On the analytic equivalence of branches in (n+1)-space

Marcelo Escudeiro Hernandes; Maria Elenice Rodrigues Hernandes; Welinton Anderson Rocha

arxiv: 2606.17211 · v1 · pith:7MDW7UZZnew · submitted 2026-06-15 · 🧮 math.AG

On the analytic equivalence of branches in (n+1)-space

Marcelo Escudeiro Hernandes , Welinton Anderson Rocha , Maria Elenice Rodrigues Hernandes This is my paper

Pith reviewed 2026-06-27 02:47 UTC · model grok-4.3

classification 🧮 math.AG

keywords Newton-Puiseux parametrizationsanalytic equivalenceMather's group Airreducible curvesspace curvessemigroup

0 comments

The pith

Criteria exist to remove parameters from Newton-Puiseux parametrizations of irreducible curves in (n+1)-space while preserving the form, when the curves lie in a fixed semigroup under Mather's group A.

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

The paper establishes criteria that permit the elimination of certain parameters from Newton-Puiseux parametrizations of irreducible curves in (n+1)-space. These criteria apply specifically when the curves belong to a fixed semigroup under the action of Mather's group A. The work extends analogous results that were previously available only for plane curves. A sympathetic reader cares because the criteria offer a systematic way to simplify analytic representations of curve branches without altering their essential form.

Core claim

We consider Newton-Puiseux parametrizations of irreducible curves in (n+1)-space, n greater than or equal to 1, that lie within a fixed semigroup under the action of Mather's group A. We establish criteria for eliminating parameters while preserving the Newton-Puiseux form, thereby extending known results for plane curves.

What carries the argument

The criteria for parameter elimination that preserve the Newton-Puiseux form for parametrizations of curves inside a fixed semigroup induced by Mather's group A.

If this is right

Newton-Puiseux parametrizations of such curves admit reduced forms with fewer parameters.
Analytic equivalence of branches in higher space can be decided using the simplified parametrizations.
The same elimination process that works for plane curves extends directly to space curves under the semigroup condition.

Where Pith is reading between the lines

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

The criteria may allow algorithmic simplification of parametrizations for computational classification of curve singularities in higher dimensions.
Similar semigroup-fixed conditions could be tested for other group actions beyond Mather's group A to see whether parameter reduction generalizes further.

Load-bearing premise

The curves are irreducible and lie inside a fixed semigroup under the action of Mather's group A.

What would settle it

An irreducible curve in (n+1)-space whose Newton-Puiseux parametrization belongs to the fixed semigroup but for which every attempt to eliminate a parameter alters the Newton-Puiseux form would falsify the criteria.

read the original abstract

In this paper, we consider Newton-Puiseux parametrizations of irreducible curves in (n+1)-space, n greater or equal to 1, within a fixed semigroup under the action of Mather's group A. We establish criteria for eliminating parameters while preserving the Newton-Puiseux form, extending known results for plane curves.

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

Extends parameter-elimination criteria for Newton-Puiseux parametrizations from plane curves to (n+1)-space under the usual semigroup and irreducibility conditions, but the abstract gives no proofs so the actual lift is impossible to check.

read the letter

The core claim is an extension of known parameter-elimination criteria from the n=1 case to irreducible curves in (n+1)-space that stay inside a fixed semigroup under the A-action. The abstract frames this directly as the goal and flags the standard conditions up front.

What the paper does well is identify a clean technical next step in the plane-curve literature and state the setting without extra fluff. The semigroup and irreducibility requirements are the usual ones in this area, so the setup feels consistent with prior work.

The obvious limitation is that only the abstract is visible here. No explicit criteria, no proof sketches, and no examples appear, so there is no evidence yet that the extension avoids new obstructions in higher dimensions. Soundness cannot be assessed from what is given.

This is for people already working on Newton-Puiseux expansions and analytic equivalence of curve branches. A reader deep in singularity theory might want to see the details if they care about higher-dimensional cases, but it is unlikely to matter outside that niche.

It should go to peer review so referees can inspect the actual arguments. The result looks incremental on its face, but if the criteria are new and correct it is worth recording.

Referee Report

0 major / 1 minor

Summary. The paper considers Newton-Puiseux parametrizations of irreducible curves in (n+1)-space (n ≥ 1) that remain inside a fixed semigroup under the action of Mather's group A. It establishes criteria for eliminating parameters while preserving the Newton-Puiseux form, extending known results for the plane-curve case.

