Table of Contents
Preface v
1 Tropical Polynomials, rationals and exponentials 1
1.1 Basic notions and elementary results for tropical polynomials and rations 1
1.1.1 Basic properties for tropical polynomials and rational functions 2
1.1.2 Equivalence classes and compact forms of tropical polynomials 7
1.1.3 Tropical version of the fundamental theorem of algebra 8
1.1.4 Tropical rational functions 13
1.2 Definitions of Nevanlinna functions 14
1.2.1 Basic Definitions 14
1.2.2 Nevanlinna functions for tropical polynomials 15
1.2.3 Order of Tropical meromorphic functions 17
1.2.4 Nevanlinna functions for tropical exponentials 19
2 Tropical entire functions 29
2.1 Definitions and basic results 30
2.1.1 Preliminaries 30
2.1.2 Growth order and tropical series expansions 34
2.1.3 Main theorem for tropical series expansions 37
2.1.4 Proof of Theorem 2.4, first part 40
2.1.5 Proof of Theorem 2.4, second part 42
2.1.6 Representations of tropical entire functions 45
2.2 Examples of tropical entire functions 53
2.2.1 Tropical entire functions of arbitrary order 53
2.2.2 A q-analogue of the exponential function 55
2.2.3 A tropical entire function related to q-difference equations 57
2.2.4 A concluding remark 62
3 One-dimensional tropical Nevanlinna theory 65
3.1 Poisson-Jensen formula in the tropical setting 65
3.2 Basic properties of Nevanlinna functions 67
3.2.1 First main theorem 69
3.2.2 Tropical Cartan identity 71
3.3 Auxiliary results from real analysis 75
3.3.1 Borel-type theorems 75
3.4 Variants of the lemma on tropical quotients 78
3.5 Second main theorem 83
3.5.1 General form of the second main theorem 83
3.5.2 Variants of the second main theorem and deficiencies 92
4 Clunie and Mohon'ko type theorems 97
4.1 Valiron Mohon'ko and Mohon'ko lemmas in tropical setting 97
4.2 Tropical Clunie lemma 104
5 Tropical holomorphic curves 111
5.1 Tropical matrixes and determinants 111
5.2 Tropical Casoratian 112
5.3 Tropical linear independence 114
5.4 Tropical holomorphic curves 121
5.5 Second main theorem for tropical holomorphic curves 131
5.6 Remification 136
5.7 Second main theorem as an application of the one-dimensional case 140
6 Representations of tropical periodic functions 145
6.1 Representations of tropical periodic functions 145
6.2 Ultra-discrete theta functions 153
7 Applications to ultra-discrete equations 157
7.1 First-order ultra-discrete equations 157
7.2 Second-order ultra-discrete equations 160
7.3 What is the general solution to ultra-discrete equations? 169
7.4 Slow growth criterion as a detector of ultra-discrete Painlevé equations 172
7.5 Tropical rational solutions to ultra-discrete Painlevé equations 175
7.6 Ultra-discrete hypergeometric solutions to Ultra-discrete Painlevé equations 177
7.7 An ultra-discrete operator 181
7.8 Ultra-discrete hypergeometric function 2Φ1 184
Appendix A Classical Nevanlinna and Cartan theories 189
A.1 Classical Nevanlinna theory 189
A.2 Difference variant of Nevanlinna theory 197
A.3 Cartan's version of Nevanlinna theory 200
A.4 Difference variant of Cartan theory 210
Appendix B Introduction to ultra-discrete Painlevé equations 219
B.1 Painlevé equations 219
B.2 Integrability of Painlevé equations and integrability testing 222
B.3 Discrete Painlevé equations 223
B.4 Discrete Painlevé equations and integrability testing 226
B.5 Ultra-discrete Painlevé equations 232
B.6 Ultra-discrete Painlevé equations and integrability testing 236
B.7 Hypergeometric solutions to Painlevé equations 242
Appendix C Some operations in complex analysis 247
C.1 Logarithmic order and type 247
C.2 Some operators and related series expansions in complex analysis 250
C.2.1 The case of difference operator 252
C.2.2 The case of q-difference operator 253
Bibliography 257
Index 265
