Ehrhart theory of cosmological polytopes
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929665717829632 |
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| author | Bruckamp, Justus Goltermann, Lina Juhnke, Martina Landin, Erik Solus, Liam |
| author_facet | Bruckamp, Justus Goltermann, Lina Juhnke, Martina Landin, Erik Solus, Liam |
| contents | The cosmological polytope of a graph $G$ was recently introduced to give a geometric approach to the computation of wavefunctions for cosmological models with associated Feynman diagram $G$. Basic results in the theory of positive geometries dictate that this wavefunction may be computed as a sum of rational functions associated to the facets in a triangulation of the cosmological polytope. The normalized volume of the polytope then provides a complexity estimate for these computations. In this paper, we examine the (Ehrhart) $h^\ast$-polynomial of cosmological polytopes. We derive recursive formulas for computing the $h^\ast$-polynomial of disjoint unions and $1$-sums of graphs. The degree of the $h^\ast$-polynomial for any $G$ is computed and a characterization of palindromicity is given. Using these observations, a tight lower bound on the $h^\ast$-polynomial for any $G$ is identified and explicit formulas for the $h^\ast$-polynomials of multitrees and multicycles are derived. The results generalize the existing results on normalized volumes of cosmological polytopes. A tight upper bound and a combinatorial formula for the $h^\ast$-polynomial of any cosmological polytope are conjectured. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01602 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ehrhart theory of cosmological polytopes Bruckamp, Justus Goltermann, Lina Juhnke, Martina Landin, Erik Solus, Liam Combinatorics The cosmological polytope of a graph $G$ was recently introduced to give a geometric approach to the computation of wavefunctions for cosmological models with associated Feynman diagram $G$. Basic results in the theory of positive geometries dictate that this wavefunction may be computed as a sum of rational functions associated to the facets in a triangulation of the cosmological polytope. The normalized volume of the polytope then provides a complexity estimate for these computations. In this paper, we examine the (Ehrhart) $h^\ast$-polynomial of cosmological polytopes. We derive recursive formulas for computing the $h^\ast$-polynomial of disjoint unions and $1$-sums of graphs. The degree of the $h^\ast$-polynomial for any $G$ is computed and a characterization of palindromicity is given. Using these observations, a tight lower bound on the $h^\ast$-polynomial for any $G$ is identified and explicit formulas for the $h^\ast$-polynomials of multitrees and multicycles are derived. The results generalize the existing results on normalized volumes of cosmological polytopes. A tight upper bound and a combinatorial formula for the $h^\ast$-polynomial of any cosmological polytope are conjectured. |
| title | Ehrhart theory of cosmological polytopes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.01602 |