Regular Decomposition of Ordinarity in Generic Exponential Sums

TL;DR

This paper generalizes existing decomposition theories for the Newton polygon associated with exponential sums over finite fields, expanding the understanding of their structure and properties.

Contribution

It introduces a unifying framework that generalizes star, parallel hyperplane, and collapsing decompositions as special cases of complete regular decompositions.

Findings

Unified decomposition framework for Newton polygons

Generalization of multiple existing decomposition methods

Enhanced understanding of exponential sums over finite fields

Abstract

In papers published in 1993 and 2004 Wan establishes a decomposition theory for the generic Newton polygon associated to a family of $L$-functions of $n$-dimensional exponential sums over finite fields. In this work we generalize the star, parallel hyperplane and collapsing decomposition, demonstrating that each is a generalization of a complete regular decomposition.