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n-th Tropical Nevanlinna Theory

Risto Korhonen, Chengliang Tan

TL;DR

The work extends tropical Nevanlinna theory from piecewise linear to piecewise polynomial functions by constructing the $n$-th Poisson-Jensen formula and defining $n$-th integrated max-plus counting functions $N^{(j)}(r,f)$ alongside the $n$-th max-plus characteristic $T(r,f)$. It develops a pointwise tropical logarithmic derivative estimate under a growth condition, enabling a tropical analog of the logarithmic derivative lemma. The paper then formulates and proves $n$-th second main theorems for tropical homogeneous and Fermat-type polynomials within tropical projective space, and examines the interplay with tropical Casoratian structures. Additionally, it establishes a ramification-based relation that implies there is no natural tropical truncated second main theorem for shift operators in this $n$-th setting, highlighting fundamental differences from the classical theory and clarifying the role of ramification in tropical difference equations.

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.

n-th Tropical Nevanlinna Theory

TL;DR

The work extends tropical Nevanlinna theory from piecewise linear to piecewise polynomial functions by constructing the -th Poisson-Jensen formula and defining -th integrated max-plus counting functions alongside the -th max-plus characteristic . It develops a pointwise tropical logarithmic derivative estimate under a growth condition, enabling a tropical analog of the logarithmic derivative lemma. The paper then formulates and proves -th second main theorems for tropical homogeneous and Fermat-type polynomials within tropical projective space, and examines the interplay with tropical Casoratian structures. Additionally, it establishes a ramification-based relation that implies there is no natural tropical truncated second main theorem for shift operators in this -th setting, highlighting fundamental differences from the classical theory and clarifying the role of ramification in tropical difference equations.

Abstract

In this paper, the tropical Nevanlinna theory is extended for piecewise polynomial continuous functions. By constructing the -th Poisson-Jensen formula, the -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 -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 and the ramification term , implying that there is no natural tropical truncated version of the second main theorem for shift operators.
Paper Structure (6 sections, 22 theorems, 167 equations, 1 table)

This paper contains 6 sections, 22 theorems, 167 equations, 1 table.

Table of Contents

  1. Introduction
  2. $n$-th tropical meromorphic functions
  3. $n$-th tropical Nevanlinna theory
  4. $n$-th tropical version of logarithmic derivative lemma
  5. $n$-th tropical holomorphic curve
  6. A relationship about tropical truncated second main theorem

Key Result

Proposition 2.2

Let $n\in\mathbb{N}$, then for any $n$-th tropical entire function $f(x)$, we have for all $r> 0$. In particular, the equality holds when $f(x)$ is tropical nowhere vanishing entire.

Theorems & Definitions (50)

  • Example 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 40 more