Vector-valued Laurent polynomial equations, toric vector bundles and matroids

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 Laurent polynomial system. 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. The talk is based on the recent joint work with A. G. Khovanskii and Hunter Spink.