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Published July 23, 2015 | Published + Submitted
Journal Article Open

From Polygons to Ultradiscrete Painlevé Equations


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.

Additional Information

© 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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August 22, 2023
August 22, 2023