Tropical geometric interpretation of ultradiscrete singularity confinement

TL;DR

This paper uses tropical geometry to interpret ultradiscrete singularities as points of non-differentiability, establishing a link between discrete and ultradiscrete integrable systems' singularity confinement.

Contribution

It introduces a tropical geometric framework to understand ultradiscrete singularities, connecting non-Archimedean valuation with singularity confinement.

Findings

Roots and poles correspond to non-differentiability points in piece-wise linear systems

A correspondence between discrete and ultradiscrete singularity confinement is demonstrated

Tropical geometry provides a new perspective on ultradiscrete integrability

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.