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On vertices and inflections of singular plane curves

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Pith's one-line read A singular point of a plane curve carries a determinate number of inflections and vertices, fixed by the Milnor number and the parametrisation's Wronskian orders.

desk verdict The core relations I_f=I_γ+3μ and V_f=V_γ+6μ hold up, but Theorem 4.3 drops a −3, so the bounds are wrong and the paper needs revision before I'd use it. read the letter →

arxiv 2505.21601 v1 pith:JSUIONLZ submitted 2025-05-27 math.DG

James William Bruce , Marco Antônio do Couto Fernandes , Farid Tari This is my paper
classification math.DG MSC 53A0453A5558K05
keywords planecurvesinflectionsverticesMilnornumberosculatingcircleWronskianinvariantssimplesingularitiesTeissier'sLemma
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper defines two numbers attached to a singular point of a plane curve germ $f(x,y)=0$ with an isolated singularity: $I_f$, the number of inflections concentrated at the point, and $V_f$, the number of vertices concentrated there, counted through intersection multiplicities of $f$ with its inflection and vertex loci. Its central claim is that these numbers are not arbitrary: for an irreducible branch with parametrisation $\gamma$, $I_f = I_\gamma + 3\mu(f)$ and $V_f = V_\gamma + 6\mu(f)$, where $I_\gamma,V_\gamma$ are the corresponding orders for the parametrised curve and $\mu(f)$ is the Milnor number. For a reducible curve the same bookkeeping holds, with pairwise intersection numbers added and $\mu(f)+n-1$ replacing $\mu(f)$. The paper also proves $V_f = I_f + 3\mu(f) + \lambda(f)$, where $\lambda(f)$ is the maximal order of contact of the branch with an osculating circle or line, and tabulates the possible values of $I_f$ and $V_f$ for Arnold's simple singularities.

What carries the argument

The machinery is the pair of Wronskians that detect inflections and vertices: for a parametrisation $\gamma(t)=(x(t),y(t))$, the inflection order is the order of $i_\gamma=x'y''-y'x''$ and the vertex order is the order of $v_\gamma=(x'^2+y'^2)(x'y'''-x'''y')+3(x'x''+y'y'')(x''y'-x'y'')$; the implicit invariants are defined as intersection numbers $I_f=m(f,i_f)$ and $V_f=m(f,v_f)$ with the corresponding expressions in the partial derivatives of $f$. The bridge between the two worlds is Teissier's Lemma, $m(f,f_y)=\mu(f)+m-1$, which fixes the exponent $k$ in the factorisation $f_x=u\,t^k y'$, $f_y=-u\,t^k x'$ along the parametrisation to be exactly $\mu(f)$. Differentiating $f\circ\gamma=0$ then gives $i_f\circ\gamma=-u^3 t^{3\mu}i_\gamma$ and $v_f\circ\gamma=-u^6 t^{6\mu}v_\gamma$, so the orders add as $3\mu(f)$ and $6\mu(f)$.

What would settle it

Take the cusp $f=y^2-x^3$, whose parametrisation is $\gamma(t)=(t^2,t^3)$ and whose Milnor number is $\mu=2$. Directly compute the intersection numbers $I_f=m(f,i_f)$ and $V_f=m(f,v_f)$; the theorem predicts $I_f=I_\gamma+3\mu=2+6=8$ and $V_f=V_\gamma+6\mu=3+12=15$. A reader who finds different values, or who checks the intermediate identity $m(f,f_y)=3$ and finds it false for this germ, has refuted the central claim.

Watch

Extended reading notes

Core claim

The discovery is an exact additive relation between the implicit and parametric counting of inflections and vertices at a singular point. For an irreducible germ $f$, the inflection invariant satisfies $I_f = I_\gamma + 3\mu(f)$ and the vertex invariant satisfies $V_f = V_\gamma + 6\mu(f)$; when $f=f_1\cdots f_n$, the totals are $I_f = \sum_i I_{\gamma_i} + 3(\mu(f)+n-1)$ and $V_f = \sum_i V_{\gamma_i} + 6(\mu(f)+n-1)$. In words, the singularity absorbs exactly $3\mu(f)$ inflections and $6\mu(f)$ vertices (plus interactions between branches) on top of what a generic perturbation of the parametrisation would already produce. The vertex count further obeys $V_f = I_f + 3\mu(f) + \lambda(f)$, tying vertices to the contact order of the curve with its osculating circle or line, and for simple singularities the paper computes the full range of attainable values.

