Open Access
2014 Index of maximality and Goss zeta function
Victor Bautista-Ancona, Javier Diaz-Vargas
Rocky Mountain J. Math. 44(6): 1763-1780 (2014). DOI: 10.1216/RMJ-2014-44-6-1763


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.


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Victor Bautista-Ancona. Javier Diaz-Vargas. "Index of maximality and Goss zeta function." Rocky Mountain J. Math. 44 (6) 1763 - 1780, 2014.


Published: 2014
First available in Project Euclid: 2 February 2015

zbMATH: 1310.11095
MathSciNet: MR3310947
Digital Object Identifier: 10.1216/RMJ-2014-44-6-1763

Primary: 11M38
Secondary: 06A07 , 11R58 , 14H05

Keywords: Goss zeta function , Newton polygon , Riemann hypothesis

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

Vol.44 • No. 6 • 2014
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