Semi-definite representations for sets of cubics on the 2-sphere

TL;DR

This paper establishes semi-definite representations for sets of ternary cubics on the 2-sphere, extending known results for quadratics and quartics, and provides explicit descriptions for these polynomial norm balls.

Contribution

It introduces explicit semi-definite representations for the cone of nonnegative ternary cubics on the sphere, generalizing previous results for quadratics and quartics.

Findings

Semi-definite representation of quadratic polynomials on the sphere

Semi-definite representation of quartic polynomials on the sphere

Explicit semi-definite description of ternary cubics nonnegative on the sphere

Abstract

The compact set of homogeneous quadratic polynomials in $n$ real variables with modulus bounded by 1 on the unit sphere $S^{n-1}$ is trivially semi-definite representable. The compact set of homogeneous ternary quartics with modulus bounded by 1 on the unit sphere $S^2$ is also semi-definite representable. This suggests that the compact set of homogeneous ternary cubics with modulus bounded by 1 on $S^2$ is semi-definite representable. We deduce an explicit semi-definite representation of this norm ball. More generally, we provide a semi-definite description of the cone of inhomogeneous ternary cubics which are nonnegative on $S^2$.