Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 056, 36 pages      arXiv:1408.5643      https://doi.org/10.3842/SIGMA.2015.056

From Polygons to Ultradiscrete Painlevé Equations

Christopher Michael Ormerod a and Yasuhiko Yamada b
a) Department of Mathematics, California Institute of Technology, 1200 E California Blvd, Pasadena, CA, 91125, USA
b) Department of Mathematics, Kobe University, Rokko, 657-8501, Japan

Received January 29, 2015, in final form July 10, 2015; Published online July 23, 2015

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.

Key words: ultradiscrete; tropical; Painlevé; QRT; Cremona.

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