Separation Theorems for Group Invariant Polynomials

We study the existence of separation theorems by polynomials that are invariant under a group action. We show that if G is a finite subgroup of (textit{GL}(n,{mathbb {C}})), K is a set in ({mathbb {C}}^{n}) that is invariant under the action of G and z is a point in ({mathbb {C}}^{n}setminus K) that can be separated from K by a polynomial Q, then z can be separated from K by a G-invariant polynomial P. Furthermore, if Q is homogeneous then P can be chosen to be homogeneous. As a particular case, if K is a symmetric polynomially convex compact set in ({mathbb {C}}^{n}) and (znotin K) then there exists a symmetric polynomial that separates z and K.