Discrete Equidecomposability and Ehrhart Theory of Polygons
Author(s)
Turner, Paxton; Wu, Yuhuai
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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.
Date issued
2020-06Department
Massachusetts Institute of Technology. Department of MathematicsJournal
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
ISSN
0179-5376
1432-0444