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.

• Tropical Polynomials, Rationals and Exponentials
• Tropical Entire Functions
• One-dimensional Tropical Nevanlinna Theory
• Clunie and Mohon'ko Type Theorems
• Tropical Holomorphic Curves
• Representations of Tropical Periodic Functions
• Applications to Ultra-Discrete Equations
• Appendices:
  • Classical Nevanlinna and Cartan Theories
  • Introduction to Ultra-Discrete Painlevé Equations
  • Some Operators in Complex Analysis

• First monograph type presentation in this topic
• Some of the material is new in contents, which follows in the presentation
• The contributors have been active in value distribution theory, including its tropical variant