Tropical Nevanlinna theory and second main theorem
Tropical Nevanlinna theory and second main theorem
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DOI：
10.1112/plms/pdq049
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发表时间：
2009-10
影响因子：
1.8
通讯作者：
I. Laine;K. Tohge
中科院分区：
文献类型：
--
作者：
I. Laine;K. Tohge
热带Nevanlinna理论描述了具有任意真实的斜率的真实的变量的连续分段线性函数的值分布，称为热带亚纯函数，类似于经典Nevanlinna理论中描述的亚纯函数。在以前的两篇论文中，由于Halburd和绍瑟尔以及Laine和Yang，只允许整数斜率;这两篇论文中给出的热带Nevanlinna理论的基本结果在我们的扩展设置中也仍然有效。在本文中，我们提出了一个热带版本的第二个主要定理想起相应的结果在经典的Nevanlinna理论。我们还分析了热带周期函数和热带超指数函数，这两个函数都具有关于第二主要定理的某种极值行为。还给出了对一阶和二阶超离散方程的应用。我们分别用热带周期函数和热带超指数函数明确地表示这些方程的解。限制超阶至多为1似乎对这些方程的热带亚纯解的存在性至关重要，而不是限制阶为有限。
Tropical Nevanlinna theory describes value distribution of continuous piecewise linear functions of a real variable with arbitrary real slopes, called tropical meromorphic functions, similarly as meromorphic functions are described in the classical Nevanlinna theory. In two previous papers, due to Halburd and Southall and to Laine and Yang, integer slopes only had been permitted; basic results of tropical Nevanlinna theory given in these two papers continue to be valid in our extended setting as well. In this paper, we present a tropical version of the second main theorem reminiscent of the corresponding result in the classical Nevanlinna theory. We also give an analysis of tropical periodic functions as well as of tropical hyper‐exponential functions, both of which have a certain extremal behaviour with respect to the second main theorem. Application to some ultra‐discrete equations of first and second order is also given. We explicitly express the solutions of these equations in terms of tropical periodic and of tropical hyper‐exponential functions, respectively. Restriction of the hyper‐order being at most 1 seems to be crucial for the existence of tropical meromorphic solutions of these equations rather than restricting the order to be finite.