Local $h$-polynomials, uniform triangulations and real-rootedness
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911026739412992 |
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| author | Athanasiadis, Christos A. |
| author_facet | Athanasiadis, Christos A. |
| contents | The local $h$-polynomial was introduced by Stanley as a fundamental enumerative invariant of a triangulation $Δ$ of a simplex. This polynomial is known to have nonnegative and symmetric coefficients and is conjectured to be $γ$-positive when $Δ$ is flag. This paper shows that the local $h$-polynomial has the stronger property of being real-rooted when $Δ$ is the barycentric subdivision of an arbitrary geometric triangulation $Γ$ of the simplex. An analogous result for edgewise subdivisions is proven. The proofs are based on a new combinatorial formula for the local $h$-polynomial of $Δ$, which is valid when $Δ$ is any uniform triangulation of $Γ$. A combinatorial interpretation of the local $h$-polynomial of the second barycentric subdivision of the simplex is deduced. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local $h$-polynomials, uniform triangulations and real-rootedness Athanasiadis, Christos A. Combinatorics 05E45 The local $h$-polynomial was introduced by Stanley as a fundamental enumerative invariant of a triangulation $Δ$ of a simplex. This polynomial is known to have nonnegative and symmetric coefficients and is conjectured to be $γ$-positive when $Δ$ is flag. This paper shows that the local $h$-polynomial has the stronger property of being real-rooted when $Δ$ is the barycentric subdivision of an arbitrary geometric triangulation $Γ$ of the simplex. An analogous result for edgewise subdivisions is proven. The proofs are based on a new combinatorial formula for the local $h$-polynomial of $Δ$, which is valid when $Δ$ is any uniform triangulation of $Γ$. A combinatorial interpretation of the local $h$-polynomial of the second barycentric subdivision of the simplex is deduced. |
| title | Local $h$-polynomials, uniform triangulations and real-rootedness |
| topic | Combinatorics 05E45 |
| url | https://arxiv.org/abs/2402.06219 |