# Tropical Nevanlinna theory of several variables

## TL;DR

This paper extends Nevanlinna theory into tropical geometry for functions of several variables, establishing key theorems and concepts like the proximity, counting, and characteristic functions in higher dimensions.

## Contribution

It develops a higher-dimensional tropical Nevanlinna theory, including the first main theorem and second main theorem for tropical holomorphic maps.

## Key findings

- Established tropical versions of the logarithmic derivative lemmas.
- Proved a second main theorem for algebraically nondegenerate tropical maps.
- Extended classical Nevanlinna theory to several variables in tropical geometry.

## Abstract

The main goal of this paper is to establish the higher-dimensional Nevanlinna theory in tropical geometry. We first develop a theory of tropical meromorphic functions ( holomorphic maps) in several variables, such as the proximity function, counting function and characteristic function, the first main theorem, higher-dimensional tropical versions of the logarithmic derivative lemmas. Based on this, for algebraically nondegenerate tropical holomorphic maps $f$ with subnormal growth from $\mathbb{R}^n$ into tropical projective space $\mathbb{TP}^{m}$ intersecting tropical hypersurfaces $\{V_{P_j}\}_{j=1}^{q}$ with degree $d_{j},$ we then obtain the Second Main Theorem $$\|\,\,\, (q-M-1-\lambda)T_f(r) \leq \sum_{j=M+2}^q \tfrac{1}{d_j}N(r,1_{\mathbb{T}} \oslash P_j \circ f) + o(T_f(r)),$$ where $d=lcd(d_{1}, \ldots, d_{q})$ and $M=(_d^{m+d})-1.$