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!



Nontrivial algorithm to check for polynomial symmetry?
One needs only check $n$ transpositions, since if each of the transpositions $(12),(23), \dots$ preserves a polynomial, then every permutation preserves that polynomial.