Load-bearing premise

The argument rests on Teissier's Lemma, the identity $m(f(x,y),f_y(x,y))=\mu(f)+m-1$ for an irreducible germ, which is used to identify the integer $k$ in the gradient factorisation along the parametrisation with the Milnor number; if the singularity were not isolated, or if $f$ had repeated factors, the relations $I_f=I_\gamma+3\mu(f)$ and $V_f=V_\gamma+6\mu(f)$ would not follow.

Editorial extensions

If this is right

  • For any irreducible singular branch, the invariants are bounded below by explicit functions of the multiplicity, the first Puiseux exponent, and the Milnor number, so high-Milnor-number singularities force many inflections and vertices to appear in generic deformations.
  • The identity $V_f=I_f+3\mu(f)+\lambda(f)$ gives a new way to compute vertices from inflection counts plus osculating-circle contact, avoiding a direct computation of the vertex locus.
  • The additivity formula over branches reduces the computation of $I_f$ and $V_f$ to irreducible germs plus pairwise intersection numbers, which is what makes the tables for Arnold's simple singularities possible.
  • For real curves, the complex invariants serve as upper bounds for the number of real inflections and vertices; the $A_1^-$ calculation shows that at least 4 inflections and 6 vertices are inherently complex and never appear in a real deformation of a hyperbolic double point.
  • For semi-quasihomogeneous germs with weights $w_1,w_2$ and degree $d$, the inflection invariant attains the exact value $d(3d-2w_1-2w_2)/(w_1w_2)$, which is also the maximum over the $\mathcal{K}$-orbit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same mechanism should produce invariants for higher-order differential-geometric features: if a Wronskian identity analogous to Teissier's Lemma holds, each new feature would be absorbed in multiples of $\mu(f)$ determined by its parametric order.
  • The difference between the complex invariant and the maximal real count, such as the gap of 4 inflections for $A_1^-$, may itself be a topological invariant of the real singularity, related to the sign of the quadratic part, rather than a crude gap.
  • The expression $V_f=I_f+3\mu(f)+\lambda(f)$ suggests that $\lambda(f)$ can be read from Puiseux data, making $V_f$ computable from a Newton polygon alone in many cases.
  • Since $I_f$ and $V_f$ are finite and bounded whenever the curve has no smooth components, they could serve as discrete invariants for classifying singularities up to $\mathcal{K}$-equivalence, complementing the Milnor number.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies germs of plane curves with an isolated singularity, defining invariants I_f and V_f that count inflections and vertices concentrated at the singular point for curves given implicitly by f=0, alongside the parametric counterparts I_γ and V_γ. The main results include additivity formulas for unions of branches (Theorem 3.4), relations I_f=I_γ+3μ(f) and V_f=V_γ+6μ(f) for irreducible germs (Theorem 4.1), a formula linking V_γ to I_γ and the contact order λ(γ) with an osculating circle (Theorem 3.5), and bounds on I_f and V_f in terms of multiplicity, Puiseux exponent, and Milnor number (Theorem 4.3). The paper also tabulates possible values for Arnold's simple singularities and discusses the real case.

Significance. If the stated results are correct, the paper provides a systematic and computable way to count inflections and vertices that emerge from a singular point under deformations, unifying parametric and implicit approaches and connecting these counts to the Milnor number and contact with osculating circles. The proofs of the additivity theorem and Theorem 4.1 are written out in detail and rely on standard tools (intersection multiplicities, Teissier's Lemma, Nakayama's lemma). The paper is self-contained and includes explicit computations for the simple singularities, which are valuable for applications. However, several of the central numerical identities and bounds contain sign errors that affect the stated relationship with the osculating circle and the resulting table, so the current version needs correction before the results can be relied upon.

