On {\L}ojasiewicz Ideals and Flatness for Zero Sets with Infinite Tangential Geometry

Abdelhafed El Khadiri

arxiv: 2512.11432 · v2 · submitted 2025-12-12 · 🧮 math.AG

On {L}ojasiewicz Ideals and Flatness for Zero Sets with Infinite Tangential Geometry

Abdelhafed El Khadiri This is my paper

Pith reviewed 2026-05-16 23:01 UTC · model grok-4.3

classification 🧮 math.AG

keywords Łojasiewicz idealsflat functionsHawaiian earringzero setssmooth functionstangential geometrydegenerate inequalities

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The pith

Any smooth function whose zero set is the Hawaiian earring must be flat at the origin.

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

This paper examines whether the existence of an open dense set of smooth points in the zero set of a finitely generated ideal in the ring of smooth functions implies that the ideal is Łojasiewicz. It shows that the converse fails in general and isolates geometric obstructions tied to infinite tangential accumulation. As a concrete case, any smooth real-valued function of two variables whose zero set is exactly the classical Hawaiian earring must vanish to infinite order at the origin. The authors introduce a geometric criterion for families of smooth arcs with unbounded or infinitely varying curvature and obtain from it a degenerate version of the Łojasiewicz inequality suited to this non-analytic setting.

Core claim

Any smooth real-valued function of two variables whose zero set coincides with the classical Hawaiian earring must be flat at the origin. More generally, when a finitely generated ideal in the smooth functions has a zero set containing an open dense set of smooth points, the ideal need not be Łojasiewicz; a geometric criterion on families of smooth arcs tangent at a point with unbounded or infinitely varying curvature is sufficient to force flatness and a degenerate Łojasiewicz inequality.

What carries the argument

A geometric criterion on families of smooth arcs tangent at a point with unbounded or infinitely varying curvature, which forces the corresponding ideal to be flat at the accumulation point and yields a degenerate Łojasiewicz inequality.

If this is right

Thom's theorem on Łojasiewicz ideals has no full converse in the smooth category for finitely generated ideals.
Smooth functions with zero sets exhibiting infinite tangential accumulation, such as the Hawaiian earring, are necessarily flat at the accumulation point.
A degenerate Łojasiewicz inequality holds under the stated geometric curvature conditions even though the functions are non-analytic.
The obstructions to the converse are realized concretely by zero sets whose tangent directions accumulate with unbounded curvature variation.

Where Pith is reading between the lines

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

Analogous flatness conclusions may hold for other plane curves whose tangent directions accumulate infinitely at a point, such as certain logarithmic spirals.
The criterion could be used to classify which non-analytic zero sets in the plane force their defining ideals to be flat.
In higher dimensions the same geometric condition on curvature variation might produce examples of non-Łojasiewicz ideals with dense smooth points.

Load-bearing premise

The geometric criterion on families of smooth arcs with unbounded or infinitely varying curvature is sufficient to force flatness and the degenerate Łojasiewicz inequality.

What would settle it

Constructing a non-flat smooth function of two variables whose zero set is exactly the Hawaiian earring would disprove the claim that every such function is flat at the origin.

read the original abstract

Let $\Omega \subset \mathbb{R}^n $ be an open set, and let $\mathcal{E}(\Omega)$ be the ring of infinitely differentiable functions on $\Omega$. For an ideal $I \subset \mathcal{E}(\Omega)$, we denote by $Z(I)$ its zero set. A classical result of Ren\'e Thom asserts that if $I$ is a finitely generated {\L}ojasiewicz ideal, then $Z(I)$ contains an open dense subset of smooth points. The goal of this note is to examine a converse question: does the existence of an open dense set of smooth points in $Z(I)$ ( $I\subset\mathcal{E}(\Omega) $ is a finitely generated ideal) imply that the ideal $I$ is \L{}ojasiewicz? We analyze obstructions to such a converse and identify geometric conditions under which it fails. As an application, we study smooth real-valued functions of two variables whose zero set coincides with the classical Hawaiian earring, the union of infinitely many tangent circles accumulating at the origin. We show that any such function must be flat at the origin, in the sense that all partial derivatives vanish there. We formulate a geometric criterion concerning families of smooth arcs tangent at a point with unbounded or infinitely varying curvature, and derive from it a degenerate form of the {\L}ojasiewicz inequality adapted to this non-analytic setting.

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.

Desk Editor's Note private letter to a colleague

The paper shows any C^infty function vanishing exactly on the Hawaiian earring must be flat at the origin and gives geometric obstructions to the converse of Thom's theorem.

read the letter

The punchline is that any infinitely differentiable function on the plane whose zero set is exactly the Hawaiian earring has to be flat at the origin. The paper also looks at when the converse to Thom's theorem on Lojasiewicz ideals fails, using geometric conditions on families of tangent arcs with unbounded or infinitely varying curvature. That gives a degenerate form of the inequality adapted to these non-analytic settings. The earring example is the clearest part of the note and feels like the real payoff. It takes the classical Thom result about finitely generated Lojasiewicz ideals having dense smooth points in their zero sets and turns the question around to find when the smooth-point condition does not force the ideal to be Lojasiewicz. The geometric criterion on the arcs seems a natural way to encode the infinite tangential accumulation at the origin, and the argument that this forces all partial derivatives to vanish is the main new claim. The paper does a decent job keeping the scope narrow and concrete rather than trying to build a full new theory. The soft spot is the step from the curvature condition to flatness. Without an explicit uniform rate or modulus controlling how the curvature blows up or oscillates across the countable family of arcs, it is not obvious that every possible non-flat perturbation is ruled out. A small bump supported along one of the circles might still satisfy the zero-set condition if the oscillation is not controlled tightly enough. If the full proof supplies that control, the claim holds; otherwise the derivation needs tightening. This is for people working in real smooth geometry and singularity theory who already know Thom's theorem and care about infinite accumulation cases. The earring application is specific but clean, so it could be useful to that group. It deserves a serious referee to check the details of the flatness argument and the obstructions. I would send it to peer review.

