2024/05/22 by Martin Bohnert, Bohnert, Martin
Computer Science · Engineering · Physics and Astronomy · #Polynomial and algebraic computation #Elasticity and Wave Propagation #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2405.13404
A half-integral polygon with quasi-period collapse behaves similarly to a lattice polygon in the sense that the number of lattice points in its integer dilates can be calculated as values of a polynomial, its Ehrhart polynomial. As a main result, we classify the Ehrhart polynomials of all half-integral non-lattice polygons with quasi-period collapse. In particular, we obtain that for any positive integer i, the polynomial (4i+5)/(2)t2+(2i+7)/(2)t+1∈ ℚ[t] is an Ehrhart polynomial of a rational polygon, which was an open question for i>1. We also study some extreme cases in detail. In particular, we show that up to affine unimodular equivalence there exist exactly 30 half-integral non-lattice polygons with quasi-periodic collapse with exactly one interior lattice point, which are the dual polygons of the 30 LDP polygons of Gorenstein index 2. Furthermore, we classify all half-integral polygons with quasi-period collapse with at most 6 interior lattice points or with i≥ 1 interior lattice points and the maximum possible number 2i+7 of boundary lattice points.