Stratification theorems for exponential sums in families

TL;DR

This paper reviews stratification theorems for exponential sums over finite fields, discusses their applications, and proves uniform variants in families, with an introductory appendix on trace functions.

Contribution

It introduces uniform variants of stratification theorems in families, extending previous results both algebraically and analytically.

Findings

Stratification theorems by Katz-Laumon and Fouvry-Katz are surveyed.

Uniform variants of these theorems in families are established.

An elementary introduction to trace functions in multiple variables is included.

Abstract

We survey some of the stratification theorems concerning exponential sums over finite fields, especially those due to Katz-Laumon and Fouvry-Katz, as well as some of their applications. Moreover, motivated partly by recent work of Bonolis, Pierce and Woo (arXiv:2505.11226), we prove that these stratification statements admit uniform variants in families, both algebraically and analytically. The paper includes an Appendix by Forey, Fres\'an and Kowalski (excerpted from arXiv:2109.11961), which provides an elementary intuitive introduction to trace functions in more than one variable over finite fields.