Logarithmic models and meromorphic functions in dimension two

Jane Bretas; Rog\'erio Mol

arxiv: 2110.07637 · v3 · submitted 2021-10-14 · 🧮 math.CV · math.CA· math.DS

Logarithmic models and meromorphic functions in dimension two

Jane Bretas , Rog\'erio Mol This is my paper

Pith reviewed 2026-05-24 13:29 UTC · model grok-4.3

classification 🧮 math.CV math.CAmath.DS

keywords logarithmic modelsmeromorphic 1-formsCamacho-Sad indicesseparatricesanalytic foliationsBendixson sectorsreduction of singularitiesindeterminacy structure

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The pith

A closed meromorphic 1-form with simple poles can be constructed from prescribed dicritical components, separatrices, and Camacho-Sad indices.

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

The paper establishes a method to construct logarithmic models in the plane, consisting of closed meromorphic one-forms with simple poles, directly from chosen data on the blown-up space. The data includes which components of the exceptional divisor are dicritical, a finite list of separatrices, and the Camacho-Sad indices along those separatrices. Such models then produce meromorphic functions whose indeterminacy points, zeros, and poles match the input data. In the real setting the models also generate analytic vector fields whose trajectories near the origin divide the neighborhood into any prescribed collection of hyperbolic, parabolic, and elliptic sectors.

Core claim

The central discovery is that for any choice of dicritical components in the exceptional divisor, any finite set of separatrices, and any assignment of Camacho-Sad indices to those separatrices that satisfy the usual compatibility relations, there exists a germ of closed meromorphic 1-form with simple poles realizing exactly that data after reduction of singularities. This form defines a foliation whose separatrices and indices match the prescription. The construction applies equally in the complex and real analytic categories and directly yields the stated applications to meromorphic functions and to Bendixson decompositions of vector fields.

What carries the argument

The logarithmic model: a germ of closed meromorphic 1-form with simple poles produced from the given dicritical structure, separatrices, and indices.

Load-bearing premise

Any prescribed set of dicritical components, separatrices, and Camacho-Sad indices that meets the local compatibility conditions can be realized by a closed meromorphic 1-form with simple poles.

What would settle it

An explicit configuration of dicritical components together with separatrices and indices for which no closed meromorphic 1-form with simple poles exists that reproduces the data after reduction of singularities.

read the original abstract

In this article we describe the construction of logarithmic models in both real and complex cases. A logarithmic model is a germ of closed meromorphic 1-form with simple poles - and the analytic foliation defined by it - produced upon some specified geometric data: the structure of dicritical (non-invariant) components in the exceptional divisor of its reduction of singularities, a prescribed finite set of separatrices - invariant analytic branches at the origin - and Camacho-Sad indices with respect to these separatrices. As an application, we use logarithmic models in order to construct real and complex germs of meromorphic functions with a given indeterminacy structure and prescribed sets of zeroes and poles. Also, in the real case, in the specific case where all trajectories accumulating at the origin are contained in analytic curves, logarithmic models are used in order to build germs of analytic vector fields with a given Bendixson's sectorial decomposition of a neighborhood of $0 \in \R^{2}$ into hyperbolic, parabolic and elliptic sectors. As a consequence, we can produce real meromorphic functions with prescribed sectorial decompositions.

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 gives an explicit construction of closed meromorphic 1-forms realizing given dicritical components, separatrices, and Camacho-Sad indices after reduction in dimension two.

read the letter

The main takeaway is that Bretas and Mol supply a direct way to produce a logarithmic model — a germ of closed meromorphic 1-form with simple poles — from prescribed data on the exceptional divisor after blow-ups, a finite collection of separatrices, and the Camacho-Sad indices along them. They carry this out in both the complex and real settings by building the form on a suitable resolution and checking closedness and the index conditions from the input data alone. The stress-test note indicates that no hidden global obstruction appears in the argument, which matches the abstract's claim of a workable construction. This then yields meromorphic functions with controlled indeterminacy and zero/pole sets, plus real analytic vector fields with specified Bendixson sectors in the case where trajectories accumulate along analytic curves. The construction itself is the new piece; prior work on these indices and reductions is cited but the systematic realization from the geometric list is presented as the contribution. It stays strictly in dimension two, which keeps the arguments manageable but also caps the scope. One minor limitation is that the data must satisfy the compatibility built into the construction; the paper does not claim to handle arbitrary indices or separatrices, only those for which the form exists. The applications follow directly once the model is in hand. This is useful reading for people already working on plane foliations and real or complex singularities who want concrete examples with prescribed local behavior. It is a technical tool rather than a broad theory paper. I would send it to peer review because the construction is stated clearly enough to be checked and the applications are spelled out without overclaiming.

Referee Report

0 major / 3 minor

Summary. The paper constructs germs of closed meromorphic 1-forms with simple poles (logarithmic models) in the real and complex settings from prescribed geometric data after reduction of singularities: the dicritical structure of the exceptional divisor, a finite set of separatrices, and the associated Camacho-Sad indices. These models are applied to realize meromorphic functions with given indeterminacy loci and prescribed zeros/poles, and (in the real case) analytic vector fields whose trajectories realize a prescribed Bendixson sectorial decomposition into hyperbolic, parabolic, and elliptic sectors.

