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

Abstract

Let $L\subset {\mathbb{C}}^r\otimes \mathbb{C}[x_1^{\pm },\dots ,x_n^{\pm }]$ be a finite dimensional subspace of vector-valued Laurent polynomials invariant under the action of torus $({\mathbb{C}}^{\ast }{)}^n$. We study subvarieties in the torus, defined by equations $f=0$ for generic $f\in L$. We generalize the BKK theorem, that counts the number of solutions of a system of Laurent polynomial equations generic for their Newton polytopes, to this setting. The answer is in terms of mixed volume of certain virtual polytopes encoding discrete invariants of $L$ which involves matroid data. Moreover, we prove an Alexandrov-Fenchel type inequality for these virtual polytopes. Finally, we extend this inequality to non-representable polymatroids. This extends the usual Alexandrov-Fenchel inequality for polytopes as well as log-concavity results related to matroids.

25 pages