Counting fibres of the Hadamard product using Bergman fans
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909930903044096 |
|---|---|
| author | Clarke, Oliver Dewar, Sean Gallet, Matteo Grasegger, Georg Tripp, Daniel Green Smith, Ben |
| author_facet | Clarke, Oliver Dewar, Sean Gallet, Matteo Grasegger, Georg Tripp, Daniel Green Smith, Ben |
| contents | We study the generic fibre of the Hadamard product of linear spaces via matroid theory and tropical geometry. To do so, we introduce the flip product, a numerical invariant associated to a pair of matroids defined via the stable intersection of their (flipped) Bergman fans. Our first main result is that the cardinality of a generic fibre for the Hadamard product of linear spaces is exactly the flip product of their matroids. We also provide a recursive algorithm for computing the flip product of any pair of matroids. As an application of our techniques, we extend the notion of realisation numbers from rigidity theory to rotational-symmetric and periodic realisation numbers and we provide combinatorial algorithms to compute them. Finally, we show a number of existing matroid invariants are specialisations of the flip product, including the beta invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22646 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting fibres of the Hadamard product using Bergman fans Clarke, Oliver Dewar, Sean Gallet, Matteo Grasegger, Georg Tripp, Daniel Green Smith, Ben Combinatorics Algebraic Geometry 14T15 (Primary) 14T90, 05B35, 52C25 (Secondary) We study the generic fibre of the Hadamard product of linear spaces via matroid theory and tropical geometry. To do so, we introduce the flip product, a numerical invariant associated to a pair of matroids defined via the stable intersection of their (flipped) Bergman fans. Our first main result is that the cardinality of a generic fibre for the Hadamard product of linear spaces is exactly the flip product of their matroids. We also provide a recursive algorithm for computing the flip product of any pair of matroids. As an application of our techniques, we extend the notion of realisation numbers from rigidity theory to rotational-symmetric and periodic realisation numbers and we provide combinatorial algorithms to compute them. Finally, we show a number of existing matroid invariants are specialisations of the flip product, including the beta invariant. |
| title | Counting fibres of the Hadamard product using Bergman fans |
| topic | Combinatorics Algebraic Geometry 14T15 (Primary) 14T90, 05B35, 52C25 (Secondary) |
| url | https://arxiv.org/abs/2511.22646 |