Restricted Log-Exp-Analytic Power Functions

TL;DR

This paper establishes a preparation theorem for compositions of restricted log-exp-analytic functions with power functions, generalizing Tamm's theorem to a broader class of definable functions.

Contribution

It introduces a new preparation theorem for these functions, extending Tamm's theorem to include power functions within the log-exp-analytic framework.

Findings

Provides a parametric version of Tamm's theorem for the class

Generalizes the theorem for n functions to include power functions

Enhances understanding of the structure of restricted log-exp-analytic functions

Abstract

A preparation theorem for compositions of restricted log-exp-analytic functions and power functions of the form $$h: \mathbb{R} \to \mathbb{R}, x \mapsto \left\{\begin{array}{ll} x^r, & x > 0, \\ 0, & \textnormal{ else, } \end{array}\right.$$ for $r \in \mathbb{R}$ is given. Consequently we obtain a parametric version of Tamm's theorem for this class of functions which is indeed a full generalisation of the parametric version of Tamm's theorem for $\mathbb{R}_{\textnormal{an}}^{\mathbb{R}}$-definable functions.