On the number of connected components of the parabolic curve

TL;DR

This paper constructs specific polynomial examples demonstrating that the parabolic curve of a smooth real algebraic surface in projective 3-space can have up to d(d-2)^2 connected components, revealing new geometric complexity.

Contribution

It introduces a method using Viro patchworking to explicitly construct polynomials with maximally many connected components in their Hessian curves.

Findings

Hessian curve of degree d polynomial can have (d-2)^2 connected components

Existence of smooth real algebraic surfaces with d(d-2)^2 parabolic components

Demonstrates geometric complexity of parabolic curves in real algebraic geometry

Abstract

We construct a polynomial of degree d in two variables whose Hessian curve has (d-2)^2 connected components using Viro patchworking. In particular, this implies the existence of a smooth real algebraic surface of degree d in RP^3 whose parabolic curve is smooth and has d(d-2)^2 connected components.