Addendum to paper on constant terms

In the paper Constant terms in powers of a Laurent polynomial we proved the conjecture of Mathieu for commutative K. In response to a question of Springer we now observe that we may drop the condition that K be connected. So the result still holds for a commutative compact group K. The reason is very simple. For every connected component one can build a generating function F as in the paper. The generating function for the full group is simply the sum of the generating functions for the components. For one component the generating function is either zero (when the origin is outside the Newton polytope) or it is asymptotic to -1/t by the arguments of the paper. Here we have normalized the Haar measure so that a component has measure one. The total generating function is thus asymptotic to m times -1/t, where m is the number of components for which the origin lies on the Newton polytope. The reasoning ends as before. It is amusing to check what this proof amounts to when K is finite. It also becomes clear that there are variations in which K is replaced by a space with a group acting on it.

October 1998
Wilberd van der Kallen