The classical Newton polyhedra theory gives formulas for discrete geometric and topological invariants of subvarieties in the algebraic torus defined by generic Laurent polynomial equations. The answers are in terms of combinatorics and geometry of convex polytopes. In this talk, far generalizing the above, we consider subvarieties in the algebraic torus, defined by generic vector-valued Laurent polynomials. We give a generalization of the famous BKK theorem for number of solutions of a generic Laurent polynomial system to this setting. The answer is in terms of mixed volume of certain virtual polytopes encoding matroid data. This is related to torus equivariant vector bundles on toric varieties and their equivariant Chern classes. Moreover, we prove an Alexandrov-Fenchel type inequality for these virtual polytopes. Finally, we extend this Alexandrov-Fenchel inequality to non-representable matroids (extending a famous log-concavity result of June Huh and collaborators). The talk is based on a joint work in progress with A. G. Khovanskii and Hunter Spink.