$R$-analytic functions

TL;DR

This paper introduces and studies $R$-analytic functions within o-minimal structures over real closed fields, extending properties of real analytic functions to a broader logical and algebraic setting.

Contribution

It defines $R$-analytic and strongly $R$-analytic functions in o-minimal expansions of real closed fields, establishing their properties and applicability in non-standard models.

Findings

$R$-analytic functions share properties with real analytic functions.

Strongly $R$-analytic functions are abundant in models of o-minimal theories.

Analytic cell decomposition extends to non-standard models.

Abstract

We introduce the notion of $R$-analytic functions. These are definable in an o-minimal expansion of a real closed field $R$ and are locally the restriction of a $K$-differentiable function (defined by Peterzil and Starchenko) where $K=R[\sqrt{-1}]$ is the algebraic closure of $R$. The class of these functions in this general setting exhibits the nice properties of real analytic functions. We also define strongly $R$-analytic functions. These are globally the restriction of a $K$-differentiable function. We show that in arbitrary models of important o-minimal theories strongly $R$-analytic functions abound and that the concept of analytic cell decomposition can be transferred to non-standard models.