Referee Report

1 major / 1 minor

Summary. The paper examines the converse to Thom's theorem: whether a finitely generated ideal I in the ring of C^∞ functions on an open set in R^n, whose zero set Z(I) contains an open dense subset of smooth points, must be a Łojasiewicz ideal. It identifies geometric obstructions to this converse and, as an application, proves that any C^∞ function of two variables whose zero set is exactly the classical Hawaiian earring (countably many circles tangent at the origin) must be flat at the origin. A geometric criterion is formulated for families of smooth arcs tangent at a point with unbounded or infinitely varying curvature; from this criterion a degenerate form of the Łojasiewicz inequality is derived in the non-analytic setting.

Significance. If the central claims are established rigorously, the work clarifies the relationship between the tangential geometry of zero sets with infinite accumulation and the algebraic properties of ideals in C^∞ rings, providing a concrete obstruction to the converse of Thom's result. The Hawaiian earring example is a natural and interesting test case that forces flatness, and the geometric criterion offers a potentially useful tool for analyzing singularities with unbounded curvature variation.

major comments (1)

[application to the Hawaiian earring] The derivation that any C^∞ function with zero set exactly the Hawaiian earring is flat at the origin (stated in the abstract and developed in the application section) rests on the geometric criterion for families of arcs with unbounded or infinitely varying curvature. However, the passage from this criterion to vanishing of all partial derivatives at the origin appears to lack explicit rate estimates or a uniform modulus controlling the curvature blow-up across the countable family of circles; without such control it is not clear that non-flat perturbations supported along the arcs are excluded. This is load-bearing for the main application and the claimed degenerate Łojasiewicz inequality.

minor comments (1)

The abstract refers to 'a degenerate form of the Łojasiewicz inequality adapted to this non-analytic setting' but does not display the precise statement; including the explicit inequality (with any necessary constants or moduli) would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that merits clarification in the Hawaiian earring application. We address the concern directly below and will incorporate additional explicit estimates in the revised manuscript.

read point-by-point responses

Referee: [application to the Hawaiian earring] The derivation that any C^∞ function with zero set exactly the Hawaiian earring is flat at the origin (stated in the abstract and developed in the application section) rests on the geometric criterion for families of arcs with unbounded or infinitely varying curvature. However, the passage from this criterion to vanishing of all partial derivatives at the origin appears to lack explicit rate estimates or a uniform modulus controlling the curvature blow-up across the countable family of circles; without such control it is not clear that non-flat perturbations supported along the arcs are excluded. This is load-bearing for the main application and the claimed degenerate Łojasiewicz inequality.

Authors: The geometric criterion is formulated precisely so that the infinite accumulation and unbounded curvature variation across the countable family already encode the required uniform control. In the proof, one proceeds by contradiction: suppose a non-flat function f vanishes exactly on the Hawaiian earring. Then there exists a finite order k such that the k-jet at the origin is nonzero. The Taylor polynomial of order k would then dominate near the origin, forcing the zero set to be locally a finite union of analytic arcs (or empty). This contradicts the presence of infinitely many distinct tangent circles whose curvatures increase without bound. The specific radii chosen for the Hawaiian earring (decreasing faster than any polynomial) ensure that the curvature blow-up is uniform across the family in the sense that any fixed-order jet cannot accommodate all of them simultaneously. We agree that the passage can be made more explicit and will add a short paragraph in the application section supplying the modulus of continuity for the curvature and the resulting jet-vanishing argument. revision: partial

Circularity Check

0 steps flagged

No circularity: theoretical derivation self-contained without reductions to inputs or self-citations

full rationale

The paper develops a purely theoretical argument in real analytic geometry: it defines a geometric criterion on families of smooth arcs with unbounded or infinitely varying curvature, then proves that this forces flatness at the origin for any C^∞ function whose zero set is the Hawaiian earring, together with a degenerate Łojasiewicz inequality. No equations, parameters, or fitted quantities appear; the steps are deductive from stated geometric assumptions rather than tautological. No self-citations are invoked as load-bearing premises, no ansatz is smuggled, and no known result is merely renamed. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are visible from the abstract; the work relies on standard properties of C^∞ rings and zero sets.

pith-pipeline@v0.9.0 · 5550 in / 1030 out tokens · 24371 ms · 2026-05-16T23:01:40.311271+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2: ... families of smooth arcs tangent at a point with curvature tending to ∞ or oscillating without bound ... |f(x)| ≤ C_N d(x,Γ)^N for all N ≥ 1

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Oxford University Press, Oxford, 1966

[Mal66] Bernard Malgrange.Ideals of Differentiable Functions. Oxford University Press, Oxford, 1966. [Tho67] Ren´ e Thom. On some ideals of differentiable functions.Journal of the Mathematical Society of Japan, 19(2):255–259, 1967. [Tou72] Jean-Claude Tougeron.Id´ eaux de fonctions diff´ erentiables, volume 71 ofLecture Notes in Mathematics. Springer, Ber...

work page 1966

[1] [1]

Oxford University Press, Oxford, 1966

[Mal66] Bernard Malgrange.Ideals of Differentiable Functions. Oxford University Press, Oxford, 1966. [Tho67] Ren´ e Thom. On some ideals of differentiable functions.Journal of the Mathematical Society of Japan, 19(2):255–259, 1967. [Tou72] Jean-Claude Tougeron.Id´ eaux de fonctions diff´ erentiables, volume 71 ofLecture Notes in Mathematics. Springer, Ber...

work page 1966