Goss polynomials, q-adic expansions, and Sheats compositions

Abstract

The zeroes of Goss polynomials $G_{k,\Lambda }(X)$ for $Λ= A \defeq \mathbb{F}_{q}[T]$ and similar lattices $\Lambda$ are studied. Generically, the zero distribution follows a simple pattern governed by the $q$-adic expansion of $k-1$. However, if $q=p^f$ with $f\ge 2$ 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 $G_{k,\Lambda }$ 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 $G_{k,A}$ and a formula for the vanishing order of $G_{k,A}$ at $x=0$.

48 pages