Copositive matrices with circulant zero pattern

Abstract

Let n≥5 and let u1,…,un be nonnegative real n-vectors such that the indices of their positive elements form the sets {1,2,…,n−2},{2,3,…,n−1},…,{n,1,…,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 u1,…,un is a face of the copositive cone Cn. We give an explicit semi-definite description of this face and of its subface consisting of positive semi-definite matrices, and study their properties. If the vectors u1,…,un and their positive multiples exhaust the zero set of an exceptional copositive form belonging to this face, then we call this form regular, otherwise degenerate. We show that degenerate forms are always extremal, and regular forms can be extremal only if n is odd. We construct explicit examples of extremal degenerate forms for any order n≥5, and examples of extremal regular forms for any odd order n≥5. The set of all degenerate forms, i.e., defined by different collections u1,…,un of zeros, is a submanifold of codimension 2n, the set of all regular forms a submanifold of codimension n.