Papers › Quasi-period collapse in half-integral polygons
Quasi-period collapse in half-integral polygons
Martin Bohnert
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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)/2t²+(2i+7)/2t+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.
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