Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yayi Fu

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 $\mathbb{R}_{an,\exp,\varphi}$.

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

Abstract

We add an analytic trans-exponential function to . We reduce the o-minimality of to the existence of "many" regular values for some definable systems of functions, which is a necessary condition for the o-minimality of .

Paper Structure

This paper contains 18 sections, 15 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Existence of a Trans-Exponential Function
  4. Bounding # of Solutions
  5. O-minimal Preliminaries
  6. The Equation System and Related Sets
  7. Finite # of Non-singular Solutions
  8. Remarks on the Regularity Assumption
  9. O-minimality of $\mathbb{R}_{an,\exp,\varphi}$
  10. Wilkie's Test of o-Minimality
  11. Reducing the Quantifier-Free Formulas
  12. Weak Morse Inequalities
  13. Reducing to Compact Smooth Manifolds
  14. Reducing to Compact Sets
  15. Reducing to Compact Smooth Manifolds
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose that $\varphi$ is an analytic trans-exponential function satisfying the regularity assumption in Theorem rtb. Then $\mathbb{R}_{an,\exp,\varphi}$ is o-minimal and trans-exponential. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (48)

  • Theorem 1.1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.2.1
  • proof
  • Lemma 1.3
  • Definition 2.1
  • Definition 2.2
  • proof
  • ...and 38 more