The algebraic matroid of the Heron variety
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916225884356608 |
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| author | Asante, Seth K. Brysiewicz, Taylor Hatzel, Michelle |
| author_facet | Asante, Seth K. Brysiewicz, Taylor Hatzel, Michelle |
| contents | We introduce the n-th Heron variety as the realization space of the (squared) volumes of faces of an n-simplex. Our primary goal is to understand the extent to which Heron's formula, which expresses the area of a triangle as a function of its three edge lengths, can be generalized. Such a formula for one face volume of an n-simplex in terms of other face volumes expresses a dependence in the algebraic matroid of the Heron variety. Whether the volume is expressible in terms of radicals is controlled by the monodromy groups of the coordinate projections of the Heron variety onto coordinates of bases. We discuss a suite of algorithms, some new, for determining these matroids and monodromy groups. We apply these algorithms toward the smaller Heron varieties, organize our findings, and interpret the results in the context of our original motivation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06286 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The algebraic matroid of the Heron variety Asante, Seth K. Brysiewicz, Taylor Hatzel, Michelle Algebraic Geometry Combinatorics 14D05 (Primary) 65H14, 51M25, 51K05 (Secondary) We introduce the n-th Heron variety as the realization space of the (squared) volumes of faces of an n-simplex. Our primary goal is to understand the extent to which Heron's formula, which expresses the area of a triangle as a function of its three edge lengths, can be generalized. Such a formula for one face volume of an n-simplex in terms of other face volumes expresses a dependence in the algebraic matroid of the Heron variety. Whether the volume is expressible in terms of radicals is controlled by the monodromy groups of the coordinate projections of the Heron variety onto coordinates of bases. We discuss a suite of algorithms, some new, for determining these matroids and monodromy groups. We apply these algorithms toward the smaller Heron varieties, organize our findings, and interpret the results in the context of our original motivation. |
| title | The algebraic matroid of the Heron variety |
| topic | Algebraic Geometry Combinatorics 14D05 (Primary) 65H14, 51M25, 51K05 (Secondary) |
| url | https://arxiv.org/abs/2401.06286 |