Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero

TL;DR

This paper develops a uniform theory of motivic integration over all non-archimedean local fields of characteristic zero, introducing a class of functions stable under key operations and proving uniform rationality results.

Contribution

It introduces a new class of motivic exponential functions stable under integration and Fourier transform, extending previous theories to all characteristic zero fields.

Findings

Functions of motivic exponential class are stable under integration.

Established uniform rationality results across all non-archimedean local fields.

Developed a refined Denef-Pas quantifier elimination for definable sets.

Abstract

Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue field characteristics. We define a class of functions, called functions of motivic exponential class, which we show to be stable under integration and under Fourier transformation, extending results and definitions from previous papers. We prove uniform results related to rationality and to various kinds of loci. A key ingredient is a refined form of Denef-Pas quantifier elimination which allows us to understand definable sets in the value group and in the valued field.