Expressive 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.