Copositive matrices with circulant zero support set

TL;DR

This paper characterizes a specific face of the copositive cone defined by circulant zero support sets, providing explicit semi-definite descriptions and analyzing extremality conditions for different zero support configurations.

Contribution

It offers an explicit semi-definite description of faces of the copositive cone associated with circulant zero support sets and classifies extremal forms based on minimality and parity of order.

Findings

Forms with non-minimal circulant zero support are always extremal.

Extremal forms with minimal circulant zero support exist only for odd n.

The set of forms with non-minimal circulant zero support has codimension 2n.

Abstract

Let $n \geq 5$ and let $u^1,\dots,u^n$ be nonnegative real $n$-vectors such that the indices of their positive elements form the sets $\{1,2,\dots,n-2\},\{2,3,\dots,n-1\},\dots,\{n,1,\dots,n-3\}$, respectively. Here each index set is obtained from the previous one by a circular shift. The set of copositive forms which vanish on the vectors $u^1,\dots,u^n$ is a face of the copositive cone ${\cal C}^n$. We give an explicit semi-definite description of this face and of its subface consisting of positive semi-definite forms, and study their properties. If the vectors $u^1,\dots,u^n$ and their positive multiples exhaust the zero set of an exceptional copositive form belonging to this face, then we say it has minimal circulant zero support set, and otherwise non-minimal circulant zero support set. We show that forms with non-minimal circulant zero support set are always extremal, and forms with minimal circulant zero support sets can be extremal only if $n$ is odd. We construct explicit examples of extremal forms with non-minimal circulant zero support set for any order $n \geq 5$, and examples of extremal forms with minimal circulant zero support set for any odd order $n \geq 5$. The set of all forms with non-minimal circulant zero support set, i.e., defined by different collections $u^1,\dots,u^n$ of zeros, is a submanifold of codimension $2n$, the set of all forms with minimal circulant zero support set a submanifold of codimension $n$.