×

zbMATH — the first resource for mathematics

Ultra-discrete equations and tropical counterparts of some complex analysis results. (English) Zbl 1403.39004
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. [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.
MSC:
39A12 Discrete version of topics in analysis
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Citations:
Zbl 1348.30004
PDF BibTeX XML Cite
Full Text: DOI
References:
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.