Georgia Institute of Technology College of Sciences

Using some classical results of invariant theory of finite reflection groups, and Lagrange multipliers, we prove that low degree or sparse real homogeneous polynomials which are invariant under the action of a finite reflection group $G$ are nonnegative if they are nonnegative on the hyperplane arrangement $H$ associated to $G$. That makes $H$ a test set for the above kind of polynomials. We also prove that under stronger sparsity conditions, for the symmetric group and other reflection groups, the test set can be much smaller. One of the main questions is deciding if certain intersections of some simply constructed real $G$-invariant varieties are empty or not.