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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.04204 |
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| _version_ | 1866910425349619712 |
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| author | Song, JuAe |
| author_facet | Song, JuAe |
| contents | We define tropical rational function semifields $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ and prove that a tropical curve $\varGamma$ is realized (except for points at infinity) as the congruence variety $V \subset \boldsymbol{R}^n$ associated with a congruence on $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ by giving a specific map $\varGamma \to V$. Also, we shed light on the relation between congruences $E$ on $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ and congruence varieties associated with them and reveal the quotient semifield $\overline{\boldsymbol{T}(X_1, \ldots, X_n)} / E$ to play the role of coordinate rings that determine isomorphism classes of affine varieties in the classical algebraic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04204 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Congruences on tropical rational function semifields and tropical curves Song, JuAe Algebraic Geometry 14T10, 14T20 (Primary) 15A80 (Secondary) We define tropical rational function semifields $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ and prove that a tropical curve $\varGamma$ is realized (except for points at infinity) as the congruence variety $V \subset \boldsymbol{R}^n$ associated with a congruence on $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ by giving a specific map $\varGamma \to V$. Also, we shed light on the relation between congruences $E$ on $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ and congruence varieties associated with them and reveal the quotient semifield $\overline{\boldsymbol{T}(X_1, \ldots, X_n)} / E$ to play the role of coordinate rings that determine isomorphism classes of affine varieties in the classical algebraic geometry. |
| title | Congruences on tropical rational function semifields and tropical curves |
| topic | Algebraic Geometry 14T10, 14T20 (Primary) 15A80 (Secondary) |
| url | https://arxiv.org/abs/2305.04204 |