Copositive matrices with circulant zero support set

Let n≥5n≥5 and let u1,…,unu1,…,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}{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,…,unu1,…,un is a face of the copositive cone CnCn. 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 u1,…,unu1,…,un 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≥5n≥5, and examples of extremal forms with minimal circulant zero support set for any odd order n≥5n≥5. The set of all forms with non-minimal circulant zero support set, i.e., defined by different collections u1,…,unu1,…,un of zeros, is a submanifold of codimension 2n, the set of all forms with minimal circulant zero support set a submanifold of codimension n.