n-th Tropical Nevanlinna Theory

TL;DR

This paper extends tropical Nevanlinna theory to piecewise polynomial functions, establishing new second main theorems and relationships between counting functions and ramification, advancing the understanding of tropical value distribution.

Contribution

It introduces the $n$-th Poisson-Jensen formula and develops second main theorems for tropical polynomials, including Fermat type, with novel properties and growth estimates.

Findings

Established the $n$-th Poisson-Jensen formula.

Derived second main theorems for tropical homogeneous and Fermat type polynomials.

Proved a strong equality relating counting functions and ramification terms.

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.