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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
Zbl 1348.30004
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