Geometry of tropical extensions of hyperfields
Fuente:
arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
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| _version_ | 1866929604418076672 |
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| author | Maxwell, James Smith, Ben |
| author_facet | Maxwell, James Smith, Ben |
| contents | We study the geometry of tropical extensions of hyperfields, including the ordinary, signed and complex tropical hyperfields. We introduce the framework of 'enriched valuations' as hyperfield homomorphisms to tropical extensions, and show that a notable family of them are relatively algebraically closed. Our main results are hyperfield analogues of Kapranov's theorem and the Fundamental theorem of tropical geometry. Utilising these theorems, we introduce fine tropical varieties and prove a structure theorem for them in terms of their initial ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_17302 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometry of tropical extensions of hyperfields Maxwell, James Smith, Ben Algebraic Geometry Rings and Algebras 12K99, 14T10 (Primary) 12J25, 12D10 (Secondary) We study the geometry of tropical extensions of hyperfields, including the ordinary, signed and complex tropical hyperfields. We introduce the framework of 'enriched valuations' as hyperfield homomorphisms to tropical extensions, and show that a notable family of them are relatively algebraically closed. Our main results are hyperfield analogues of Kapranov's theorem and the Fundamental theorem of tropical geometry. Utilising these theorems, we introduce fine tropical varieties and prove a structure theorem for them in terms of their initial ideals. |
| title | Geometry of tropical extensions of hyperfields |
| topic | Algebraic Geometry Rings and Algebras 12K99, 14T10 (Primary) 12J25, 12D10 (Secondary) |
| url | https://arxiv.org/abs/2309.17302 |