Towards Trans-Exponential O-minimal Expansion of $(\mathbb{R},+,\cdot, 0, 1 <)$

TL;DR

This paper explores extending o-minimal structures on the real numbers by adding a trans-exponential function, linking o-minimality to the existence of many regular values in definable systems.

Contribution

It introduces a new analytic trans-exponential function to the structure and reduces o-minimality to a condition on regular values, advancing understanding of such expansions.

Findings

Reduced o-minimality to regular value conditions

Linked o-minimality of the expansion to definable system properties

Proposed a framework for analyzing trans-exponential expansions

Abstract

We add an analytic trans-exponential function $\varphi$ to $\mathbb{R}_{an,\exp}$. We reduce the o-minimality of $\mathbb{R}_{an,\exp,\varphi}$ to the existence of "many" regular values for some definable systems of functions, which is a necessary condition for the o-minimality of $\mathbb{R}_{an,\exp,\varphi}$.