Tropical value distribution theory and ultra-discrete equations

"This is the first textbook-type presentation of tropical value distribution theory. It provides a detailed introduction of the tropical version of the Nevanlinna theory, describing growth and value distribution analysis of continuous, piecewise linear functions on the real axis. The book also includes applications of this theory to ultra-discrete equations. Three appendices are given to compare the contents of the theory with the classical counterparts in complex analysis. Detailed presentation of the proofs makes the book accessible for lecture courses and independent studies at the graduate and post-doctoral level."--

Machine generated contents note: 1.1. Basic notions and elementary results for tropical polynomials and rationals -- 1.1.1. Basic properties for tropical polynomials and rational functions -- 1.1.2. Equivalence classes and compact forms of tropical polynomials -- 1.1.3. Tropical version of the fundamental theorem of algebra -- 1.1.4. Tropical rational functions -- 1.2. Definitions of Nevanlinna functions -- 1.2.1. Basic definitions -- 1.2.2. Nevanlinna functions for tropical polynomials -- 1.2.3. Order of tropical meromorphic functions -- 1.2.4. Nevanlinna functions for tropical exponentials -- 2.1. Definitions and basic results -- 2.1.1. Preliminaries -- 2.1.2. Growth order and tropical series expansions -- 2.1.3. Main theorem for tropical series expansions -- 2.1.4. Proof of Theorem 2.4, first part -- 2.1.5. Proof of Theorem 2.4, second part -- 2.1.6. Representations of tropical entire functions -- 2.2. Examples of tropical entire functions

Note continued: 2.2.1. Tropical entire functions of arbitrary order -- 2.2.2.A q-analogue of the exponential function -- 2.2.3.A tropical entire function related to q-difference equations -- 2.2.4.A concluding remark -- 3.1. Poisson[--]Jense…