Stratification theorems for exponential sums in families

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, Fresán and Kowalski (excerpted from arXiv:2109.11961), which provides an elementary intuitive introduction to trace functions in more than one variable over finite fields.