Goss polynomials, q-adic expansions, and Sheats compositions

Abstract

The zeroes of Goss polynomials Gk,Λ(X) for Λ=A:=Fq[T] and similar lattices Λ are studied. Generically, the zero distribution follows a simple pattern governed by the q-adic expansion of k−1. However, if q=pf with f≥2 is a proper power of the prime p, irregularities of the q-adic sum-of-digits function may lead to deviations from this pattern, to irregular zeroes, and an abundance of trivial zeroes of Gk,Λ compared to the generic formula. These phenomena are related to properties of the Sheats compositions of natural numbers n divisible by q−1. Among other things, we give a necessary and sufficient condition for the existence of irregular zeroes of Gk,A and a formula for the vanishing order of Gk,A at x=0.