Power Sum Decompositions of Defining Equations of Reflection Arrangements

We determine the Waring rank and a Waring decomposition of the fundamental skew invariant of any complex reflection group whose highest degree is a regular number. This includes all irreducible real reflection groups. Given a homogeneous polynomial f of degree d, the Waring rank of f , denoted r(f), is the smallest positive integer r such that there exist linear forms l1, . . . , lr with f = l d 1+· · ·+lr , and a Waring decomposition is such an expression with length r = r(f). For example, xy = 1 4 ((x+ y) − (x− y)), so r(xy) ≤ 2, and