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A quasi-polynomial is a function defined of the form q(k) = c d (k) k d + c d−1 (k) k d−1 + · · · + c0(k), where c0, c1,. .. , c d are periodic functions in k ∈ Z. Prominent examples of quasi-polynomials appear in Ehrhart's theory as integer-point counting functions for rational polytopes, and McMullen gives upper bounds for the periods of the cj (k) for… (More)

The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in P 7. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Göpel variety, and over the reflection representation of type E 7. We develop classical themes such as… (More)

1. INTRODUCTION. Enumerative combinatorics is a rich and vast area of study. Particularly interesting in this subject are families of objects parameterized by the positive integers Z >0 with an associated counting function f (t) that is polynomial; this last statement means that there is some polynomial p(t) such that p(t) = f (t) for all t ∈ Z >0. It is a… (More)

We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Goren-stein. We also introduce the notion of domino stackings and present some results… (More)

The image of the complement of a hyperplane arrangement under a monomial map can be tropicalized combinatorially using matroid theory. We apply this to classical moduli spaces that are associated with complex reflection arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa quartic, and moduli of marked del Pezzo surfaces of degrees… (More)

A rational polytope is the convex hull of a finite set of points in R d with rational coordinates. Given a rational polytope P ⊆ R d , Ehrhart proved that, for t ∈ Z ≥0 , the function #(tP ∩ Z d) agrees with a quasi-polynomial L P (t), called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a… (More)

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