Hypergraph characterizations of copositive tensors

A real symmetric tensor $${\mathscr{A}} = ({a_{{i_1} \ldots {i_m}}}) \in {\mathbb{R}^{[m,n]}}$$ is copositive (resp., strictly copositive) if $${\mathscr{A}}\;{x^m} \geqslant 0$$ (resp., $${\mathscr{A}}\;{x^m} > 0$$ ) for any nonzero nonnegative vector x ∈ ℝn. By using the associated hypergraph of $${\mathscr{A}}$$ , we give necessary and sufficient conditions for the copositivity of $${\mathscr{A}}$$ . For a real symmetric tensor $${\mathscr{A}}$$ satisfying the associated negative hypergraph $${H_ - }({\mathscr{A}})$$ and associated positive hypergraph $${H_ + }({\mathscr{A}})$$ are edge disjoint subhypergraphs of a supertree or cored hypergraph, we derive criteria for the copositivity of $${\mathscr{A}}$$ . We also use copositive tensors to study the positivity of tensor systems.