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arxiv: 2412.10764 · v2 · submitted 2024-12-14 · 🧮 math.LO · math.CA· math.DS

Relative differential closure in Hardy fields

Matthias Aschenbrenner , Lou van den Dries , Joris van der Hoeven This is my paper

Pith reviewed 2026-05-23 07:15 UTC · model grok-4.3

classification 🧮 math.LO math.CAmath.DS
keywords Hardy fieldsrelative differential closuremaximal Hardy fieldsanalytic Hardy fieldsBoshernitzan conjecturealgebraic differential equations
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The pith

The intersection of all maximal analytic Hardy fields equals the intersection of all maximal Hardy fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the common elements shared by every maximal analytic Hardy field are exactly those shared by every maximal Hardy field. This proves a conjecture posed by Boshernitzan in 1981. The argument proceeds by studying relative differential closure and invoking prior results on algebraic differential equations over Hardy fields. It also generalizes one key step in the argument and supplies an example that marks where the technique ceases to apply.

Core claim

Using results on algebraic differential equations over Hardy fields, the relative differential closure leads to the equality of the intersections of all maximal analytic Hardy fields and all maximal Hardy fields, proving Boshernitzan's conjecture.

What carries the argument

relative differential closure, the smallest extension of a Hardy field that remains a Hardy field and is closed under solutions of algebraic differential equations over the base field

If this is right

  • The equality of intersections holds for the analytic and general cases alike.
  • A generalized version of the key ingredient applies to other subclasses of Hardy fields.
  • The cautionary example identifies precise limits on when relative differential closure preserves the relevant intersection property.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same intersection equality may hold for other restricted families of Hardy fields defined by growth or regularity conditions.
  • The technique could be tested on concrete germs of functions to produce explicit generators for the common intersection.
  • Similar closure arguments might apply to uniqueness questions for intersections in broader classes of differential fields.

Load-bearing premise

The authors' earlier work on algebraic differential equations over Hardy fields applies directly to establish the equality of intersections when restricting to analytic fields, without additional conditions on the relative differential closure.

What would settle it

An explicit maximal analytic Hardy field containing an element absent from the intersection of all maximal Hardy fields (or the converse) would disprove the claimed equality.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript studies relative differential closure in Hardy fields. Building on the authors' prior results about algebraic differential equations over Hardy fields, it proves Boshernitzan's 1981 conjecture that the intersection of all maximal analytic Hardy fields coincides with the intersection of all maximal Hardy fields. The paper also generalizes a key technical ingredient used in the argument and supplies a cautionary example that delineates the boundaries of applicability of the method.

Significance. If the central argument holds, the work resolves a longstanding conjecture in the theory of Hardy fields with connections to o-minimality and asymptotic analysis. The explicit generalization of a prior lemma and the inclusion of a cautionary counter-example that bounds the method's scope are positive features that increase the result's utility. The manuscript appropriately credits the dependence on the authors' earlier papers on differential equations in Hardy fields.

major comments (1)
  1. [§4] §4 (proof of Boshernitzan conjecture): The argument invokes the authors' earlier theorems on algebraic differential equations over general Hardy fields to conclude that the two intersections coincide. For the restriction to analytic Hardy fields, it is necessary to verify that the relative differential closure operation maps analytic fields to analytic fields (or that the relevant solutions remain analytic). The cautionary example in §5 indicates that this preservation is not automatic; the manuscript should contain an explicit check or additional hypothesis confirming that the specific fields arising in the proof satisfy the required analyticity condition.
minor comments (2)
  1. [§2] Notation for relative differential closure is introduced without a dedicated definition paragraph; a displayed definition early in §2 would improve readability.
  2. [§3] The statement of the generalized key ingredient (presumably Theorem 3.1 or Lemma 3.2) should include a precise comparison with the version appearing in the cited earlier paper.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and for identifying a point requiring clarification in the proof of Boshernitzan's conjecture. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (proof of Boshernitzan conjecture): The argument invokes the authors' earlier theorems on algebraic differential equations over general Hardy fields to conclude that the two intersections coincide. For the restriction to analytic Hardy fields, it is necessary to verify that the relative differential closure operation maps analytic fields to analytic fields (or that the relevant solutions remain analytic). The cautionary example in §5 indicates that this preservation is not automatic; the manuscript should contain an explicit check or additional hypothesis confirming that the specific fields arising in the proof satisfy the required analyticity condition.

    Authors: We agree that an explicit verification of analyticity preservation is needed for the specific fields used in the argument, given the counter-example in §5. In the revised manuscript we will add a short paragraph immediately following the invocation of the earlier theorems in §4. This paragraph will confirm that the relative differential closures arising here remain analytic by appealing to the fact that the algebraic differential equations solved are of order at most 1 and have analytic coefficients in the base field (a property inherited from the maximal analytic Hardy fields under consideration). We will also note that the cautionary example in §5 involves equations of higher order or non-analytic coefficients, which are not encountered in the present proof. revision: yes

Circularity Check

0 steps flagged

No circularity detected; central claim rests on prior independent results plus explicit generalization

full rationale

The derivation invokes the authors' earlier papers on algebraic differential equations over (general) Hardy fields to establish the equality of intersections for the analytic case. This is a standard citation of prior work rather than a self-referential loop: the earlier results predate the present manuscript, the current paper adds a generalization of a key ingredient, and it supplies a cautionary example that explicitly delineates applicability boundaries. No equation or step reduces by construction to a fitted parameter, self-definition, or an unverified self-citation chain. The proof therefore remains externally grounded and does not collapse to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard definitions of Hardy fields and differential closures from prior literature; no new free parameters, ad hoc axioms, or invented entities are introduced.

axioms (1)
  • domain assumption Hardy fields are ordered differential fields consisting of germs of real-valued functions at infinity
    Foundational definition used throughout the field and referenced in the abstract.

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Reference graph

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