On Preparation Theorems for $\mathbb{R}_{an,exp}$-definable functions

TL;DR

This paper establishes enhanced preparation theorems for functions definable in the o-minimal structure ng,exp, utilizing model-theoretic insights to describe these functions piecewise via log-analytic and exponential compositions.

Contribution

It provides stronger versions of preparation theorems for ng,exp-definable functions, building on model theory and log-analytic function analysis.

Findings

Functions are piecewise represented by ng,exp-terms.

Log-analytic functions are key building blocks.

Enhanced preparation theorems improve understanding of ng,exp-definable functions.

Abstract

In this article we give strong versions for preparation theorems for $\mathbb{R}_{an,exp}$-definable functions outgoing from methods of Lion and Rolin ($\mathbb{R}_{an,exp}$ is the o-minimal structure generated by all restricted analytic functions and the global exponential function). By a deep model theoretic fact of Van den Dries, Macintyre and Marker every $\mathbb{R}_{an,exp}$-definable function is piecewise given by $\mathcal{L}_{an}(\exp,\log)$-terms where $\mathcal{L}_{an}(\exp,\log)$ denotes the language of ordered rings augmented by all restricted analytic functions, the global exponential and the global logarithm. The idea is to consider log-analytic functions at first, i.e. functions which are iterated compositions from either side of globally subanalytic functions and the global logarithm, and then $\mathbb{R}_{an,exp}$-definable functions as compositions of log-analytic functions and the global exponential.