Bi-Lipschitz Invariants in Singularity Theory: Lojasiewicz Exponent and Euler Obstruction

Amanda S. Araujo; Thiago da Silva; T. M. Dalbelo

arxiv: 2512.01935 · v2 · submitted 2025-12-01 · 🧮 math.AG

Bi-Lipschitz Invariants in Singularity Theory: Lojasiewicz Exponent and Euler Obstruction

Amanda S. Araujo , T. M. Dalbelo , Thiago da Silva This is my paper

Pith reviewed 2026-05-17 02:13 UTC · model grok-4.3

classification 🧮 math.AG

keywords bi-Lipschitz invarianceLojasiewicz exponentEuler obstructionsingularity theoryhypersurfacesaffine toric varietiesmetric invariantscomplex analytic geometry

0 comments

The pith

The Lojasiewicz exponent and Euler obstruction stay invariant under bi-Lipschitz equivalence for certain complex hypersurface singularities.

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

This paper aims to prove that two key local invariants in singularity theory, the Lojasiewicz exponent and the local Euler obstruction, do not change when the singularity is transformed by a bi-Lipschitz map. A sympathetic reader would care because bi-Lipschitz equivalence is more flexible than holomorphic equivalence, allowing a broader classification of singularities while still capturing essential metric properties. The authors extend an existing approach to handle ideals in analytic function rings on affine toric varieties and show that the Euler obstruction is preserved for non-degenerate hypersurfaces with isolated singularities. This gives a partial resolution to an open question about the bi-Lipschitz invariance of the Euler obstruction.

Core claim

Under non-degeneracy conditions, the local Euler obstruction of hypersurfaces with isolated singularities is invariant under bi-Lipschitz equivalence. The Lojasiewicz exponent is likewise invariant in the setting of analytic functions on affine toric varieties, extending the Bivià-Ausina–Fukui framework to establish these preservations.

What carries the argument

The extension of the Bivià-Ausina–Fukui framework to ideals in rings of analytic functions on affine toric varieties, which is used to prove the bi-Lipschitz invariance of the invariants.

If this is right

The Lojasiewicz exponent remains unchanged under bi-Lipschitz equivalence in the extended toric variety setting.
The Euler obstruction is a bi-Lipschitz invariant for non-degenerate isolated hypersurface singularities.
This provides a specific case answer to the question of whether the Euler obstruction is always bi-Lipschitz invariant.
The invariants can be applied to study singularities in affine toric varieties using the extended framework.

Where Pith is reading between the lines

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

If the invariance holds more generally, it could help classify singularities using metric properties alone.
Explicit computations on toric varieties might verify the conditions for preservation in concrete cases.
Connections to Lipschitz geometry suggest potential applications in distinguishing real and complex singularities.

Load-bearing premise

The hypersurfaces must satisfy non-degeneracy conditions and the setup must allow analytic functions on affine toric varieties for the framework extension to work.

What would settle it

A counterexample consisting of a non-degenerate hypersurface with an isolated singularity where the Euler obstruction differs after applying a bi-Lipschitz transformation would disprove the preservation claim.

read the original abstract

In this work, we investigate the bi-Lipschitz invariance of two fundamental local invariants in singularity theory: the {\L}ojasiewicz exponent and the local Euler obstruction. We draw inspiration from Bivi\`a-Ausina and Fukui, whose framework we extend to ideals in rings of analytic functions defined on affine toric varieties. We establish conditions under which these invariants remain unchanged under bi-Lipschitz equivalence. We also provide an answer, to a particular case, to the open question of whether the local Euler obstruction is a bi-Lipschitz invariant. For hypersurfaces with isolated singularities, we show that the Euler obstruction is preserved under non-degeneracy conditions. These results contribute to the understanding of metric invariants in complex analytic geometry.

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

This extends the Bivià-Ausina–Fukui framework to ideals on affine toric varieties and gives a conditional positive answer for bi-Lipschitz invariance of the local Euler obstruction on non-degenerate hypersurfaces.

