Blenders

TL;DR

This paper studies the mathematical structure of blenders, which are special convex cones of polynomials closed under linear transformations, focusing on their properties and extremal elements.

Contribution

It introduces the concept of blenders, explores their properties, and analyzes extremal elements in specific cases, advancing understanding of polynomial cones.

Findings

Non-trivial blenders only occur in even degree

Examples include cones of psd, sos, convex forms, and sums of powers

Analysis of extremal elements in specific blenders

Abstract

A blender is a closed convex cone of real homogeneous polynomials that is also closed under linear changes of variable. Non-trivial blenders only occur in even degree. Examples include the cones of psd forms, sos forms, convex forms and sums of $2u$-th powers of forms of degree $v$. We present some general properties of blenders and analyze the extremal elements of some specific blenders.