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Maximal Hardy Fields

Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven

Abstract

We show that all maximal Hardy fields are elementarily equivalent as differential fields, and give various applications of this result and its proof. We also answer some questions on Hardy fields posed by Boshernitzan.

Maximal Hardy Fields

Abstract

We show that all maximal Hardy fields are elementarily equivalent as differential fields, and give various applications of this result and its proof. We also answer some questions on Hardy fields posed by Boshernitzan.
Paper Structure (55 sections, 1328 theorems, 1776 equations)

This paper contains 55 sections, 1328 theorems, 1776 equations.

Table of Contents

  1. Preliminaries
  2. Linear Differential Operators and Differential Polynomials
  3. The Group of Logarithmic Derivatives
  4. The Valuation of Differential Polynomials at Infinity$\,({}^{*})$
  5. $\uplambda$-freeness and $\upomega$-freeness
  6. Complements on Linear Differential Operators
  7. Special Elements
  8. Differential Henselianity of the Completion
  9. Complements on Newtonianity
  10. The Universal Exponential Extension
  11. Some Facts about Group Rings
  12. The Universal Exponential Extension
  13. The Spectrum of a Differential Operator
  14. Self-Adjointness and its Variants$\,({}^{*})$
  15. Eigenvalues and Splittings
  16. ...and 40 more sections

Key Result

Theorem A

Let $P(Y)$ be a differential polynomial in a single differential indeterminate $Y$ over $H$, and let $f< g$ in $H$ be such that $P(f) <0 < P(g)$. Then there is a $y$ in a Hardy field extension of $H$ such that $f < y < g$ and $P(y)=0$.

Theorems & Definitions (2339)

  • Theorem A
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Corollary 7
  • Corollary 8
  • Example
  • ...and 2329 more