Tropical Nevanlinna Theory and Ultradiscrete Equations

Abstract

A tropical version of Nevanlinna theory is described in which the role of meromorphic functions is played by continuous piecewise linear functions of a real variable whose one-sided derivatives are integers at every point. These functions are naturally defined on the max-plus (or tropical) semi-ring. Analogues of the Nevanlinna characteristic, proximity and counting functions are defined and versions of Nevanlinna's first main theorem, the lemma on the logarithmic derivative and Clunie's lemma are proved.As well as providing another example of a tropical or dequantized analogue of an important area of complex analysis, this theory has applications to so-called ultra-discrete equations. Preliminary results are presented suggesting that the existence of finite-order max-plus meromorphic solutions can be considered to be an ultra-discrete analogue of the Painlevé property.

A note on Ultra-discrete equations

“…Finally, we end this section with some remarks about a new suggested integrability detector. There have been some encouraging numerical results suggesting that Trpoical Nevanlinna theory might be used to comment on the integrability of ultra-discrete equations [40]. We end this note by giving some summarized remarks in section 5.…”

“…This term was first used by French mathematicians to honour their Brazilian colleague Imre Simon according to the authors of [41]. This semi-ring was first appeared in middle of the last century in Kleene's paper on nerve sets and automata according to [40]. Another semi-ring that is Idempotent and Tropical the min-plus semi-ring where…”

“…Some of these systems appeared in the middle of the last century e.g. the max-plus algebra (appeared first in Kleene's paper on nerve sets and automata as it was stated in [40]), the min-plus algebra and the minimax algebra [11]. There are lots of fields (e.g.…”

Tropical variants of some complex analysis results

“…Tropical Nevanlinna theory may be understood as a cross-road between tropical mathematics and the classical Nevanlinna theory, see [17] for a general background of tropical mathematics and [8], [12] and [13] for the tropical setting of the Nevanlinna theory. Indeed, tropical Nevanlinna theory provides the flexibility of applying complex analysis methods to considering real functions.…”

“…Indeed, tropical Nevanlinna theory provides the flexibility of applying complex analysis methods to considering real functions. Recall that Halburd and Southall [8] described continuous piecewise linear functions on R with one-side integer derivatives as tropical meromorphic functions, and established tropical versions of the first main theorem and the lemma on the logarithmic derivative. Laine and Tohge [13] then showed that tropical Nevanlinna theory also holds to piecewise linear functions with arbitrary real slopes, and obtained a tropical version of the second main theorem.…”