Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the elementary theory of the real exponential field

Alessandro Berarducci, Francesco Gallinaro

Abstract

Assuming Schanuel's conjecture, we prove that the complete theory $T_{\exp}$ of the real exponential field is axiomatized by the axioms of definably complete exponential fields satisfying $\exp' = \exp$. This implies the result of Macintyre and Wilkie that, under the same conjecture, $T_{\exp}$ is decidable. Our approach is based on the model completeness of a similar set of axioms for the exponential function restricted to $(-1,1)$, which we prove unconditionally.

On the elementary theory of the real exponential field

Abstract

Assuming Schanuel's conjecture, we prove that the complete theory of the real exponential field is axiomatized by the axioms of definably complete exponential fields satisfying . This implies the result of Macintyre and Wilkie that, under the same conjecture, is decidable. Our approach is based on the model completeness of a similar set of axioms for the exponential function restricted to , which we prove unconditionally.
Paper Structure (16 sections, 39 theorems, 59 equations)

This paper contains 16 sections, 39 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Model-theoretic preliminaries
  4. Definable completeness and o-minimality
  5. Exponential fields
  6. Convex valuation rings
  7. Tangent spaces and regular points
  8. Restricted exponential polynomials
  9. Exponential algebraicity
  10. Germs of polynomially bounded functions
  11. Ax's theorem
  12. Khovanskii points are bounded
  13. Model completeness
  14. Lifting Khovanskii points from the residue field
  15. Completeness
  16. ...and 1 more sections

Key Result

Proposition 3.1

Let $M$ be a definably complete expansion of an ordered field and let $f:(a,b) \to M$ be a definable function of class $C^n$. Then for every $x, x_0\in (a,b)$ for some $\xi$ in the interval between $x$ and $x_0$.

Theorems & Definitions (112)

  • Proposition 3.1: Servi2008
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Definition 4.4
  • Definition 4.5
  • Remark 4.6
  • Definition 4.7
  • ...and 102 more