Index of maximality and Goss zeta function

Abstract

In this article, we define the index of maximality $\mathfrak{m}(y)$ of a positive integer $y$, associated with the vanishing of certain power sums over $\mathbb{F}_q[T]$, related to the set $V_m(y)$ of ``valid'' decompositions of $y=X_1+\cdots+X_m$ of length~$m$. The index of maximality determines the maximum positive integer $m$ for which the sets $V_m(y)$ are not empty. An algorithm is provided to find $V_i(y)$, $1\leq i\leq \mathfrak{m}(y)$ explicitly.

The invariance, under some action, of the index of maximality $\mathfrak{m}(y)$ and of the property of divisibility by $q-1$ of $\ell_q (y)$, the sum of the $q$-adic digits of $y$, implies the invariance of the degree of Goss zeta function; it is illustrated here for two cases.

Finally, we generalize, to all $q$, the properties of an equivalence relation on $\mathbb{Z}_p$, which depends on the Newton polygon of the Goss zeta function.