A vector-valued BKK theorem: generic zeros of a torus-invariant vector-valued Laurent polynomial are counted by the mixed volume of virtual polytopes delta_i - delta_{i-1}, and the associated mixed volumes satisfy an Alexandrov-Fenchel inequality extended to polymatroids.
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Vector-valued Laurent polynomial equations, toric vector bundles and matroids
A vector-valued BKK theorem: generic zeros of a torus-invariant vector-valued Laurent polynomial are counted by the mixed volume of virtual polytopes delta_i - delta_{i-1}, and the associated mixed volumes satisfy an Alexandrov-Fenchel inequality extended to polymatroids.