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Relative differential closure in Hardy fields

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

TL;DR

The paper resolves Boshernitzan's conjecture by showing that the intersection of all maximal analytic Hardy fields equals the intersection of all maximal Hardy fields, i.e., $\text{E}=\text{E}^ ext{infty}=\text{E}^\omega=\text{D}(\mathbb{Q})$. It develops a framework of relative differential closure in Hardy fields, supported by a Hensel-type lemma for Hahn fields and by leveraging differential-intermediate-value properties (DIVP) together with $\omega$-free, newtonian structure. The core idea is to relate differential-closure inside extensions to conditions on constants and exponential closures, enabling a precise description of when a Hardy-field extension is differential-closed. The results yield a canonical view of tameness and differential-algebraic closure in Hardy-field extensions and delineate the boundaries of these techniques via explicit counterexamples. Altogether, the work provides a robust, model-theoretic and differential-algebraic foundation for understanding maximal Hardy-field extensions and their intersections.

Abstract

We study relative differential closure in the context of Hardy fields. Using our earlier work on algebraic differential equations over Hardy fields, this leads to a proof of a conjecture of Boshernitzan (1981): the intersection of all maximal analytic Hardy fields agrees with that of all maximal Hardy fields. We also generalize a key ingredient in the proof, and describe a cautionary example delineating the boundaries of its applicability.

Relative differential closure in Hardy fields

TL;DR

The paper resolves Boshernitzan's conjecture by showing that the intersection of all maximal analytic Hardy fields equals the intersection of all maximal Hardy fields, i.e., . It develops a framework of relative differential closure in Hardy fields, supported by a Hensel-type lemma for Hahn fields and by leveraging differential-intermediate-value properties (DIVP) together with -free, newtonian structure. The core idea is to relate differential-closure inside extensions to conditions on constants and exponential closures, enabling a precise description of when a Hardy-field extension is differential-closed. The results yield a canonical view of tameness and differential-algebraic closure in Hardy-field extensions and delineate the boundaries of these techniques via explicit counterexamples. Altogether, the work provides a robust, model-theoretic and differential-algebraic foundation for understanding maximal Hardy-field extensions and their intersections.

Abstract

We study relative differential closure in the context of Hardy fields. Using our earlier work on algebraic differential equations over Hardy fields, this leads to a proof of a conjecture of Boshernitzan (1981): the intersection of all maximal analytic Hardy fields agrees with that of all maximal Hardy fields. We also generalize a key ingredient in the proof, and describe a cautionary example delineating the boundaries of its applicability.

Paper Structure

This paper contains 6 sections, 52 theorems, 42 equations.

Table of Contents

  1. Preliminaries
  2. Hensel's Lemma for Analytic Functions on Hahn Fields
  3. Relative Differential Closure
  4. Revisiting [ADH, 16.0.3]
  5. Relative Differential Closure in $H$-fields
  6. Relative Differential Closure in Hardy Fields

Key Result

Theorem 1

Given any Hardy field $H$, polynomial $P\in H[Y_0,\dots,Y_n]$ and $f,g \in H$ with $f<g$ and $P(f,f',\dots,f^{(n)})<0<P(g,g',\dots,g^{(n)})$, there is a $y$ in a Hardy field extension of $H$ such that $f<y<g$ and $P(y,y',\dots,y^{(n)})=0$.

Theorems & Definitions (82)

  • Theorem
  • Lemma 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 1.5
  • Lemma 1.6
  • proof
  • Lemma 1.7
  • Proposition 1.8
  • ...and 72 more