major comments (3)
  1. [Theorem 4.3(1)] Theorem 4.3(1) is false as stated. Combining Theorem 4.1 (V_f=V_γ+6μ(f), I_f=I_γ+3μ(f)) with Theorem 3.5(3) (V_γ=I_γ+λ(γ)-3) yields V_f=I_f+3μ(f)+λ(f)-3, not I_f+3μ(f)+λ(f). The missing -3 is not cosmetic: for the cusp A2, with f=y^2-x^3, γ=(t^2,t^3), one computes I_f=8, V_f=15, μ=2, λ=4, so the printed identity would give 18, contradicting the direct computation and the paper's own Table 1 (which lists V_f=15 for A2). This error also propagates into the V_f bounds in Theorem 4.3(2), which for A2 predict V_f≥18.
  2. [Theorem 3.5(4)] Theorem 3.5(4) has an off-by-three error in the n=2m case. From Theorem 3.5(3) and the expression for λ in the proof of Theorem 3.5(1), one obtains V_γ=I_γ+λ-3=(m+n-3)+(2m+ord(a_{2m}t^{2m}(a_{2m}+y_1(t))^2-y_1(t)))-3=3m+n+ord(...)-6, not -3. For a concrete check, take γ=(t^2,t^4+t^5), so m=2, n=4, and ord(...)=1; direct differentiation gives v_γ(t)=-360t^5+..., hence V_γ=5, while the printed formula gives 8. Since Theorem 4.3(2) uses these estimates, its V_f bounds are also affected.
  3. [Table 1 and Theorem 5.1] Table 1 is inconsistent with the corrected formulas. For A4 (the case k=2 in A_{2k}), the table lists V_f=28 or 29, but the corrected identity forces V_γ≥5 (since m=2 and the relevant instances are n=2m or n=β=5, both giving V_γ=5), so V_f=V_γ+6μ=29 only; the value 28 would require V_γ=4, which is impossible for a reduced parametrization with m=2. The same discrepancy appears for A2 in Theorem 4.3(2), where the printed lower bound 18 clashes with the table's value 15. Because the proof of Theorem 5.1 is not given, the source of these numerical errors cannot be checked from the text, and the table should be revised after correcting Theorems 3.5(4) and 4.3.
minor comments (4)
  1. [Theorem 3.5(1)] The terminology in Theorem 3.5(1) is ambiguous: the text refers to a unique osculating circle or line, but λ(γ) is later used as the maximal contact with a genuine circle, and in the case n>2m the maximal contact with a line is larger than the maximal circle contact. Please clarify the definition of λ(γ) to avoid the apparent contradiction with the proof of (3).
  2. [Theorem 2.2(3)] There is a typo in the proof of Theorem 2.2(3): 'the the curve' should read 'the curve'.
  3. [Theorem 2.2(7)] In the statement of Theorem 2.2(7), 'Assume also that that' contains a duplicated 'that'.
  4. [Proof of Theorem 3.3(5)] In the proof of Theorem 3.3(5), the sentence 'and the the formula for V_f follows in the same way' contains a duplicated 'the'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 4.1 is derived from independent lemmas; Theorem 4.3(1) shows a non-circular arithmetic inconsistency.

full rationale

The central relations I_f = I_gamma + 3mu(f) and V_f = V_gamma + 6mu(f) are not assumed, fitted, or imported from a self-citation. I_f and V_f are defined directly as intersection numbers m(f,i_f) and m(f,v_f), and the proof of Theorem 4.1 obtains the Milnor-number corrections from the chain-rule identities (5), the factorization in (6), and Teissier's Lemma (7), which is a standard external result. The attribution of Theorem 4.1(1) to Wall [18] is accompanied by an independent proof, so that citation is not load-bearing. The real-case discussion in Section 6 cites the authors' earlier work [5,8], but only for comparison bounds on real deformations, not as the source of the complex invariants or the main formulas. The paper is therefore self-contained against external benchmarks. The manuscript does contain an apparent inconsistency: Theorem 4.3(1) prints V_f = I_f + 3mu(f) + lambda(f), while combining Theorem 4.1 and Theorem 3.5(3) gives V_f = I_f + 3mu(f) + lambda(f) - 3; for the cusp Y^2 - X^3, direct computation gives V_f = 15 but I_f + 3mu + lambda = 18. This is a correctness / sign-error issue in a derived statement, not a circular reduction of a prediction to an input, so it does not raise the circularity score. Proposition 6.1 is stated without proof, but it is peripheral and not a circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or geometric objects beyond the invariants themselves; all assumptions are standard singularity theory. The entries above are the background facts the central formulas depend on, all drawn from published sources. No free parameters are fitted.

assumptions (5)
  • standard math Intersection multiplicity m(f,g) is a symmetric, additive bifurcation invariant with the properties listed in Section 2.2 (from [10]).
    Used throughout Proposition 2.1 and Theorem 3.4 to compute I_f and V_f as intersection numbers.
  • standard math Teissier's Lemma: for an irreducible germ f, m(f(x,y), f_y) = mu(f) + m - 1.
    Invoked in Theorem 4.1 proof, equation (7), citing [4]; it converts the order of f_y along the parametrisation into the Milnor number.
  • standard math Milnor number additivity: mu(f) = sum_i mu(f_i) + 2 sum_{i<j} m(f_i,f_j) - n + 1 (Wall, Theorem 6.5.1 of [17]).
    Used in Theorem 4.1(3) to pass from irreducible branches to reducible f.
  • standard math Finite determinacy and Nakayama's lemma for K-finite germs (Wall [15]).
    Used in Theorem 3.3(4) to show invariants depend only on finite jets.
  • domain assumption All defining functions are K-finite, i.e., f has isolated singularity and no repeated factors.
    Stated in the Introduction as the standing hypothesis; it guarantees the curve is a union of reduced branches and the invariants are defined.

