Special homogeneous curves

TL;DR

This paper classifies all special homogeneous curves formed by hyperbolic points of certain homogeneous polynomials, focusing on their symmetry properties under linear transformations.

Contribution

It provides a complete classification of special homogeneous curves with transitive linear group actions, expanding understanding of their geometric and algebraic structure.

Findings

Complete classification of special homogeneous curves

Identification of their symmetry groups

Characterization of hyperbolic point components

Abstract

We classify all special homogeneous curves. A special homogeneous curve $\mathcal{H}$ consists of connected components of the hyperbolic points in the level set $\{h=1\}$ of a homogeneous polynomial $h$ in two real variables of degree at least three, and admits a transitive group action of a subgroup $G\subset\mathrm{GL}(2)$ on $\mathcal{H}$ that acts via linear coordinate change.