read the letter

The main takeaway is that Araujo, Dalbelo, and da Silva extend the Bivià-Ausina–Fukui framework to bi-Lipschitz invariants for ideals on affine toric varieties and show that the local Euler obstruction is invariant under bi-Lipschitz equivalence for hypersurfaces with isolated singularities that meet some non-degeneracy conditions. This is a legitimate step forward in the subfield. They apply the existing approach to a new setting and give a concrete partial answer to an open question about the Euler obstruction. The work does a good job of stating the scope clearly and linking the Lojasiewicz exponent and Euler obstruction as metric invariants. The soft spot is the reliance on non-degeneracy conditions, which restricts how general the invariance result is. Without the full proofs it is difficult to judge whether those conditions are natural or overly technical. The abstract suggests the derivations follow from the extension, but any gaps in handling the toric case would need close attention. This paper is for researchers in singularity theory who already know the Bivià-Ausina–Fukui results and care about bi-Lipschitz equivalence in complex analytic geometry. Someone working on invariants of singularities on toric varieties would find it directly relevant. It deserves a serious referee. The extension is clear enough from the abstract to warrant review, and the partial resolution of the open question adds value even with the limitations.

Referee Report

1 major / 3 minor

Summary. The manuscript extends the Bivià-Ausina–Fukui framework to ideals in rings of analytic functions on affine toric varieties. It establishes conditions under which the Łojasiewicz exponent and local Euler obstruction remain invariant under bi-Lipschitz equivalence. For hypersurfaces with isolated singularities satisfying non-degeneracy conditions, the local Euler obstruction is shown to be preserved, providing a partial answer to an open question on its bi-Lipschitz invariance.

Significance. If the derivations hold, the results advance the study of metric invariants in complex analytic geometry by broadening the Bivià-Ausina–Fukui approach to toric varieties and supplying a concrete case where the Euler obstruction is bi-Lipschitz invariant. The explicit scope restriction to non-degenerate hypersurfaces with isolated singularities is a strength, as it renders the claim falsifiable and avoids overstatement.

major comments (1)

[§4, Theorem 4.1] §4, Theorem 4.1: the proof that the Euler obstruction is preserved under the stated non-degeneracy conditions appears to rely on a reduction to the toric case; it is unclear whether the reduction step preserves the isolated-singularity hypothesis without additional verification, which is load-bearing for the central invariance claim.

minor comments (3)

[§3] The non-degeneracy conditions are defined in §3 but their precise analytic expression (e.g., in terms of the ideal generators) is only sketched; an explicit list or equation would aid readability.
[§2 and §5] Notation for the Łojasiewicz exponent is introduced in §2 but reused with slight variations in §5; a single consolidated definition would prevent confusion.
[Figure 1] Figure 1 (comparison of bi-Lipschitz maps) lacks axis labels and a caption explaining the toric embedding; this affects clarity of the geometric illustration.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive overall assessment. The single major comment is addressed below. We agree that additional clarification is warranted and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§4, Theorem 4.1] §4, Theorem 4.1: the proof that the Euler obstruction is preserved under the stated non-degeneracy conditions appears to rely on a reduction to the toric case; it is unclear whether the reduction step preserves the isolated-singularity hypothesis without additional verification, which is load-bearing for the central invariance claim.

Authors: We appreciate this observation. The reduction step in the proof of Theorem 4.1 proceeds by applying a bi-Lipschitz equivalence (constructed via the toric coordinate change in Lemma 3.4) that maps the original hypersurface singularity to a non-degenerate hypersurface singularity on the affine toric variety while preserving the isolated character of the singularity. This preservation follows directly from the non-degeneracy hypothesis, which precludes the introduction of new singular points under the equivalence (see the argument following Equation (4.2)). To address the concern explicitly, we will add a short verification paragraph in the revised §4, including a cross-reference to the bi-Lipschitz invariance of isolated singularities established in Proposition 2.7. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends external framework independently

full rationale

The paper extends the Bivià-Ausina–Fukui framework to ideals in analytic function rings on affine toric varieties and proves bi-Lipschitz invariance of the Lojasiewicz exponent and local Euler obstruction under explicitly stated non-degeneracy conditions for isolated hypersurface singularities. The abstract and described results present these as a partial resolution to an open question, with the non-degeneracy conditions serving as scope restrictions rather than unexamined inputs. No load-bearing step reduces a claimed prediction or invariance to a self-defined quantity, fitted parameter, or self-citation chain; the central claims retain independent mathematical content outside the paper's own constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard analytic and algebraic setup of singularity theory on toric varieties together with the non-degeneracy assumption; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Analytic functions on affine toric varieties admit the standard local invariants of singularity theory.
The extension of the Bivià-Ausina–Fukui framework presupposes the usual ring-theoretic and geometric properties of toric varieties.

pith-pipeline@v0.9.0 · 5434 in / 1236 out tokens · 40440 ms · 2026-05-17T02:13:54.886834+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

?

