Extremal cubics on the circle and the 2-sphere

TL;DR

This paper investigates the geometric structure of the space of homogeneous cubics on the circle and the 2-sphere, providing a complete description for the circle and partial classification for the sphere.

Contribution

It offers a complete characterization of the facial structure for cubics on the circle and classifies extremal points for cubics on the sphere, advancing understanding of these polynomial spaces.

Findings

Complete description of facial structure for cubics on the circle

Classification of extremal points for cubics on the sphere

Identification of families of faces in the cubic norm ball

Abstract

We study balls of homogeneous cubics on $\mathbb R^n$, $n = 2,3$, which are bounded by unity on the unit sphere. For $n = 2$ we completely describe the facial structure of this norm ball, while for $n = 3$ we classify all extremal points and describe some families of faces.