On the recognition of S-systems

Abstract A sequence (A1,…,An) of nonempty subsets of R m is an S-system if for each choice of a1∈A1,…,an∈An, the nullspace of the matrix [a1,…,an] is a line that penetrates the strictly positive orthant of R n. For the case in which each Aj is a sign cone, S-systems are basic to the study of the sign-solvability of systems of linear equations; in that case, they can be recognized in time O(m2). The present paper provides several characterizations of general S-systems and describes a polynomial-time algorithm (based on linear programming) for the recognition of such systems when the Aj are arbitrary finitely presented convex cones.