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On positivity of Ehrhart polynomials. (English) Zbl 1435.52007
Barcelo, Hélène (ed.) et al., Recent trends in algebraic combinatorics. Cham: Springer. Assoc. Women Math. Ser. 16, 189-237 (2019).
A well-known theorem in convex polytope theory (due to Ehrhart) states that, for any integral convex \(d\)-polytope \(P\), the function that counts the number of integer lattice points in the dilated copy \(tP\) of \(P\) is a real polynomial in \(t\). This polynomial is called the Ehrhart polynomial of \(P\). The polytope \(P\) is said to be Ehrhart positive if all coefficients of its Ehrhart polynomial are positive real numbers. The paper surveys interesting families of convex polytopes that are known to be Ehrhart positive, and explains the reason why their polytopes are Ehrhart positive. Also discussed are examples of convex polytopes that have some negative Ehrhart coefficients, as well as families of convex polytopes that are conjectured to be Ehrhart positive. The author also poses some open problems.
For the entire collection see [Zbl 1410.05001].

MSC:
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52C05 Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry)
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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