n-th Tropical Nevanlinna Theory

Abstract

In this paper, the tropical Nevanlinna theory is extended for piecewise polynomial continuous functions. By constructing the $n$-th Poisson-Jensen formula, the $n$-th tropical counting, proximity, and characteristic functions are introduced, which have some different properties compared to the classical tropical setting. Then, not only is the $n$-th version of the second main theorem for tropical homogeneous polynomials obtained, but also a tropical second main theorem for ordinary Fermat type polynomials is acquired. Moreover, by estimating the tropical logarithmic derivative with a growth assumption pointwise, a strong equality is proved. This equality illustrates the relationship between ${\sum }_{i=0}^{m}N(r,1_0\oslash f_i)$ and the ramification term $N(r,C_0(f_0,\cdots ,f_m))$, implying that there is no natural tropical truncated version of the second main theorem for shift operators.

48 pages