Submission date: 15 December 2021

Abstract:

In this article we give strong versions for preparation theorems for ℝ_{an,exp}-definable functions outgoing from methods of Lion and Rolin (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 ℝ_{an,exp}-definable function is piecewise given by L_{an}(exp,log)-terms where L_{an}(exp,log) denotes the language of ordered rings augmented by all restricted analytic functions, the global exponential and the global logarithm. So our 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 ℝ_{an,exp}-definable functions as compositions of log-analytic functions and the global exponential.

Mathematics Subject Classification: 03C64, 32B20, 33B10, 26A09

Keywords and phrases:

Full text arXiv 2112.08161: pdf, ps.