Significance. If the criteria are correctly derived and the proofs are valid, the work would provide a concrete generalization of parameter-elimination techniques from plane-curve singularities to higher-dimensional branches, under the standard restriction of a fixed A-semigroup. This could facilitate the study of analytic equivalence classes in (n+1)-space.

minor comments (1)

The abstract states the main claim but supplies no indication of the form of the criteria or the key technical steps; a reader cannot assess soundness from the abstract alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript. The report summarizes the paper accurately and notes its potential significance if the criteria and proofs hold, but provides no specific major comments and issues an 'uncertain' recommendation. We address this below.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained extension of prior results

full rationale

The provided abstract and context describe a standard mathematical result: criteria for parameter elimination in Newton-Puiseux parametrizations of irreducible curves in (n+1)-space that stay in a fixed semigroup under Mather's A-action, extending the n=1 case. No equations, self-citations, fitted parameters, or derivation steps are exhibited that reduce by construction to the inputs. The conditions (irreducibility + fixed semigroup) are stated as the explicit setting rather than derived from the result itself. No load-bearing self-citation chain or ansatz smuggling is visible. This matches the default expectation of no circularity when the paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details available from the abstract alone to identify free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5582 in / 913 out tokens · 51026 ms · 2026-06-27T02:47:52.199901+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 1 canonical work pages

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Hernandes, M. E., and Rodrigues Hernandes, M. E.,The analytic classification of plane curves. Comp. Math. 160(4), 915-944 (2024)

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E., Rocha, W

Hernandes, M. E., Rocha, W. A., and Rodrigues Hernandes, M. E.,Normal forms of parametrized curves of multiplicity 4. In preparation
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Hernandes, M. E., Rodrigues Hernandes, M. E., and Ruas, M. A. S.,Ae-codimension of germs of analytic curves. Manusc. Math. 124, 237-246 (2007)

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A., and Sadykov, R

Kolgushkin, P. A., and Sadykov, R. R.,Simple singularities of multigerms of curves. Rev. Mat. Complutense 2, 311-344 (2001)

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Semigroup Forum

Montgomery, R.,The honest embedding dimension of a numerical semigroup. Semigroup Forum. https://doi.org/10.1007/s00233-026-10637-3, (2026)

work page doi:10.1007/s00233-026-10637-3 2026
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E., and Ruas, M

Rodrigues Hernandes, M. E., and Ruas, M. A. S.,Join operation andA-finite map-germs. Quart. J. Math., 74(4), 1415-1434 (2023)

2023
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C., and García-Sánchez, P

Rosales, J. C., and García-Sánchez, P. A.,Numerical Semigroups. Developments in Mathe- matics 20, Springer (2009)

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Journal of Singularities 12, 191-206 (2015)

Stevens, J.,Simple curve singularities. Journal of Singularities 12, 191-206 (2015)

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[20]

Wall, C. T. C.,Finite determinacy of smooth map-germs. Bull. London Math. Soc. 13(6), 481-539 (1981)

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[21]

Zariski, O.,On the topology of algebroid singularities. Amer. Journal Math. 54(32), 453-465 (1932)

1932
[22]

Zariski, O.,Characterization of plane algebroid curves whose module of differentials has maximum torsion. Proc. National Academy of Sciences 56(3), 781-786 (1966). 22

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[23]

Zariski, O.,General theory of saturation and of saturated local rings II: Saturated local rings of dimension 1. Am. J. Math., 93(4), 872-964 (1971)

1971
[24]

Zariski, O.,The Moduli Problem for Plane Branches.University Lecture Series 39, AMS, (2006)

2006
[25]

Zhitomirskii, M.,Fully simple singularities of plane and space curves. Proc. London Math. Soc. 96(3), 792-812 (2008). 23

2008

[1] [1]

Birkhäuser Verlag, 1986

Brieskorn, E., and Knörrer, H.,Plane Algebraic Curves. Birkhäuser Verlag, 1986

1986

[2] [2]

Delorme, C.,Sur les modules des singularités des courbes planes. Bull. Soc. Math. France 106, 417-446 (1978)

1978

[3] [3]

Ebey, S.,The classification of singular points of algebraic curves. Trans. Amer. Math. Soc. 118, 454-471 (1965)

1965

[4] [4]

Fernandes, A. C. G.,Topological equivalence of complex curves and bi-Lipschitz homeomor- phisms. Michigan Math. J. 51(3), 593-606 (2003). 21

