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arxiv: 2604.06640 · v1 · submitted 2026-04-08 · 🧮 math.DS · math.CA

Curves of tangencies of foliation pairs and normalizing transformations

Jessica Ang\'elica Jaurez-Rosas , Laura Ortiz-Bobadilla , Sergei Voronin This is my paper

Pith reviewed 2026-05-10 18:29 UTC · model grok-4.3

classification 🧮 math.DS math.CA
keywords foliation pairscurves of tangenciesnormalizing transformationsanalytic differential equationsdegenerate singularitiesnon-dicritical foliationsdicritical foliations
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The pith

Germs of analytic foliation pairs with degenerate singularities induce curves of tangencies that admit complete description and realization under genericity assumptions via normalizing transformations.

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

The paper establishes a full classification of the loci where two foliations touch along tangent directions for pairs defined by analytic differential equations near degenerate singularities. It proceeds by reducing the pairs to local models through analytic coordinate changes that produce k-normal forms for the transformations. These forms then yield explicit parametrizations of the tangency branches up to any prescribed finite jet order. The authors further prove a realization result showing that every analytic curve whose branches are smooth and meet transversally arises as the tangency curve of some suitable foliation pair, again under natural genericity conditions on the singularities.

Core claim

Using local models and analytic normalizing transformations, the authors give a complete description of the collection of curves of tangencies induced by germs of foliation pairs, both non-dicritical and dicritical, arising from analytic differential equations with degenerate singularities that satisfy genericity assumptions; they obtain k-normal forms for the normalizing transformations that parametrize the branches of these curves up to finite jets and prove that any germ of analytic curve with pairwise transversal smooth branches is realized as such a tangency curve for appropriate non-dicritical and dicritical foliation pairs.

What carries the argument

Analytic normalizing transformations reduced to k-normal forms that bring foliation pairs to standard local models and thereby parametrize the branches of their tangency curves up to finite order.

If this is right

  • Every tangency curve of a generic foliation pair admits a parametrization of its branches up to any finite jet order via the k-normal forms.
  • The collection of possible tangency curves is exhausted by those arising from the local models after normalization.
  • The realization theorem supplies a construction that produces a foliation pair realizing any given transversal analytic curve as its tangency locus.
  • Both non-dicritical and dicritical cases are covered uniformly by the same normalizing procedure.

Where Pith is reading between the lines

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

  • The normal-form technique may allow systematic computation of tangency loci for concrete polynomial or power-series examples.
  • The realization result suggests a way to embed arbitrary transversal curves into foliated plane structures for studying global dynamics.
  • Extensions to higher-dimensional foliations or to non-analytic categories could be tested by checking whether the same normalizing steps survive without analyticity.

Load-bearing premise

The singularities and foliation pairs must satisfy genericity assumptions that guarantee the existence and utility of the normalizing transformations.

What would settle it

An explicit analytic foliation pair near a degenerate singularity whose induced tangency curve fails to match any of the parametrized forms, or a concrete analytic curve with pairwise transversal smooth branches that cannot be obtained as the tangency locus of any generic foliation pair.

Figures

Figures reproduced from arXiv: 2604.06640 by Jessica Ang\'elica Jaurez-Rosas, Laura Ortiz-Bobadilla, Sergei Voronin.

Figure 1
Figure 1. Figure 1: Local models and normalizing transformations with respect to (Fµ, Gr). Biholomorphism Ψi is not defined at the singular point (0, pi), biholomorphism Φj is not defined at the tangency point (0, qj ); thus, biholomorphism Hn is not defined neither at singular points nor at tangency points. If, additionally, the normalizing transformations at the singular points and tangency points satisfy the factorization … view at source ↗
Figure 2
Figure 2. Figure 2: Foliation pairs (F, G) and (F ′ , G ′ ) are strictly analytically equivalent if and only if they have the same local models with respect to (Fµ, Gr). We begin by giving normal forms for the normalizing transformations satisfying that, for each k ≥ 1, the normalizing transformations Hn, Hpi , Hqj with respect to the foliation pair (Fµ, Gr) can be simultaneously simplified, up to order k, by a suitable tange… view at source ↗
Figure 3
Figure 3. Figure 3: The factorization equations Hpi = Hn ◦ ξi ◦ Ψi and Hqj = Hn ◦ ζj ◦ Φj , give a decomposition of (F˜, G˜) by local analytic represen￾tatives. In (3.5) the biholomorphism Ψi is defined in a neighborhood in C 2 of the annulus Dpi ∖ {(0, pi)}, and the biholomorphism ξi : (C 2 , Dpi ) → (M, Dpi ) satisfies (see Proposition 3.1 in [JOV]) the equalities Ψi = [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Realization of curves as curves of tangencies of foliation pairs. parametrizations of their branches with respect to x coincide up to their (m+n−1)- jet, then there exists a change of coordinates tangent to the identity, which sends the curve C to the curve D (see also [V, Subsec. 2.3]). We will now state Theorem 5.2. This theorem is a reformulation of Theorem 1.7 and it will be proved by using Theorem 3.7… view at source ↗
read the original abstract

