Goss polynomials, q-adic expansions, and Sheats compositions

TL;DR

This paper investigates the zero distribution of Goss polynomials over function fields, revealing how $q$-adic expansions and Sheats compositions influence zero patterns, especially in cases with irregularities when $q$ is a prime power.

Contribution

It provides a detailed analysis of zero distributions of Goss polynomials, including conditions for irregular zeroes and formulas for their vanishing orders, advancing understanding of their arithmetic properties.

Findings

Zero distribution follows a pattern governed by $q$-adic expansion of $k-1$.

Irregularities occur when $q=p^{f}$ with $f extgreater 1$, affecting zero patterns.

Necessary and sufficient conditions for irregular zeroes are established.

Abstract

The zeroes of Goss polynomials $G_{k, \Lambda}(X)$ for $\Lambda = A \defeq \mathbb{F}_{q}[T]$ 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 $f \geq 2$ 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 $x=0$.