Nonnegative polynomials and their Carath\'eodory number

TL;DR

This paper investigates the structure of nonnegative polynomials, focusing on their sum-of-squares representations and the Carathéodory number within Hilbert cones, especially for quadratic and binary forms.

Contribution

It computes the Carathéodory number for Hilbert cones of nonnegative quadratic and binary forms, extending classical results on polynomial nonnegativity.

Findings

Carathéodory number for quadratic forms computed

Carathéodory number for binary forms computed

Provides canonical decompositions for these forms

Abstract

In 1888 Hilbert showed that every nonnegative homogeneous polynomial with real coefficients of degree $2d$ in $n$ variables is a sum of squares if and only if $d=1$ (quadratic forms), $n=2$ (binary forms) or $(n,d)=(3,2)$ (ternary quartics). In these cases, it is interesting to compute canonical expressions for these decompositions. Starting from Carath\'eodory's Theorem, we compute the Carath\'eodory number of Hilbert cones of nonnegative quadratic and binary forms.