Tropical geometric interpretation of ultradiscrete singularity confinement

Abstract

Abstract. Using the interpretation of the ultradiscretization procedure as a non-Archimedean valuation, we use results of tropical geometry to show how roots and poles manifest themselves in piece-wise linear systems as points of non-differentiability. This will allow us to demonstrate a correspondence between singularity confinement for discrete integrable systems and ultradiscrete singularity confinement for ultradiscrete integrable systems.

Integrability criterion in terms of coprime property for the discrete Toda equation

“…By applying the co-prime criterion and by investigating the common factors even more closely, we expect to obtain convincing results on the integrability of these equations in future works. It is also a good idea to investigate the relation of our results with other integrability criteria such as the p-adic number theoretic interpretation of the confined singularities [9], and the singularity confinement for ultra-discrete systems [8], which has recently been studied in relation to the tropical geometry [12].…”

From Polygons to Ultradiscrete Painlevé Equations

“…In the integrable community a nonanalytic limit known as ultradiscretization is used as a way of obtaining new and interesting piecewise linear integrable systems [52]. Relating tropicalization with ultradiscretization gives us a way of understanding the geometry of ultradiscrete systems [36].…”

“…for all subtraction free functions [34,36]. The above extension, given by (3.5), is one of a number of ways to incorporate a version of subtraction into the ultradiscretization procedure [12,22,23,32].…”

“…The ultradiscrete Painlevé equations are second order non-linear difference equations defined over the max-plus semifield that are integrable in the sense that they possess many of same properties of the continuous and discrete Painlevé equations that are associated with integrability, albeit, in some tropical form. These properties include tropical Lax representations [15,35] and tropical singularity confinement [14,36]. They also admit symmetry groups of affine Weyl type [18,19] and special solutions of rational and hypergeometric type [26,34,50].…”

“…Since every non-degenerate level set of a tropical biquadratic function is a tropical genus one curve, one may say that the ultradiscrete QRT maps can be lifted to automorphisms of tropical elliptic surfaces. Given the geometric interpretation of tropical singularity confinement [36], the positions of the rays in any pencil of tropical genus one curves play the same role as the positions of the base points in a pencil of genus one curves. In this way, there is an analogous constraint on the positions of the rays of any pencil of tropical genus one curves, which when removed, results in curves that are no longer closed.…”