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

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.