Integration and Cell Decomposition in $P$-minimal Structures

TL;DR

This paper proves that in $P$-minimal structures, the class of constructible functions remains closed under integration, extending known results and establishing weak cell decomposition and function preparation without relying on Skolem functions.

Contribution

It introduces a weak form of cell decomposition and function preparation for $P$-minimal structures, enabling integration closure results without Skolem functions.

Findings

Closure of constructible functions under integration in $P$-minimal structures.

A weak cell decomposition theorem independent of Skolem functions.

Rationality of Poincaré series in $P$-minimal expansions.

Abstract

We show that the class of $\mathcal{L}$-constructible functions is closed under integration for any $P$-minimal expansion of a $p$-adic field $(K,\mathcal{L})$. This generalizes results previously known for semi-algebraic and sub-analytic structures. As part of the proof, we obtain a weak version of cell decomposition and function preparation for $P$-minimal structures, a result which is independent of the existence of Skolem functions. %The result is obtained from weak versions of cell decomposition and function preparation which we prove for general $P$-minimal structures. A direct corollary is that Denef's results on the rationality of Poincar\'e series hold in any $P$-minimal expansion of a $p$-adic field $(K,\mathcal{L})$.