Triangulations of simplicial complexes and theta polynomials
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913724681420800 |
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| author | Athanasiadis, Christos A. |
| author_facet | Athanasiadis, Christos A. |
| contents | An enumerative theory of triangulations of simplicial complexes has been developed by Stanley. A key role in his theory is played by the local $h$-polynomial of a triangulation of a simplex. This paper develops a parallel theory, in which the role of the local $h$-polynomial is played by a simpler invariant, namely the theta polynomial. This allows one to deduce unimodality and gamma-positivity properties of $h$-polynomials of triangulations of simplicial complexes from corresponding properties of theta polynomials, which are studied here in some detail. To mention one concrete application, the $h$-polynomial of the antiprism triangulation of any simplicial homology sphere is shown to be gamma-positive, thus confirming Gal's conjecture in a new special case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_01674 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Triangulations of simplicial complexes and theta polynomials Athanasiadis, Christos A. Combinatorics 05E45 An enumerative theory of triangulations of simplicial complexes has been developed by Stanley. A key role in his theory is played by the local $h$-polynomial of a triangulation of a simplex. This paper develops a parallel theory, in which the role of the local $h$-polynomial is played by a simpler invariant, namely the theta polynomial. This allows one to deduce unimodality and gamma-positivity properties of $h$-polynomials of triangulations of simplicial complexes from corresponding properties of theta polynomials, which are studied here in some detail. To mention one concrete application, the $h$-polynomial of the antiprism triangulation of any simplicial homology sphere is shown to be gamma-positive, thus confirming Gal's conjecture in a new special case. |
| title | Triangulations of simplicial complexes and theta polynomials |
| topic | Combinatorics 05E45 |
| url | https://arxiv.org/abs/2209.01674 |