Ultra-discrete equations and tropical counterparts of some complex analysis results
type: Item
instance of: scholarly article
title: Ultra-discrete equations and tropical counterparts of some complex analysis results
zbMATH Open document ID: 1403.39004
author: Minfeng Chen
author: Zongsheng Gao
author: Jilong Zhang
published in: Kodai Mathematical Journal
publication date: 2018-09-14T00:00:00Z
review text: The authors consider a tropical version of Nevanlinna theory which describes the value distribution of continuous piecewise linear functions of a real variable, see e.g. [\textit{R. Korhonen} et al., Tropical value distribution theory and ultra-discrete equations. Hackensack, NJ: World Scientific (2015; Zbl 1348.30004)]. The tropical semi-ring is the set \(\mathbb R\cup\{-\infty\}\) with tropical addition and tropical multiplication defined by \(x\oplus y=\max(x,y)\) and \(x\otimes y=x+y\), respectively. Further, \(x^{\otimes \alpha}=\alpha x\) for all \(\alpha\in\mathbb R\). The authors obtain the following result: let \(\alpha>0\), \(P(x,f)\) be a tropical difference Laurent polynomial with tropical meromorphic functions of finite order as coefficients and \(\deg(P)>0\). If \(f(x)\) is a tropical entire function of infinite order such that \(\rho_2(f)<1\), then \(f(x)^{\otimes \alpha}\otimes P(x,f)\) cannot be a non-constant tropical meromorphic function of finite order. The authors also consider the uniqueness theory of tropical entire functions and ultra-discrete equations.
reviewed by: Katsuya Ishizaki
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 6936452
zbMATH Keywords: tropical Nevanlinna theory
zbMATH Keywords: tropical entire functions
zbMATH Keywords: value distribution
zbMATH Keywords: uniqueness
zbMATH Keywords: ultra-discrete equations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1530496839
OpenAlex ID: W2810402893
Wikidata QID: Q129607801
DOI: 10.2996/KMJ/1530496839
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description: scientific article