Discrete Equidecomposability and Ehrhart Theory of Polygons
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Discrete Equidecomposability and Ehrhart Theory of Polygons Paxton Turner1
· Yuhuai Wu2
Received: 7 February 2018 / Revised: 25 February 2020 / Accepted: 21 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Combinatorics · Discrete geometry · Dissections and valuations · Lattice polytopes Mathematics Subject Classification 52B20 · 52B45
Editor in Charge: János Pach This material is based upon work supported by the National Science Foundation under Grant No. DMS0931908 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI during the Summer@ICERM 2014 program. Paxton Turner [email protected] Yuhuai Wu [email protected] 1
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2
Computer Science, University of Toronto, Toronto, ON M5S2E4, Canada
123
Discrete & Computational Geometry
1 Introduction 1.1 Ehrhart Theory Ehrhart theory is the study of counting the number of integer lattice points in integral dilates of polytopes [3]. Let P denote a polytope in Rk . Given t ∈ N, the t th dilate of P is the set t P = {t p | p ∈ P}. Here t p denotes scalar multiplication of the point p by t. With this set-up, we can define the central object of Ehrhart theory, the function ehr P (t) that counts the number of lattice points in t P: ehr P (t) := |{t P ∩ Zk }|.
(1)
We say that P is an integral polytope, or simply, P is integral if its vertices lie in the lattice Zk . Similarly, a rational polytope has all of its vertices given by points whose coordinates are rational. If P is rational, the denominator of P is defined to be the least positive integer d such that d P is an integral polytope. A fundamental theorem due to Ehrhart states that if P is a rational polytope with denominator d, then ehr P (t) is a quasi-polynomial of period d [2,3,11,12]. Polygons P and Q are defined to be Ehrhart equivalent if ehr P (t) = ehr Q (t). 1.2 Discrete Equidecomposability Unless expli
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