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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.03500 |
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| _version_ | 1866915770619920384 |
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| author | Korhonen, Risto Tan, Chengliang |
| author_facet | Korhonen, Risto Tan, Chengliang |
| contents | In this paper, the tropical Nevanlinna theory is extended for piecewise polynomial continuous functions. By constructing the $n$-th Poisson-Jensen formula, the $n$-th tropical counting, proximity, and characteristic functions are introduced, which have some different properties compared to the classical tropical setting. Then, not only is the $n$-th version of the second main theorem for tropical homogeneous polynomials obtained, but also a tropical second main theorem for ordinary Fermat type polynomials is acquired. Moreover, by estimating the tropical logarithmic derivative with a growth assumption pointwise, a strong equality is proved. This equality illustrates the relationship between $\sum_{i=0}^{m}N(r, 1_{0}\oslash f_{i})$ and the ramification term $N(r, C_{0}(f_{0}, \cdots, f_{m}))$, implying that there is no natural tropical truncated version of the second main theorem for shift operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03500 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | n-th Tropical Nevanlinna Theory Korhonen, Risto Tan, Chengliang Algebraic Geometry 14T90, 30D35, 32H30 In this paper, the tropical Nevanlinna theory is extended for piecewise polynomial continuous functions. By constructing the $n$-th Poisson-Jensen formula, the $n$-th tropical counting, proximity, and characteristic functions are introduced, which have some different properties compared to the classical tropical setting. Then, not only is the $n$-th version of the second main theorem for tropical homogeneous polynomials obtained, but also a tropical second main theorem for ordinary Fermat type polynomials is acquired. Moreover, by estimating the tropical logarithmic derivative with a growth assumption pointwise, a strong equality is proved. This equality illustrates the relationship between $\sum_{i=0}^{m}N(r, 1_{0}\oslash f_{i})$ and the ramification term $N(r, C_{0}(f_{0}, \cdots, f_{m}))$, implying that there is no natural tropical truncated version of the second main theorem for shift operators. |
| title | n-th Tropical Nevanlinna Theory |
| topic | Algebraic Geometry 14T90, 30D35, 32H30 |
| url | https://arxiv.org/abs/2602.03500 |