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On vertices and inflections of singular plane curves

James William Bruce, Marco Antônio do Couto Fernandes, Farid Tari

TL;DR

This work introduces two local invariants, $I_f$ and $V_f$, that count inflections and vertices concentrated at isolated singularities of plane curves, with $I_f$ affine-invariant and $V_f$ similarity-invariant (including a complex analogue). It develops a framework relating these counts to classical singularity data, notably the multiplicity, Puiseux exponents, and Milnor number, and gives exact deformation formulas: for irreducible $f$, $I_f=I_\gamma+3\mu(f)$ and $V_f=V_\gamma+6\mu(f)$, with generalizations to multi-branch cases via cross-terms. The authors compute the ranges of $(I_f,V_f)$ for Arnold’s ${\\cal K}$-simple singularities and connect the invariants to the contact with the osculating circle, providing a concrete bridge between singularity theory and local differential geometry. They also address the real case, deriving bounds and index interpretations from the degree of the maps $(f,i_f)$ and $(f,v_f)$, and outlining how many inflections/vertices survive under perturbations. These results deepen the understanding of local curvature-type features at singularities and furnish explicit counts usable in deformation theory and geometric analysis.

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.

On vertices and inflections of singular plane curves

TL;DR

This work introduces two local invariants, and , that count inflections and vertices concentrated at isolated singularities of plane curves, with affine-invariant and similarity-invariant (including a complex analogue). It develops a framework relating these counts to classical singularity data, notably the multiplicity, Puiseux exponents, and Milnor number, and gives exact deformation formulas: for irreducible , and , with generalizations to multi-branch cases via cross-terms. The authors compute the ranges of for Arnold’s -simple singularities and connect the invariants to the contact with the osculating circle, providing a concrete bridge between singularity theory and local differential geometry. They also address the real case, deriving bounds and index interpretations from the degree of the maps and , and outlining how many inflections/vertices survive under perturbations. These results deepen the understanding of local curvature-type features at singularities and furnish explicit counts usable in deformation theory and geometric analysis.

Abstract

Given the germ of a smooth plane curve , with an isolated singularity, we define two invariants and , 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 , and their analogue for . 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 -simple singularities. We also establish a relationship between these invariants, the Milnor number of and the contact of the curve germ with its \lq osculating circle\rq.

Paper Structure

This paper contains 9 sections, 12 theorems, 34 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vertices and inflections of regular curves
  4. Multiplicity and intersection number
  5. Contact between singular and smooth curves
  6. Geometric invariants of singular plane curves
  7. Computing $I_f$ and $V_f$ and their ranges
  8. $I_f$ and $V_f$ for simple singularities
  9. Vertices and inflections of curves in $\hbox{\boldmath $\mathbb{R}$}^2$

Key Result

Proposition 2.1

(1) Let $\gamma:(\hbox{\boldmath $\mathbb{C}$},0)\to (\hbox{\boldmath $\mathbb{C}$}^2,0)$ be a parametrisation of a germ of an irreducible curve $C_f$. Then $m(f,g)=\hbox{\rm ord} (g\circ \gamma)$ (Hefez). If $f=0$ is reducible with $r$ branches parametrised by $\gamma_j$, then $m(f,g)=\sum_{j=1}^r

Figures (1)

  • Figure 1: The degree of an ordinary inflection, $+1$ (first figure) and $-1$ (second figure); inward vertex (third figure) and outward vertex (last figure). The curve in red is the evolute of the curve in black.

Theorems & Definitions (14)

  • Proposition 2.1
  • Theorem 2.2
  • Definition 3.1
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Corollary 3.7
  • Theorem 4.1
  • Theorem 4.3
  • Theorem 4.4
  • ...and 4 more