Tropical Nevanlinna theory and second main theorem

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.