Expressive curves

TL;DR

This paper introduces the concept of expressive real plane algebraic curves, characterized by minimal critical points, and provides conditions, constructions, and examples illustrating their properties and classifications.

Contribution

It defines expressive curves, establishes necessary and sufficient conditions for their existence, and explores various explicit constructions and examples.

Findings

Expressive curves have minimal critical points dictated by their topology.

Conditions for a curve to be expressive include real polynomial parametrizations, hyperbolic nodes, and connected real points.

Multiple explicit examples of expressive curves are provided, including classical and special curves.

Abstract

We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve $C$ is expressive if (a) each irreducible component of $C$ can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of $C$ in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of $C$ in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)-(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.