From Polygons to Ultradiscrete Painlev\'e Equations

TL;DR

This paper explores the geometric and algebraic structures underlying ultradiscrete Painlevé equations through the study of spiraling polygons and their associated piecewise linear transformations, revealing connections to affine Weyl groups.

Contribution

It introduces a novel geometric framework involving spiraling polygons and demonstrates their link to ultradiscrete Painlevé equations via tropical rational transformations.

Findings

Spiraling polygons correspond to ultradiscrete Painlevé equations.

Piecewise linear transformations form tropical rational groups.

Connections to affine Weyl groups are established.

Abstract

The rays of tropical genus one curves are constrained in a way that defines a bounded polygon. When we relax this constraint, the resulting curves do not close, giving rise to a system of spiraling polygons. The piecewise linear transformations that preserve the forms of those rays form tropical rational presentations of groups of affine Weyl type. We present a selection of spiraling polygons with three to eleven sides whose groups of piecewise linear transformations coincide with the B\"acklund transformations and the evolution equations for the ultradiscrete Painlev\'e equations.