Quadratically Enriched Tropical Intersections
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866914957375832064 |
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| author | Puentes, Andrés Jaramillo Pauli, Sabrina |
| author_facet | Puentes, Andrés Jaramillo Pauli, Sabrina |
| contents | Using tropical geometry one can translate problems in enumerative geometry to combinatorial problems. Thus tropical geometry is a powerful tool in enumerative geometry over the complex and real numbers. Results from $\mathbb{A}^1$-homotopy theory allow to enrich classical enumerative geometry questions and get answers over an arbitrary field. In the resulting area, $\mathbb{A}^1$-enumerative geometry, the answer to these questions lives in the Grothendieck-Witt ring of the base field $k$. In this paper, we use tropical methods in this enriched set up by showing Bézout's theorem and a generalization, namely the Bernstein-Kushnirenko theorem, for tropical hypersurfaces enriched in $\operatorname{GW}(k)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_00240 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quadratically Enriched Tropical Intersections Puentes, Andrés Jaramillo Pauli, Sabrina Algebraic Geometry 14N10 (Primary) 14T25, 14G27 (Secondary) Using tropical geometry one can translate problems in enumerative geometry to combinatorial problems. Thus tropical geometry is a powerful tool in enumerative geometry over the complex and real numbers. Results from $\mathbb{A}^1$-homotopy theory allow to enrich classical enumerative geometry questions and get answers over an arbitrary field. In the resulting area, $\mathbb{A}^1$-enumerative geometry, the answer to these questions lives in the Grothendieck-Witt ring of the base field $k$. In this paper, we use tropical methods in this enriched set up by showing Bézout's theorem and a generalization, namely the Bernstein-Kushnirenko theorem, for tropical hypersurfaces enriched in $\operatorname{GW}(k)$. |
| title | Quadratically Enriched Tropical Intersections |
| topic | Algebraic Geometry 14N10 (Primary) 14T25, 14G27 (Secondary) |
| url | https://arxiv.org/abs/2208.00240 |