Vector-valued Laurent polynomial equations, toric vector bundles and matroids
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| Format: | Preprint |
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2025
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| _version_ | 1866915388051161088 |
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| author | Kaveh, Kiumars Khovanskii, Askold Spink, Hunter |
| author_facet | Kaveh, Kiumars Khovanskii, Askold Spink, Hunter |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_09793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vector-valued Laurent polynomial equations, toric vector bundles and matroids Kaveh, Kiumars Khovanskii, Askold Spink, Hunter Algebraic Geometry Combinatorics 14M25, 52B40 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. |
| title | Vector-valued Laurent polynomial equations, toric vector bundles and matroids |
| topic | Algebraic Geometry Combinatorics 14M25, 52B40 |
| url | https://arxiv.org/abs/2507.09793 |