Significance. If the explicit construction holds, the work supplies a direct realization procedure for a range of local singularity configurations of foliations and vector fields in dimension two. The absence of hidden global obstructions in the argument, together with the direct verification of closedness and index conditions from the input data, strengthens the result and makes the models usable for further classification or deformation problems.

minor comments (3)

The abstract and introduction would benefit from a brief statement of the precise compatibility conditions imposed on the input data (dicritical components, separatrices, and indices) before the construction begins.
Notation for the resolved surface and the pull-back of the 1-form should be introduced once and used consistently; several passages reuse symbols for both the original and resolved objects.
The real-case application to vector fields would be clearer if the correspondence between the logarithmic model and the sectorial decomposition were summarized in a short table or diagram.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation to accept the manuscript. There are no major comments requiring response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents an explicit construction realizing prescribed geometric data (dicritical components in the exceptional divisor, finite set of separatrices, and Camacho-Sad indices) by a closed meromorphic 1-form with simple poles after reduction of singularities. The abstract and skeptic summary describe building the form on a suitable resolution and verifying closedness and index conditions directly from the data, with no equations, parameters, or steps reducing by construction to fitted inputs or self-citations. The central claims are applications to meromorphic functions and vector fields, all framed as forward realizations from independent geometric inputs rather than self-referential derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard background results in complex geometry rather than new free parameters or invented entities.

axioms (1)

domain assumption Existence of reduction of singularities for germs of meromorphic 1-forms
Referenced when describing the exceptional divisor and dicritical components after reduction.

pith-pipeline@v0.9.0 · 5722 in / 1275 out tokens · 24815 ms · 2026-05-24T13:29:07.028622+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

A logarithmic model is a germ of closed meromorphic 1-form with simple poles produced upon some specified geometric data: the structure of dicritical components... Camacho-Sad indices... (Theorem A, Sections 2–4)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Existence of consistent data of residues and of a logarithmic 1-form... (Proposition 3.6, 3.7; amalgamation in proof of Theorem A)

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

18 extracted references · 18 canonical work pages

[1]

Bendixson

I. Bendixson. Sur les courbes d´ eﬁnies par des ´ equations diﬀ´ erentielles.Acta Math. , 24(1):1–88, 1901

work page 1901
[2]

Brunella

M. Brunella. Some remarks on indices of holomorphic vect or ﬁelds. Publ. Mat. , 41(2):527–544, 1997

work page 1997
[3]

C. Camacho. Quadratic forms and holomorphic foliations on singular surfaces. Math. Ann., 282(2):177– 184, 1988

work page 1988
[4]

Camacho, A

C. Camacho, A. Lins Neto, and P. Sad. Topological invaria nts and equidesingularization for holomor- phic vector ﬁelds. J. Diﬀerential Geom. , 20(1):143–174, 1984

work page 1984
[5]

Camacho and P

C. Camacho and P. Sad. Invariant varieties through singu larities of holomorphic vector ﬁelds. Ann. of Math. (2) , 115(3):579–595, 1982

work page 1982
[6]

Cano and N

F. Cano and N. Corral. Dicritical logarithmic foliation s. Publ. Mat. , 50(1):87–102, 2006

work page 2006
[7]

F. Cano, R. Moussu, and F. Sanz. Oscillation, spiralemen t, tourbillonnement. Comment. Math. Helv. , 75(2):284–318, 2000

work page 2000
[8]

N. Corral. Sur la topologie des courbes polaires de certa ins feuilletages singuliers. Ann. Inst. Fourier (Grenoble), 53(3):787–814, 2003

work page 2003
[9]

Corral and F

N. Corral and F. Sanz. Real logarithmic models for real an alytic foliations in the plane. Rev. Mat. Complut., 25(1):109–124, 2012

work page 2012
[10]

Ilyashenko and S

Y. Ilyashenko and S. Yakovenko. Lectures on analytic diﬀerential equations , volume 86 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2008. LOGARITHMIC MODELS AND MEROMORPHIC FUNCTIONS IN DIMENSION TWO 40

work page 2008
[11]

H. Laufer. Normal two-dimensional singularities . Princeton University Press, Princeton, N.J.; Univer- sity of Tokyo Press, Tokyo, 1971. Annals of Mathematics Stud ies, No. 71

work page 1971
[12]

Lins Neto

A. Lins Neto. Algebraic solutions of polynomial diﬀere ntial equations and foliations in dimension two. In Holomorphic dynamics (Mexico, 1986) , volume 1345 of Lecture Notes in Math. , pages 192–232. Springer, Berlin, 1988

work page 1986
[13]

R. Mol. Meromorphic ﬁrst integrals: some extension res ults. Tohoku Math. J. (2) , 54(1):85–104, 2002

work page 2002
[14]

J.-J. Risler. Invariant curves and topological invari ants for real plane analytic vector ﬁelds. J. Diﬀer- ential Equations , 172(1):212–226, 2001

work page 2001
[15]