In this work we give a complete description of the collection of curves of tangencies induced by germs of foliation pairs -- non dicritical and dicritical -- given by analytic differential equations with degenerated non dicritical and dicritical singularities, satisfying some genericity assumptions. To this purpose we use local models and analytic normalizing transformations. Moreover, for each natural number $k$ we obtain $k$-normal forms for the normalizing transformations. These normal forms are used to give parametrizations, up to a finite jet, of the branches of the curves of tangencies. We also prove that under natural genericity assumptions any germ of analytic curve having pairwise transversal smooth branches is realized as curve of tangencies of a -- non dicritical and dicritical -- foliation pair.

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 / 2 minor

Summary. The paper claims to provide a complete description of the curves of tangencies induced by germs of foliation pairs (both non-dicritical and dicritical) arising from analytic differential equations with degenerate singularities, under genericity assumptions. It employs local models and analytic normalizing transformations to derive k-normal forms for these transformations, enabling parametrizations of the tangency curve branches up to finite jets. The work also establishes a realization theorem: under natural genericity assumptions, any germ of an analytic curve with pairwise transversal smooth branches can be realized as the curve of tangencies for such a foliation pair.

Significance. If the central claims hold, the results offer a systematic classification of tangency loci for degenerate foliation singularities in the analytic category, extending standard techniques for normal forms and local models. The k-normal forms and finite-jet parametrizations provide concrete tools for analyzing branches, while the realization theorem demonstrates that a wide class of transversal curve germs arise naturally as tangency sets. This could facilitate further study of dicritical and non-dicritical cases in complex dynamics, with potential applications to classification problems.

minor comments (2)
  1. The genericity assumptions invoked throughout (e.g., on singularities and foliation pairs) should be collected and stated explicitly in a dedicated subsection or remark early in the paper to clarify their scope and restrictiveness for readers.
  2. Notation for the foliation pairs and their local models could be standardized more consistently across sections to avoid minor ambiguities when transitioning between non-dicritical and dicritical cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for recognizing its significance in classifying tangency loci for degenerate foliation singularities. The referee's description aligns closely with the abstract and main results. Since the report lists no specific major comments, we have no individual points to address point-by-point. We accept the recommendation of minor revision and remain available to incorporate any additional minor suggestions from the referee or editor.

Circularity Check

0 steps flagged

No significant circularity detected in derivation

full rationale

The paper derives a complete description of tangency curves for non-dicritical and dicritical foliation pairs at degenerate singularities via local models, analytic normalizing transformations, and k-normal forms for parametrizations up to finite jets, plus a realization theorem for germs with pairwise transversal smooth branches under explicit genericity assumptions. No load-bearing step reduces by construction to its own inputs: there are no self-definitional equivalences, fitted parameters renamed as predictions, or uniqueness theorems imported solely via self-citation chains. The methods are standard in analytic foliation theory and singularity classification, with all claims resting on independent analytic constructions rather than tautological renamings or ansatzes smuggled through prior author work. The genericity conditions are invoked openly as hypotheses, not as hidden circular justifications.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard assumptions from complex analysis together with genericity conditions introduced to make the classification and realization hold; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Foliations are given by germs of analytic differential equations
    Required for the local analytic setting and normalizing transformations to apply.
  • ad hoc to paper Genericity assumptions on the singularities and foliation pairs
    Invoked to guarantee the complete description and the realization of arbitrary transversal curves.

pith-pipeline@v0.9.0 · 5433 in / 1466 out tokens · 50441 ms · 2026-05-10T18:29:53.224539+00:00 · methodology

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