From Polygons to Ultradiscrete Painlevé Equations

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 Bäcklund transformations and the evolution equations for the ultradiscrete Painlevé equations.

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© 2015 The authors. The authors retain the copyright for their papers published in SIGMA under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. Received January 29, 2015, in final form July 10, 2015; Published online July 23, 2015. Christopher M. Ormerod would like to acknowledge Eric Rains for his helpful discussions. Y. Yamada is supported by JSPS KAKENHI Grant Number 26287018.

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Submitted - 1408.5643v2.pdf

Published - sigma15-056.pdf

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