Goss polynomials, q-adic expansions, and Sheats compositions

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