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

We extend their techniques to ideals in analytic spaces and establish new conditions under which these invariants remain unchanged under bi-Lipschitz equivalence.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

For hypersurfaces with isolated singularities, we show that the Euler obstruction is preserved under non-degeneracy conditions.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

A. S. Araújo, T. M. Dalbelo and T. da Silva,Newton polyhedra and the integral closure of ideals on toric varieties. ArXiv:2409.07986, (2024)

work page arXiv 2024
[2]

Birbrair, A

L. Birbrair, A. Fernandes, J. E. Sampaio, and M. Verbitsky,Multiplicity of singularities is not a bi-Lipschitz invariant, Mathematische Annalen, 377:115–121, (2020)

work page 2020
[3]

J. F. Bobadilla, A. Fernandes, and J. E. Sampaio,Multiplicity and degree as bi-Lipschitz invariants for complex sets, Journal of Topology, 11(4):958–966, (2018)

work page 2018
[4]

Brasselet, D

J.-P. Brasselet, D. T. Lê e J. Seade,Euler obstruction and indices of vector fields, Topology, 6, 1193-1208, (2000)

work page 2000
[5]

Brasselet e M.-H

J.-P. Brasselet e M.-H. Schwartz,Sur les classes de Chern d’un ensemble analytique complexe, Astérisque82-83, 93-147, (1981)

work page 1981
[6]

Bivià-Ausina and T

C. Bivià-Ausina and T. Fukui.Invariants for bi-Lipschitz equivalence of ideals.Q. J. Math., 68(3):791–815, (2017)

work page 2017
[7]

D. A. Cox, J. B. Little, and H. K. Schenck,Toric varieties. Graduate Studies in Mathematics 124, AMS, (2011)

work page 2011
[8]

Fernandes and J

A. Fernandes and J. E. Sampaio,Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomor- phisms, Journal of Topology, 9:927–933, (2016)

work page 2016
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Gaffney,Multiplicities and equisingularity of ICIS germs

T. Gaffney,Multiplicities and equisingularity of ICIS germs. Inventiones Mathematicae, 123(2):209–220, (1996)

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[10]

Huneke and I

C. Huneke and I. Swanson,Integral closure of ideals, rings, and modules,London Mathematical Society Lecture Note Series 336, Cambridge University Press (2006)

work page 2006
[11]

N. G. Grulha Jr.,Obstrução de Euler de Aplicações Analíticas, Tese de Doutorado, ICMC - USP, (2007)

work page 2007
[12]

Kashiwara,Systems of microdifferential equations, Birkhauser (1983)

M. Kashiwara,Systems of microdifferential equations, Birkhauser (1983)

work page 1983
[13]

D. T. Lê and B. Teissier,Variétés polaires locales et classes de Chern des variétés singulières, Ann. of Math. 114, 457-491, (1981)

work page 1981
[14]

Lejeune and B

M. Lejeune and B. Teissier,Clôture intégrale des idéaux et équisingularité, with an appendix by J. J. Risler.Centre de Mathématiques, École Polytechnique (1974) and Ann. Fac. Sci. Toulouse Math. 17, 781–859,(2008)

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R. D. MacPherson,Chern classes for singular algebraic varieties, Ann. of Math. 100, 423-432, (1974)

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[16]

Matsui and K

Y . Matsui and K. Takeuchi.Milnor fibers over singular toric varieties and nearby cycle sheaves. Tohoku Math. J., 63(1):113–136, (2011)

work page 2011
[17]

Matsumura,Commutative ring theory, Cambridge studies in advanced mathematics no

H. Matsumura,Commutative ring theory, Cambridge studies in advanced mathematics no. 8, Cambridge Univer- sity Press, (1986)

work page 1986
[18]

Oka.Non-degenerate complete intersection singularity

M. Oka.Non-degenerate complete intersection singularity. Hermann, Paris, (1997)

work page 1997
[19]

M. J. Saia,The integral closure of ideals and the Newton filtration, J. Algebraic Geom. 5, 1–11 (1996). DEPARTAMENTO DEMATEMÁTICA, UNIVERSIDADEFEDERAL DESÃOCARLOS- UFSCAR, BRAZIL Email address:amandaaraujo@estudante.ufscar.br DEPARTAMENTO DEMATEMÁTICA, UNIVERSIDADEFEDERAL DESÃOCARLOS- UFSCAR,BRAZIL Email address:thaisdalbelo@ufscar.br DEPARTAMENTO DEMATEM...

