Chow polynomials of rank-uniform labeled posets
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_ | 1866912775462191104 |
|---|---|
| author | Coron, Basile Ferroni, Luis Li, Shiyue |
| author_facet | Coron, Basile Ferroni, Luis Li, Shiyue |
| contents | We introduce and develop the theory of UMEL-shellable posets. These are posets equipped with an edge-lexicographical labeling satisfying certain uniformity and monotonicity properties. This framework encompasses classical families of combinatorial geometries, including uniform matroids, projective and affine geometries, braid matroids of type A and B, and all Dowling geometries. It also comprises all rank-uniform supersolvable lattices, and therefore also all rank-uniform distributive lattices. Our main result establishes real-rootedness phenomena for the Chow polynomials, the augmented Chow polynomials, and the chain polynomials associated with those posets, thus making simultaneous progress towards conjectures by Ferroni--Schröter, Huh--Stevens, and Athanasiadis--Kalampogia-Evangelinou. In the special case of lattices of flats of matroids, the (augmented) Chow polynomials coincide with the Hilbert--Poincaré series of the Chow ring associated to the smooth and generally noncompact toric varieties of the (augmented) Bergman fan of the matroid, whereas the chain polynomial encodes the Hilbert--Poincaré series of the Stanley--Reisner ring of the Bergman complex of the matroid. Therefore, these real-rootedness results are tightly linked to the study of these algebro-geometric structures in matroid theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13819 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chow polynomials of rank-uniform labeled posets Coron, Basile Ferroni, Luis Li, Shiyue Combinatorics 05E14, 05E45, 14N10 We introduce and develop the theory of UMEL-shellable posets. These are posets equipped with an edge-lexicographical labeling satisfying certain uniformity and monotonicity properties. This framework encompasses classical families of combinatorial geometries, including uniform matroids, projective and affine geometries, braid matroids of type A and B, and all Dowling geometries. It also comprises all rank-uniform supersolvable lattices, and therefore also all rank-uniform distributive lattices. Our main result establishes real-rootedness phenomena for the Chow polynomials, the augmented Chow polynomials, and the chain polynomials associated with those posets, thus making simultaneous progress towards conjectures by Ferroni--Schröter, Huh--Stevens, and Athanasiadis--Kalampogia-Evangelinou. In the special case of lattices of flats of matroids, the (augmented) Chow polynomials coincide with the Hilbert--Poincaré series of the Chow ring associated to the smooth and generally noncompact toric varieties of the (augmented) Bergman fan of the matroid, whereas the chain polynomial encodes the Hilbert--Poincaré series of the Stanley--Reisner ring of the Bergman complex of the matroid. Therefore, these real-rootedness results are tightly linked to the study of these algebro-geometric structures in matroid theory. |
| title | Chow polynomials of rank-uniform labeled posets |
| topic | Combinatorics 05E14, 05E45, 14N10 |
| url | https://arxiv.org/abs/2511.13819 |