On vertices and inflections of singular plane curves

TL;DR

This paper introduces two invariants measuring inflections and vertices at singular points of plane curves, explores their properties, and computes their possible values for specific singularities, linking them to classical invariants like the Milnor number.

Contribution

It defines new invariants for singular plane curves, analyzes their properties, and relates them to existing invariants such as the Milnor number and contact order.

Findings

Invariants are finite and bounded when the curve has no smooth components.

Range of invariants computed for Arnold's K-simple singularities.

Established relationships between invariants, Milnor number, and contact with osculating circle.

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.