Test sets for nonnegativity of polynomials invariant under a finite reflection group

A set S⊂Rⁿ is a nonnegativity witness for a set U of homogeneous polynomials if F in U is nonnegative on Rⁿ if and only if it is nonnegative at all points of S. We prove that the union of the hyperplanes perpendicular to the elements of a root system Φ⊆Rⁿ is a witness set for nonnegativity of forms of low degree which are invariant under the reflection group defined by Φ. We prove that our bound for the degree is sharp for all reflection groups which contain multiplication by -1. We then characterize subspaces of forms of arbitrarily high degree where this union of hyperplanes is a nonnegativity witness set. Finally we propose a conjectural generalization of Timofte's half-degree principle to finite reflection groups.