Sur la structure algébrique du cône copositif

International audienceWe decompose the copositive cone $COP^n$ into a disjoint union of a finite number of open subsets $S_E$ of algebraic sets $Z_E$. Each set $S_E$ consists of interiors of faces of $COP^n$. On each irreducible component of $Z_E$ these faces generically have the same dimension. Each algebraic set $Z_E$ is characterized by a finite collection $E = {(I_\alpha, J_\alpha)}, \alpha=1,...,|E|$ of pairs of index sets. Namely, $Z_E$ is the set of symmetric matrices $A$ such that the submatrices $A_{I_\alpha\times J_\alpha}$ are rank-deficient for all $\alpha$. For every copositive matrix $A \in S_E$, the index sets $I_\alpha$ are the minimal zero supports of $A$. If $u^\alpha$ is a corresponding minimal zero of $A$, then $J_\alpha$ is the set of indices $j$ such that $(Au^\alpha)j = 0$. We call the pair $(I_\alpha, J_\alpha)$ the extended support of the zero $u^\alpha$ , and $E$ the extended minimal zero support set of $A$. We provide some necessary conditions on $E$ for $S_E$ to be non-empty, and for a subset $S_E$ to intersect the boundary of another subset $S_E$