Stratification for multiplicative character sums

TL;DR

This paper establishes a stratification framework for families of algebraic multiplicative character sums, analyzing how the sums' weights vary across parameter spaces with respect to offsets of rational functions.

Contribution

It introduces a stratification result for families of multiplicative character sums, providing bounds on the codimension of parameter space strata where sums attain maximum weight.

Findings

Stratification bounds depend on parameters n, r, and j.

Maximum weight strata have codimension at least j times a function of r and n.

Results apply to families with rational functions offset by translations.

Abstract

We prove a stratification result for certain families of $n$-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of $r$ multiplicative characters evaluated at rational functions, and the families (with $nr$ parameters) are obtained by allowing each of the $r$ rational functions to be replaced by an "offset", i.e. a translate, of itself. For very general such families, we show that the stratum of the parameter space on which the character sum has maximum weight $n+j$ has codimension at least $j\lfloor(r-1)/2(n-1)\rfloor$ for $1\le j\le n-1$ and $\lceil nr/2\rceil$ for $j=n$.