Tropical Value Distribution Theory and Ultra-Discrete Equations

The following sections are included:

• Preface
• Contents

Tropical value distribution theory, introduced by Halburd and Southall [46] and extended by Laine and Tohge [69], describes value distribution of continuous piecewise linear functions of a real variable. In many aspects, the presentation has similarities to what is known as the meromorphic function theory as described in the classical complex analysis. In these lecture notes, we start by looking at the one-dimensional setting, i.e. functions f : ℝ → ℝ. Later on in Chapter 5, we shall consider corresponding functions defined on a tropical projective space.

This chapter is devoted to studying tropical transcendental entire functions on ℝ defined by max-plus series expansions of type

This is, of course, a natural extension of tropical polynomials that we have treated in Chapter 1, see also Tsai [106] for tropical polynomials and their fundamental properties. Moreover, some of tropical exponentials considered in Chapter 1 are basic examples of transcendental tropical entire functions …

In Chapter 1 we introduced Nevanlinna functions for tropical meromorphic functions and looked at a few examples of their behavior in the case of tropical polynomials and rational functions. In this chapter we present the full details of the one-dimensional tropical Nevanlinna theory, including the first and the second main theorems.

In this chapter, we prove tropical counterparts to three lemmas from classical complex analysis, frequently used in applications of Nevanlinna theory. As to these classical results, the reader may look at a lemma due to Valiron and Mohon'ko, see [68], Theorem 2.2.5, another one due to Mohon'ko, see [68], Proposition 9.2.3, and the original Clunie lemma in [18]. Previous tropical versions for these lemmas may be found in [46], [71] and [69]. The first two of these references are treating the restricted case of integer slopes only.

Cartan's generalization of Nevanlinna's second main theorem for holomorphic curves in the finite dimensional projective space is an important extension of value distribution theory, which has surprising applications in the study of meromorphic functions in the complex plane as well. The purpose of this chapter is to introduce a tropical version of Cartan's value distribution theory. Following [67], we will define a tropical analogue of Cartan's characteristic function, and show that it is well defined and reduces exactly to the tropical characteristic function introduced in Chapter 3 in the special case where the tropical holomorphic curve is one-dimensional. The central result of this section is a tropical analogue of Cartan's second theorem. We will show that this result is a natural extension of the tropical second main theorem, Theorem 3.40, by proving that under a certain natural non-degeneracy condition on targets, the tropical Cartan theorem reduces to Theorem 3.40 in the one-dimensional case. The tropical Cartan's second main theorem also implies a second main theorem type inequality, which appears to include a previously unknown ramification type term. We will conclude the section by a discussion on ramification in the tropical meromorphic functions. In order to study the properties of tropical holomorphic curves, we first need to introduce some notions from tropical linear algebra.

In this chapter, our key result is to giving a representation of tropical meromorphic, periodic functions in terms of certain elementary tropical meromorphic functions that have, essentially, a sawtooth waveform, with their width, height and inclination depending on two parameters. Before proceeding to this representation, we first make some simple observations concerning periodic functions in the tropical setting. Without restricting generality, we assume in this chapter that tropical meromorphic functions f under consideration here are of period 1, unless otherwise specified. Then, trivially, f satisfies the elementary equation y(x + 1) ⊘ y(x) = 0, that is, y(x + 1) − y(x) = 0.

In this chapter we look into methods of solving ultra-discrete equations in terms of tropical meromorphic functions. We present general solutions of certain relatively simple classes of ultra-discrete equations, and classes of special rational and hypergeometric solutions of ultra-discrete Painlevé equations. In addition, we describe how to apply tropical Nevanlinna theory to study the growth of tropical meromorphic solutions of ultra-discrete equations.

The following sections are included:

• Appendix A: Classical Nevanlinna and Cartan theories
  • Classical Nevanlinna theory
  • Difference variant of Nevanlinna theory
  • Cartan's version of Nevanlinna theory
  • Difference variant of Cartan theory
• Appendix B: Introduction to ultra-discrete Painlevé equations
  • Painlevé equations
  • Integrability of Painlevé equations and integrability testing
  • Discrete Painlevé equations
  • Discrete Painlevé equations and integrability testing
  • Ultra-discrete Painlevé equations
  • Ultra-discrete Painlevé equations and integrability testing
  • Hypergeometric solutions to Painlevé equations
• Appendix C: Some operators in complex analysis
  • Logarithmic order and type
  • Some operators and related series expansions in complex analysis
    • The case of difference operator
    • The case of q-difference operator
• Bibliography
• Index