2003

[5] [5]

Gau, Y.-N., and Lipman, J.,Differential invariance of multiplicity on analytic varieties. Invent. math. 73, 165-188, (1983)

1983

[6] [6]

G., and Hobbs, C

Gibson, C. G., and Hobbs, C. A.,Simple singularities of space curves. Math. Proc. Camb. Phil. Soc. 113(2), 297-310 (1993)

1993

[7] [7]

In Real and Complex Singularities

Hefez, A.,Irreducible Plane Curve Singularities. In Real and Complex Singularities. Eds D. Mond and M. J. Saia. Lecture Notes in Pure and Appl. Math. 232, New York, Marcel Dekker (2003)

2003

[8] [8]

E.,Standard bases for local rings of branches and their module of differentials, J

Hefez, A., and Hernandes, M. E.,Standard bases for local rings of branches and their module of differentials, J. Symb. Comp. 42, 178-191, (2007)

2007

[9] [9]

E.,The analytic classification of plane branches

Hefez, A., and Hernandes, M. E.,The analytic classification of plane branches. Bull. London Math. Soc. 43(2), 289-298, (2011)

2011

[10] [10]

E.,The analytic classification of irreducible plane curve sin- gularities

Hefez, A., and Hernandes, M. E.,The analytic classification of irreducible plane curve sin- gularities. In: Cisneros-Molina, J. L., Lê, D. T., Seade, J. (eds) Handbook of Geometry and Topology of Singularities II. Springer, 1-65 (2021)

2021

[11] [11]

E., and Rodrigues Hernandes, M

Hernandes, M. E., and Rodrigues Hernandes, M. E.,The analytic classification of plane curves. Comp. Math. 160(4), 915-944 (2024)

2024

[12] [12]

E., Rocha, W

Hernandes, M. E., Rocha, W. A., and Rodrigues Hernandes, M. E.,Normal forms of parametrized curves of multiplicity 4. In preparation

[13] [13]

E., Rodrigues Hernandes, M

Hernandes, M. E., Rodrigues Hernandes, M. E., and Ruas, M. A. S.,Ae-codimension of germs of analytic curves. Manusc. Math. 124, 237-246 (2007)

2007

[14] [14]

A., and Sadykov, R

Kolgushkin, P. A., and Sadykov, R. R.,Simple singularities of multigerms of curves. Rev. Mat. Complutense 2, 311-344 (2001)

2001

[15] [15]

Kunz, E.,The value-semigroup of a one-dimensional Gorenstein ring. Proc. Amer. Math. Soc. 25(4), 748-751 (1970)

1970

[16] [16]

Semigroup Forum

Montgomery, R.,The honest embedding dimension of a numerical semigroup. Semigroup Forum. https://doi.org/10.1007/s00233-026-10637-3, (2026)

work page doi:10.1007/s00233-026-10637-3 2026

[17] [17]

E., and Ruas, M

Rodrigues Hernandes, M. E., and Ruas, M. A. S.,Join operation andA-finite map-germs. Quart. J. Math., 74(4), 1415-1434 (2023)

2023

[18] [18]

C., and García-Sánchez, P

Rosales, J. C., and García-Sánchez, P. A.,Numerical Semigroups. Developments in Mathe- matics 20, Springer (2009)

2009

[19] [19]

Journal of Singularities 12, 191-206 (2015)

Stevens, J.,Simple curve singularities. Journal of Singularities 12, 191-206 (2015)

2015

[20] [20]

Wall, C. T. C.,Finite determinacy of smooth map-germs. Bull. London Math. Soc. 13(6), 481-539 (1981)

1981

[21] [21]

Zariski, O.,On the topology of algebroid singularities. Amer. Journal Math. 54(32), 453-465 (1932)

1932

[22] [22]

Zariski, O.,Characterization of plane algebroid curves whose module of differentials has maximum torsion. Proc. National Academy of Sciences 56(3), 781-786 (1966). 22

1966

[23] [23]

Zariski, O.,General theory of saturation and of saturated local rings II: Saturated local rings of dimension 1. Am. J. Math., 93(4), 872-964 (1971)

1971

[24] [24]

Zariski, O.,The Moduli Problem for Plane Branches.University Lecture Series 39, AMS, (2006)

2006

[25] [25]

Zhitomirskii, M.,Fully simple singularities of plane and space curves. Proc. London Math. Soc. 96(3), 792-812 (2008). 23

2008