Index of maximality and goss zeta function

Abstract

In this article, we define the index of maximality m(y) of a positive integer y, associated with the vanishing of certain power sums over Fq[T], related to the set Vm(y) of "valid" decompositions of y=X1+...+Xm of length m. The index of maximality determines the maximum positive integer m for which the sets Vm(y) are not empty. An algorithm is provided to find Vi(y),1<=i<=m(y) explicitly.

The invariance, under some action, of the index of maximality m(y) and of the property of divisibility by q-1 of ℓq(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 Zp, which depends on the Newton polygon of the Goss zeta function.