Exponential-constructible functions in $P$-minimal structures

TL;DR

This paper refines the concept of exponential-constructible functions to ensure stability under integration within P-minimal structures, broadening applicability beyond semi-algebraic and sub-analytic contexts.

Contribution

It introduces a natural refinement of exponential-constructible functions that removes the need for definable Skolem functions, extending stability results to wider P-minimal structures.

Findings

Achieved stability of exponential-constructible functions in P-minimal structures.

Removed the requirement of definable Skolem functions from proofs.

Extended stability results to intermediate structures between semi-algebraic and sub-analytic languages.

Abstract

Exponential-constructible functions are an extension of the class of constructible functions. This extension was formulated by Cluckers-Loeser in the context of semi-algebraic and sub-analytic structures, when they studied stability under integration. In this paper we will present a natural refinement of their definition that allows for stability results to hold within the wider class of P-minimal structures. One of the main technical improvements is that we remove the requirement of definable Skolem functions from the proofs. As a result, we obtain stability in particular for all intermediate structures between the semi-algebraic and the sub-analytic languages.