Working with Tropical Meromorphic Functions of One Variable

TL;DR

This paper surveys tropical meromorphic functions, introduces practical computational techniques, and establishes foundational theorems linking tropical and classical complex analysis.

Contribution

It provides new methods for representing tropical polynomials, proves existence and uniqueness theorems, and explores the relationship between tropical and classical meromorphic functions.

Findings

Introduces maximally represented tropical polynomials

Proves existence and uniqueness of tropical meromorphic functions

Establishes analogues of complex analysis theorems in tropical setting

Abstract

In this paper, we survey and study definitions and properties of tropical polynomials, tropical rational functions and in general, tropical meromorphic functions, emphasizing practical techniques that can really carry out computations. For instance, we introduce maximally represented tropical polynomials and tropical polynomials in compact forms to quickly find roots of given tropical polynomials. We also prove the existence and uniqueness of tropical theorems for meromorphic functions with prescribed roots and poles. Moreover, we explain the relations between classical and tropical meromorphic functions. Different definitions and applications of tropical meromorphic functions are discussed. Finally, we point out the properties of tropical meromorphic functions are very similar to complex ones and prove some tropical analogues of theorems in complex analysis.