Ultra-discrete equations and tropical counterparts of some complex analysis results
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.
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