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Periods of Degree-2 Ehrhart Quasipolynomials

Periods of Degree-2 Ehrhart Quasipolynomials. Christopher O’Neill and Anastasia Chavez. Polytopes. Vertex Description: P = conv{v 1 , ... , v n } Hyperplane Description: P = P(A, z) These two definitions are equivalent P is integral if vertices are integers

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Periods of Degree-2 Ehrhart Quasipolynomials

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  1. Periods of Degree-2 Ehrhart Quasipolynomials • Christopher O’Neill and Anastasia Chavez

  2. Polytopes • Vertex Description: P = conv{v1, ... , vn} • Hyperplane Description: P = P(A, z) • These two definitions are equivalent • P is integral if vertices are integers • P is rational if vertices are rational

  3. Counting Lattice Points • Lattice Points = Integer Points • Counting integer points inside a polytope

  4. Dilates of Polytopes • Dilate: scale each vertex • P = conv{(0,0), (0,2), (2,1), (0,1)},2P = conv{(0,0), (0,4), (4,2), (0,2)},3P = conv{(0,0), (0,6), (6,3), (0,3)}, ... • Lp(t) = # integer points in tP

  5. Properties of Lp(t) • P is integral => Lp(t) is a polynomial • P is rational => Lp(t) is a quasipolynomialLp(t) = C2(t) * t^2 + C1(t) * t + C0(t) • In both cases, deg(Lp(t)) = dim(P)

  6. Our Project • Consider rational polygons (dim(P) = 2) • Fact: for rational polytopes, C2(t) = C2 • Consider the periods of C1(t), C0(t) • Which period combinations (p1, p0) can happen?

  7. Steps to Take • Study the paper “The Minimum period of the Ehrhart Quasi-polynomial of a Rational Polytope” by T. McAllister and K. Woods • Learn to use LattE • Implement an algorithm that calculates Lp(t) for rational polygons

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