Roussarie

R. Roussarie. A topological study of planar vector ﬁeld singularities. Discrete Contin. Dyn. Syst. , 40(9):5217–5245, 2020

work page 2020
[16]

Seidenberg

A. Seidenberg. Reduction of singularities of the diﬀer ential equation A dy = B dx. Amer. J. Math. , 90:248–269, 1968

work page 1968
[17]

T. Suwa. Indices of holomorphic vector ﬁelds relative t o invariant curves on surfaces. Proc. Amer. Math. Soc. , 123(10):2989–2997, 1995

work page 1995
[18]

T. Suwa. Indices of vector ﬁelds and residues of singular holomorphi c foliations . Actualit´ es Math´ ematiques. [Current Mathematical Topics]. Hermann, Paris, 1998. Departamento de Matem ´atica — Centro Federal de Educac ¸ ˜ao Tecnol ´ogica de Minas Gerais Current address : Av. Amazonas, 7675 — 30510-000 — Belo Horizonte, BRAZIL Email address : janebret...

work page 1998

[1] [1]

Bendixson

I. Bendixson. Sur les courbes d´ eﬁnies par des ´ equations diﬀ´ erentielles.Acta Math. , 24(1):1–88, 1901

work page 1901

[2] [2]

Brunella

M. Brunella. Some remarks on indices of holomorphic vect or ﬁelds. Publ. Mat. , 41(2):527–544, 1997

work page 1997

[3] [3]

C. Camacho. Quadratic forms and holomorphic foliations on singular surfaces. Math. Ann., 282(2):177– 184, 1988

work page 1988

[4] [4]

Camacho, A

C. Camacho, A. Lins Neto, and P. Sad. Topological invaria nts and equidesingularization for holomor- phic vector ﬁelds. J. Diﬀerential Geom. , 20(1):143–174, 1984

work page 1984

[5] [5]

Camacho and P

C. Camacho and P. Sad. Invariant varieties through singu larities of holomorphic vector ﬁelds. Ann. of Math. (2) , 115(3):579–595, 1982

work page 1982

[6] [6]

Cano and N

F. Cano and N. Corral. Dicritical logarithmic foliation s. Publ. Mat. , 50(1):87–102, 2006

work page 2006

[7] [7]

F. Cano, R. Moussu, and F. Sanz. Oscillation, spiralemen t, tourbillonnement. Comment. Math. Helv. , 75(2):284–318, 2000

work page 2000

[8] [8]

N. Corral. Sur la topologie des courbes polaires de certa ins feuilletages singuliers. Ann. Inst. Fourier (Grenoble), 53(3):787–814, 2003

work page 2003

[9] [9]

Corral and F

N. Corral and F. Sanz. Real logarithmic models for real an alytic foliations in the plane. Rev. Mat. Complut., 25(1):109–124, 2012

work page 2012

[10] [10]

Ilyashenko and S

Y. Ilyashenko and S. Yakovenko. Lectures on analytic diﬀerential equations , volume 86 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2008. LOGARITHMIC MODELS AND MEROMORPHIC FUNCTIONS IN DIMENSION TWO 40

work page 2008

[11] [11]

H. Laufer. Normal two-dimensional singularities . Princeton University Press, Princeton, N.J.; Univer- sity of Tokyo Press, Tokyo, 1971. Annals of Mathematics Stud ies, No. 71

work page 1971

[12] [12]

Lins Neto

A. Lins Neto. Algebraic solutions of polynomial diﬀere ntial equations and foliations in dimension two. In Holomorphic dynamics (Mexico, 1986) , volume 1345 of Lecture Notes in Math. , pages 192–232. Springer, Berlin, 1988

work page 1986

[13] [13]

R. Mol. Meromorphic ﬁrst integrals: some extension res ults. Tohoku Math. J. (2) , 54(1):85–104, 2002

work page 2002

[14] [14]

J.-J. Risler. Invariant curves and topological invari ants for real plane analytic vector ﬁelds. J. Diﬀer- ential Equations , 172(1):212–226, 2001

work page 2001

[15] [15]

Roussarie

R. Roussarie. A topological study of planar vector ﬁeld singularities. Discrete Contin. Dyn. Syst. , 40(9):5217–5245, 2020

work page 2020

[16] [16]

Seidenberg

A. Seidenberg. Reduction of singularities of the diﬀer ential equation A dy = B dx. Amer. J. Math. , 90:248–269, 1968

work page 1968

[17] [17]

T. Suwa. Indices of holomorphic vector ﬁelds relative t o invariant curves on surfaces. Proc. Amer. Math. Soc. , 123(10):2989–2997, 1995

work page 1995

[18] [18]

T. Suwa. Indices of vector ﬁelds and residues of singular holomorphi c foliations . Actualit´ es Math´ ematiques. [Current Mathematical Topics]. Hermann, Paris, 1998. Departamento de Matem ´atica — Centro Federal de Educac ¸ ˜ao Tecnol ´ogica de Minas Gerais Current address : Av. Amazonas, 7675 — 30510-000 — Belo Horizonte, BRAZIL Email address : janebret...

work page 1998