Facial structure of the cone of nonnegative ternary quartics

TL;DR

This paper thoroughly analyzes the facial structure of the cone of nonnegative ternary quartic polynomials, establishing equivalence relations, classifying faces, and presenting a complete lattice of all faces.

Contribution

It provides a complete classification and lattice structure of all faces of the cone of nonnegative ternary quartics, including equivalence relations and properties preservation.

Findings

Complete list of face equivalence classes

Dimension and inclusion relations among faces

Construction of the face lattice

Abstract

In this work we will discuss the facial structure of the cone of nonnegative ternary quartics with real coefficients. We will establish an equivalence relation on the set of all faces, which preserves certain properties like dimension or the number of common zeros. Moreover we will give a complete list of equivalence classes and we will discuss their dimension and inclusions between them. Eventually we will be able to present a complete lattice of all faces of this particular cone.