Permutation groups as symmetry groups of polynomials
If $X$ is a graph, define the polynomial $p_X$ to be the sum of the monomials $x_ix_j$, where $ij$ runs over the edges of $X$. (The polynomial in the question is the polynomial of a perfect matching.) Then the automorphism group of this polynomial (in the sense defined above) is the automorphism group of the graph $X$. It has long been known that every finite group is the full automorphism group of a finite graph. For background, see the wikipedia article on Frucht's theorem.