Hi.

As is known, a polynomial $P \in K[x_1, \dots, x_n]$ is symmetric when permuting its variables always yields the same polynomial. This immediately yields an algorithm $O(n!)$ to check for symmetry of a polynomial.

Are there known algorithms faster than $O(n!)$ (perhaps using other bounds, like the degree) to decide if a polynomial of $n$ variables is symmetric?

Thanks!