k$ is a field and I have two ring homomorphisms $f, g: k[x_1, ..., x_m] \to k[y_1, ..., y_n]$. How can I use Gröbner bases (or other computational tools) to compute the subring of elements $a$ such that $f(a)=g(a)$? I have the same question with the domain or codomain replaced by quotients of polynomial algebras
(When I say "compute", I mean actually compute, for example using Sage, Macaulay2, etc. If you can provide an algorithm, I can code it, so I'm looking for an algorithm. I actually don't care about using Gröbner bases, I just want to do the computation.)