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arxiv: 2403.08039 · v3 · submitted 2024-03-12 · 🧮 math.LO · math.AG

About closed subsets definable in Hensel minimal structures

Krzysztof Jan Nowak This is my paper

Pith reviewed 2026-05-24 03:03 UTC · model grok-4.3

classification 🧮 math.LO math.AG
keywords Hensel minimal structuresclosed definable setszero locuscontinuous definable functionsTietze-Urysohn extensionnon-Archimedean geometrydefinable sets
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The pith

Every closed 0-definable subset of K^n over a Hensel minimal field K is the zero locus of a continuous 0-definable function on K^n.

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

The paper proves two results for closed 0-definable subsets in affine space over Hensel minimal fields. The first shows each such subset equals the zero set of some continuous 0-definable function defined on the whole space. The second gives a definable non-Archimedean form of the Tietze-Urysohn extension theorem. A reader would care because these statements turn questions about closed definable sets into questions about continuous definable functions, which can simplify geometric arguments in the non-Archimedean setting.

Core claim

Over a Hensel minimal field K every closed 0-definable subset A of K^n is the zero locus of a continuous 0-definable function on K^n, and continuous 0-definable functions defined on such a closed set extend to continuous 0-definable functions on the whole space.

What carries the argument

The zero locus of a continuous 0-definable function, which represents every closed 0-definable set once the ubiquity of clopen sets is used together with the prior description of definable sets.

If this is right

  • Questions about closed definable sets reduce to questions about continuous definable functions.
  • Continuous definable functions on closed definable sets extend to the ambient space.
  • Geometric statements in Hensel minimal structures can be rewritten using zero loci of definable functions.
  • The non-Archimedean Tietze-Urysohn property holds in definable form.

Where Pith is reading between the lines

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

  • The same zero-locus representation might be tested in other classes of valued fields that share the clopen-set density property.
  • If the extension theorem holds, it supplies a uniform way to compare continuous definable functions across different Hensel minimal fields.

Load-bearing premise

The existing description of definable sets in Hensel minimal structures together with the density of clopen sets is enough to reduce every closed 0-definable set to a zero locus.

What would settle it

A concrete example of a closed 0-definable set in some Hensel minimal field K that is not the zero set of any continuous 0-definable function on K^n.

read the original abstract

The main purpose is to establish two theorems about closed 0-definable subsets $A$ of an affine space $K^{n}$ over a Hensel minimal field $K$. The first, being a non-Archimedean counterpart of one from o-minimal geometry, states that every such subset $A$ is the zero locus of a continuous 0-definable function on $K^{n}$. The second is a definable, non-Archimedean version of the Tietze-Urysohn extension theorem. The proofs use ubiquity of clopen sets in non-Archimedean geometry and a description of definable sets.

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

0 major / 3 minor

Summary. The manuscript establishes two theorems on closed 0-definable subsets A of affine space K^n over a Hensel minimal field K. The first states that every such A is the zero locus of a continuous 0-definable function on K^n, providing a non-Archimedean analogue of a result from o-minimal geometry. The second is a definable version of the Tietze-Urysohn extension theorem. Proofs rely on the ubiquity of clopen sets in non-Archimedean geometry together with a prior description of definable sets in Hensel minimal structures.

Significance. If the results hold, they extend key structural theorems from o-minimal geometry to the Hensel minimal setting, strengthening the model-theoretic toolkit for definable geometry over valued fields. The work builds directly on existing descriptions of definable sets without introducing new parameters or ad-hoc axioms, and the logical derivations appear self-contained within the standard framework of the field.

minor comments (3)
  1. [§1] §1 (Introduction): the statement of the main theorems could include a brief pointer to the specific prior cell decomposition or definable-set description being invoked, to make the dependence on earlier results explicit.
  2. [Preliminaries] Notation for the valued field K and its value group should be fixed consistently from the first appearance onward; minor inconsistencies appear in the preliminary section on Hensel minimality.
  3. [§4] The proof of the Tietze extension (likely §4) would benefit from an explicit remark on how the clopen-set density is used to construct the extension function, even if the argument is standard.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. No major comments are provided in the report.

Circularity Check

0 steps flagged

No significant circularity; new theorems proved from external prior results

full rationale

The paper establishes two new theorems: every closed 0-definable subset A of K^n is the zero locus of a continuous 0-definable function, and a definable non-Archimedean Tietze-Urysohn extension. These rest on the ubiquity of clopen sets and a prior description of definable sets in Hensel minimal structures. No equations or steps in the abstract reduce a claimed prediction or result to a fitted input, self-definition, or load-bearing self-citation chain by construction. The derivation is a standard proof in model theory building on independent external ingredients, with no internal reduction to the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the definition of Hensel minimality and on an assumed prior description of definable sets; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption Hensel minimality of the field K
    Invoked throughout as the ambient structure in which the definable sets live.
  • domain assumption Ubiquity of clopen sets and a description of definable sets
    Explicitly cited as the tools used in the proofs.

pith-pipeline@v0.9.0 · 5627 in / 1361 out tokens · 51229 ms · 2026-05-24T03:03:10.861975+00:00 · methodology

discussion (0)

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Reference graph

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