How to find the symmetry group of a polynomial

Question

Say I have a polynomial in $n$ variables of maximum degree $m$. I define its symmetry group to be the subgroup of the permutation group which fixes the polynomial when it acts on the variables. Example: $x^2+y^2+z$ has the subgroup $\langle(x,y)\rangle$. Given such a polynomial, what is the best way to find such a polynomial.

Usage Note: I need this algorithm to terminate in "a short amount of time" for $n$ about 15, $m$ about 10. Short being, I guess, less than 1 minute.

Partial progress: If all I do is test particular elements of $S_n$, this problem is basically intractable as I need roughly $(n-2)!$ tests. Current implementation does basic checks such as the following. In my example $x$ cannot map to $z$ because $z^2$ terms do not exist. In this way we can eliminate much larger fractions of permutations. However, since $S_n$ is so large, this approach is still too slow.