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arXiv · 1408.5643

From Polygons to Ultradiscrete Painlev\'e Equations

Christopher Michael Ormerod·Yasuhiko Yamada
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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.

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Christopher Michael Ormerod, Yasuhiko Yamada. 2014-08-24. From Polygons to Ultradiscrete Painlev\'e Equations. https://doi.org/10.3842/sigma.2015.056

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