Roots of Ehrhart Polynomials of Smooth Fano Polytopes

V. Golyshev conjectured that for any smooth polytope P of dimension at most five, the roots z\in\C of the Ehrhart polynomial for P have real part equal to -1/2. An elementary proof is given, and in each dimension the roots are described explicitly. We also present examples which demonstrate that this result cannot be extended to dimension six.Comment: 10 page

Fano polytopes

Fano polytopes are the convex-geometric objects corresponding to toric Fano varieties. We give a brief survey of classification results for different classes of Fano polytopes

Reflexive polytopes of higher index and the number 12

We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a change of the underlying lattice. This allows us to efficiently classify all isomorphism classes of l-reflexive polygons up to index 200. As another application, we show that any reflexive polygon of arbitrary index satisfies the famous "number 12" property. This is a new, infinite class of lattice polygons possessing this property, and extends the previously known sixteen instances. The number 12 property also holds more generally for l-reflexive non-convex or self-intersecting polygonal loops. We conclude by discussing higher-dimensional examples and open questions.Comment: Dedicated to the memory of Maximilian Kreuzer. 23 pages, 4 figures, 4 tables, an appendix containing Magma source cod