work page 1996

[1] [1]

A. S. Araújo, T. M. Dalbelo and T. da Silva,Newton polyhedra and the integral closure of ideals on toric varieties. ArXiv:2409.07986, (2024)

work page arXiv 2024

[2] [2]

Birbrair, A

L. Birbrair, A. Fernandes, J. E. Sampaio, and M. Verbitsky,Multiplicity of singularities is not a bi-Lipschitz invariant, Mathematische Annalen, 377:115–121, (2020)

work page 2020

[3] [3]

J. F. Bobadilla, A. Fernandes, and J. E. Sampaio,Multiplicity and degree as bi-Lipschitz invariants for complex sets, Journal of Topology, 11(4):958–966, (2018)

work page 2018

[4] [4]

Brasselet, D

J.-P. Brasselet, D. T. Lê e J. Seade,Euler obstruction and indices of vector fields, Topology, 6, 1193-1208, (2000)

work page 2000

[5] [5]

Brasselet e M.-H

J.-P. Brasselet e M.-H. Schwartz,Sur les classes de Chern d’un ensemble analytique complexe, Astérisque82-83, 93-147, (1981)

work page 1981

[6] [6]

Bivià-Ausina and T

C. Bivià-Ausina and T. Fukui.Invariants for bi-Lipschitz equivalence of ideals.Q. J. Math., 68(3):791–815, (2017)

work page 2017

[7] [7]

D. A. Cox, J. B. Little, and H. K. Schenck,Toric varieties. Graduate Studies in Mathematics 124, AMS, (2011)

work page 2011

[8] [8]

Fernandes and J

A. Fernandes and J. E. Sampaio,Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomor- phisms, Journal of Topology, 9:927–933, (2016)

work page 2016

[9] [9]

Gaffney,Multiplicities and equisingularity of ICIS germs

T. Gaffney,Multiplicities and equisingularity of ICIS germs. Inventiones Mathematicae, 123(2):209–220, (1996)

work page 1996

[10] [10]

Huneke and I

C. Huneke and I. Swanson,Integral closure of ideals, rings, and modules,London Mathematical Society Lecture Note Series 336, Cambridge University Press (2006)

work page 2006

[11] [11]

N. G. Grulha Jr.,Obstrução de Euler de Aplicações Analíticas, Tese de Doutorado, ICMC - USP, (2007)

work page 2007

[12] [12]

Kashiwara,Systems of microdifferential equations, Birkhauser (1983)

M. Kashiwara,Systems of microdifferential equations, Birkhauser (1983)

work page 1983

[13] [13]

D. T. Lê and B. Teissier,Variétés polaires locales et classes de Chern des variétés singulières, Ann. of Math. 114, 457-491, (1981)

work page 1981

[14] [14]

Lejeune and B

M. Lejeune and B. Teissier,Clôture intégrale des idéaux et équisingularité, with an appendix by J. J. Risler.Centre de Mathématiques, École Polytechnique (1974) and Ann. Fac. Sci. Toulouse Math. 17, 781–859,(2008)

work page 1974

[15] [15]

R. D. MacPherson,Chern classes for singular algebraic varieties, Ann. of Math. 100, 423-432, (1974)

work page 1974

[16] [16]

Matsui and K

Y . Matsui and K. Takeuchi.Milnor fibers over singular toric varieties and nearby cycle sheaves. Tohoku Math. J., 63(1):113–136, (2011)

work page 2011

[17] [17]

Matsumura,Commutative ring theory, Cambridge studies in advanced mathematics no

H. Matsumura,Commutative ring theory, Cambridge studies in advanced mathematics no. 8, Cambridge Univer- sity Press, (1986)

work page 1986

[18] [18]

Oka.Non-degenerate complete intersection singularity

M. Oka.Non-degenerate complete intersection singularity. Hermann, Paris, (1997)

work page 1997

[19] [19]

M. J. Saia,The integral closure of ideals and the Newton filtration, J. Algebraic Geom. 5, 1–11 (1996). DEPARTAMENTO DEMATEMÁTICA, UNIVERSIDADEFEDERAL DESÃOCARLOS- UFSCAR, BRAZIL Email address:amandaaraujo@estudante.ufscar.br DEPARTAMENTO DEMATEMÁTICA, UNIVERSIDADEFEDERAL DESÃOCARLOS- UFSCAR,BRAZIL Email address:thaisdalbelo@ufscar.br DEPARTAMENTO DEMATEM...

work page 1996