Constant Terms of Powers of a Laurent Polynomial

We prove a special case of a conjecture of Mathieu ((Mat]). Conjecture 1 (Mathieu) Let K be a connected real compact Lie group. Let f and g be K-nite functions on K. Assume that for all n 1 the constant term of f n vanishes. Then for all but nitely many n the constant term of f n g also vanishes. Here the constant term Cst(f) of f is deened as the average Z K f(k) dk of f over K, the integral of f with respect to the Haar measure, normalized so that R K dk = 1. In the canonical decomposition of C K] in matrix coeecients of irreducible nite-dimensional representations of K, the constant term is given by the number Cst(f), which explains the name. In this paper we prove the conjecture for commutative K. Therefore, from now on K is a real torus and its complexiication is an algebraic torus T of rank`. The ring of K-nite functions is the aane coordinate ring C T] of T. The choice of a Z-basis z 1 ; : : : ; z ` in the character group X (T) = Hom(T; G m), where G m is the multiplicative group of the nonzero complex numbers, leads to an identiication of T with G m ` and of Z ` with X (T), under which p 2 Z ` corresponds to the character (Laurent monomial) z p := Q ` i=1 z i p i. A K-nite function h then is nothing else than a Laurent polynomial in the z i. And Cst(h) is just the constant term of the Laurent polynomial h. Or, if one thinks of C T] as the linear span of the characters, then Cst(h) is the term corresponding with the trivial character. We will actually prove that if Cst(f n) = 0 for all n 1, then the trivial character does not belong to the convex hull of the characters which occur in f with nonzero coeecients, where we view X (T) ' Z ` as a lattice in X (T) R ' R `. 1 One Variable We start with the casè = 1, when T = G m and K is the circle, as it is much more elementary and yet illustrates the method. Theorem 2 Assume that f 2 C z; z ?1 ] is neither a polynomial in z nor …