The Ehrhart polynomial ehr P (n) of a lattice polytope P gives the number of integer lattice points in the n-th dilate of P for all integers n ≥ 0. The degree of P is defined as the degree of its h * -polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory.A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree 2 thereby complementing results of Scott (1976), Treutlein (2010), and Henk-Tagami (2009. Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all 3-dimensional zonotopes of degree 2.
The Ehrhart polynomial$${\text {ehr}}_P (n)$$ ehr P ( n ) of a lattice polytope P gives the number of integer lattice points in the n-th dilate of P for all integers $$n\ge 0$$ n ≥ 0 . The degree of P is defined as the degree of its $$h^*$$ h ∗ -polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree 2 thereby complementing results of Scott (Bull Aust Math Soc 15(3), 395–399, 1976), Treutlein (J Combin Theory Ser A 117(3), 354–360, 2010), and Henk and Tagami (Eur J Combin 30(1), 70–83, 2009). Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all 3-dimensional zonotopes of degree 2.
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