Discrete Equidecomposability and Ehrhart Theory of Polygons
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Author(s)
Turner, Paxton
Wu, Yuhuai
Date Issued
June 2020
Journal
Discrete & Computational Geometry
Publisher
Springer Science and Business Media LLC
Citation
Turner, Paxton and Yuhuai Wu. "Discrete Equidecomposability and Ehrhart Theory of Polygons." Discrete & Computational Geometry 65, 1 (June 2020): 90–115 © 2020 Springer Science Business Media
Version
Author's final manuscript
Abstract
Motivated by questions from Ehrhart theory, we present new results on discrete equidecomposability. Two rational polygons P and Q are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection (equivalently, a piecewise affine-linear bijection that preserves the integer lattice Z²) from P to Q. We develop an invariant for a particular version of this notion called rational finite discrete equidecomposability. We construct triangles that are Ehrhart equivalent but not rationally finitely discretely equidecomposable, thus providing a partial negative answer to a question of Haase–McAllister on whether Ehrhart equivalence implies discrete equidecomposability. Surprisingly, if we delete an edge from each of these triangles, there exists an infinite rational discrete equidecomposability relation between them. Our final section addresses the topic of infinite equidecomposability with concrete examples and a potential setting for further investigation of this phenomenon.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00454-020-00211-1