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Pith. "Pith review of On vertices and inflections of singular plane curves." pith.science (2026). https://pith.science/paper/JSUIONLZ

@misc{pith2026250521601,
  author       = {Pith},
  title        = {Pith review of: On vertices and inflections of singular plane curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JSUIONLZ}},
  note         = {Machine review of arXiv:2505.21601}
}
abstract

Given the germ of a smooth plane curve $(\{f(x,y)=0\},0)\subset (\mathbb{K}^2,0), \mathbb{K}=\mathbb{R}, \mathbb{C}$, with an isolated singularity, we define two invariants $I_f$ and $V_f \in \mathbb{N} \cup\{\infty\}$, which count the number of inflections and vertices (suitably interpreted in the complex case) concentrated at the singular point. The first is an affine invariant, while the second is invariant under similarities of $\mathbb{R}^2$, and their analogue for $\mathbb{C}^2$. When the curve has no smooth components, these invariants are always finite and bounded. We illustrate our results by computing the range of possible values for these invariants for Arnold's ${\cal K}$-simple singularities. We also establish a relationship between these invariants, the Milnor number of $f$ and the contact of the curve germ with its \lq osculating circle\rq.

Figures

Figures reproduced from arXiv: 2505.21601 by the authors.

Figure 1
Figure 1. The degree of an ordinary inflection, +1 (first figure) and [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗

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Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [8]

    M. A. C. Fernandes and S. dos Santos, Geometric deformations of singularities of plane curves. In preparation

    work page

  2. [1]

    V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko,Singularities of differen- tiable maps. Vol. I. Monographs in Mathematics 82, 1985. 20

    work page 1985

  3. [2]

    J. W. Bruce and T. J. Gaffney, Simple singularities of mappingsC,0→C 2,0.J. London Math. Soc.26 (1982), 465–474

    work page 1982

  4. [3]

    J. W. Bruce and P. J. Giblin,Curves and Singularities. Cambridge University Press, 1992

    work page 1992

  5. [4]

    Cassou-Nogu` es and A

    P. Cassou-Nogu` es and A. Ploski, Invariants of plane curve singularities and New- ton diagram.Univ. Iagellonicae Acta Math.49 (2011), 9–34

    work page 2011

  6. [5]

    F. S. Dias and F. Tari, On vertices and inflections of plane curves.J. Singul.17 (2018), 70–80

    work page 2018

  7. [6]

    Diatta and P

    A. Diatta and P. Giblin, Vertices and inflexions of plane sections of surfaces inR 3. Real and Complex Singularities, 71–97, Trends Math., Birkh¨ auser Basel, 2007

    work page 2007

  8. [7]

    Eisenbud and H

    D. Eisenbud and H. I. Levine, An algebraic formula for the degree of aC ∞ map germ.Ann. of Math.106 (1977), 19–44

    work page 1977

Show all 18 references
  1. [9]

    D. M. Garay, On vanishing inflection points of plane curves.Ann. Inst. Fourier (Grenoble) 52 (2002), 849–880

    2002

  2. [10]

    Hefez, Irreducible plane curve singularities.Real and complex singularities, 1–120, CRC Press, 2003

    A. Hefez, Irreducible plane curve singularities.Real and complex singularities, 1–120, CRC Press, 2003

    2003

  3. [11]

    J. W. Milnor,Singular points of complex hypersurfaces. Annals of Math. Studies, Princeton University Press, 1968

    1968

  4. [12]

    I. R. Porteous, Probing singularities. In Singularities,Proceedings of Symposia in Pure Mathematics, AMS, vol. 20 - Part 2 (1983), 395–406

    1983

  5. [13]

    Salarinoghabi and F

    M. Salarinoghabi and F. Tari, Flat and round singularity theory of plane curves. Quart. J. Math.68 (2017), 1289–1312

    2017

  6. [14]

    F. Tari, M. Salarinoghabi, M. Hasegawa,Geometric Deformations of Discrim- inants and Apparent Contours.Volume 2370, Lecture Notes in Mathematics, Springer Nature, 2025

    2025

  7. [15]

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

    1981

  8. [16]

    C. T. C. Wall, Duality of singular plane curves.J. London Math. Soc.50 (1994), 265–275. 21

    1994

  9. [17]

    C. T. C. Wall,Singular points of plane curves. London Math. Soc. Student Texts, 63, Cambridge University Press, 2004

    2004

  10. [18]

    C. T. C. Wall, Flat singularity theory.J. London Math. Soc.87 (2013), 622–640. JWB: Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BXl E-mail: billbrucesingular@gmail.com MACF: Departamento de Matem´ atica, Universidade Federal de Vi¸ cosa, Av. P...

    2013

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