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Constructing $ω$-free Hardy fields

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

TL;DR

This work proves that every Hardy field can be embedded into an $\upomega$-free Hardy field, crystallizing the link between oscillation criteria for second-order linear differential equations and Hardy-field extensions. By leveraging transfinite iterates of logarithms and Riccati-type invariants, the authors build $\upomega$-free extensions and establish that maximal Hardy fields are $\upomega$-free, yielding consequences for translogarithmic germs and coinitiality results in $\operatorname{E}(H)$. The results generalize Boshernitzan’s oscillation criteria beyond differentially algebraic germs and connect to Liouville closures and the broader structure of Hardy-field extensions. Applications include answering Boshernitzan’s questions about translogarithmic elements in maximal Hardy fields and clarifying when perfect hulls coincide with their differential-algebraic counterparts. Overall, the paper deepens the structural understanding of asymptotic differential-algebraic behavior within Hardy fields and provides robust extension tools for oscillation-sensitive analyses.

Abstract

We show that every Hardy field extends to an $ω$-free Hardy field. This result relates to classical oscillation criteria for second-order homogeneous linear differential equations. It is essential in [11], and here we apply it to answer questions of Boshernitzan, and to generalize a theorem of his.

Constructing $ω$-free Hardy fields

TL;DR

This work proves that every Hardy field can be embedded into an -free Hardy field, crystallizing the link between oscillation criteria for second-order linear differential equations and Hardy-field extensions. By leveraging transfinite iterates of logarithms and Riccati-type invariants, the authors build -free extensions and establish that maximal Hardy fields are -free, yielding consequences for translogarithmic germs and coinitiality results in . The results generalize Boshernitzan’s oscillation criteria beyond differentially algebraic germs and connect to Liouville closures and the broader structure of Hardy-field extensions. Applications include answering Boshernitzan’s questions about translogarithmic elements in maximal Hardy fields and clarifying when perfect hulls coincide with their differential-algebraic counterparts. Overall, the paper deepens the structural understanding of asymptotic differential-algebraic behavior within Hardy fields and provides robust extension tools for oscillation-sensitive analyses.

Abstract

We show that every Hardy field extends to an -free Hardy field. This result relates to classical oscillation criteria for second-order homogeneous linear differential equations. It is essential in [11], and here we apply it to answer questions of Boshernitzan, and to generalize a theorem of his.
Paper Structure (8 sections, 118 theorems, 180 equations)

This paper contains 8 sections, 118 theorems, 180 equations.

Table of Contents

  1. Preliminaries on Asymptotic Fields
  2. Germs of Continuous Functions
  3. Germs of Differentiable Functions
  4. Hardy Fields
  5. Upper and Lower Bounds on the Growth of Hardian Germs
  6. Second-Order Linear Differential Equations over Hardy Fields
  7. Extending Hardy Fields to $\upomega$-free Hardy Fields
  8. A Question of Boshernitzan

Key Result

Theorem 1

Every Hardy field is contained in some $\upomega$-free $H$.

Theorems & Definitions (198)

  • Theorem
  • Corollary 1
  • Corollary 2
  • Lemma 1.1
  • proof
  • Corollary 1.2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • ...and 188 more