n-th Tropical Nevanlinna Theory

Risto Korhonen; Chengliang Tan

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arXiv · 2602.03500

n-th Tropical Nevanlinna Theory

Risto Korhonen·Chengliang Tan

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Abstract

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.

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BibTeX RIS

Risto Korhonen, Chengliang Tan. 2026-02-03. n-th Tropical Nevanlinna Theory. https://arxiv.org/abs/2602.03500

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