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

TL;DR

This paper extends classical results on Laurent polynomial equations and mixed volumes to vector-valued cases involving toric vector bundles and matroids, establishing new inequalities and counting solutions.

Contribution

It generalizes the BKK theorem to vector-valued Laurent polynomials and introduces Alexandrov-Fenchel type inequalities for virtual polytopes associated with matroids.

Findings

Generalization of BKK theorem to vector-valued Laurent polynomials

Establishment of Alexandrov-Fenchel inequality for virtual polytopes

Extension of inequalities to non-representable polymatroids

Abstract

Let $L \subset \mathbb{C}^r \otimes \mathbb{C}[x_1^\pm, \ldots, x_n^\pm]$ be a finite dimensional subspace of vector-valued Laurent polynomials invariant under the action of torus $(\mathbb{C}^*)^n$. 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.