Vertex Posets, Monotone Path Polytopes, and Chow Polynomials
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arXiv
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| Format: | Preprint |
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2026
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| author | Michałek, Mateusz Monin, Leonid Wang, Botong |
| author_facet | Michałek, Mateusz Monin, Leonid Wang, Botong |
| contents | Let $P\subset\mathbb R^n$ be a convex polytope and let $\ell$ be a linear functional which is nonconstant on every edge of $P$. The induced acyclic orientation determines positive and negative Białynicki-Birula type partitions of $P$ into unions of relative interiors of faces. Our first result establishes a duality: the positive partition is a stratification if and only if the negative one is a stratification. Our second result connects poset invariants with monotone path polytopes. Assuming the induced vertex relation admits the structure of a graded poset, we prove that the Chow polynomial of the resulting vertex poset agrees with the $h$-polynomial of a (dual) monotone path polytope. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_27515 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vertex Posets, Monotone Path Polytopes, and Chow Polynomials Michałek, Mateusz Monin, Leonid Wang, Botong Combinatorics Algebraic Geometry Primary: 52B05 Secondary: 06A11, 05E18, 05E14, 52B11 Let $P\subset\mathbb R^n$ be a convex polytope and let $\ell$ be a linear functional which is nonconstant on every edge of $P$. The induced acyclic orientation determines positive and negative Białynicki-Birula type partitions of $P$ into unions of relative interiors of faces. Our first result establishes a duality: the positive partition is a stratification if and only if the negative one is a stratification. Our second result connects poset invariants with monotone path polytopes. Assuming the induced vertex relation admits the structure of a graded poset, we prove that the Chow polynomial of the resulting vertex poset agrees with the $h$-polynomial of a (dual) monotone path polytope. |
| title | Vertex Posets, Monotone Path Polytopes, and Chow Polynomials |
| topic | Combinatorics Algebraic Geometry Primary: 52B05 Secondary: 06A11, 05E18, 05E14, 52B11 |
| url | https://arxiv.org/abs/2604.27515 |