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Unimodality problems in Ehrhart theory. (English) Zbl 1366.52014

Beveridge, Andrew (ed.) et al., Recent trends in combinatorics. Cham: Springer (ISBN 978-3-319-24296-5/hbk; 978-3-319-24298-9/ebook). The IMA Volumes in Mathematics and its Applications 159, 687-711 (2016).
This well-written survey paper gives the state of the art regarding unimodality questions of Ehrhart \(h^*\)-vectors. If \(P \subset\mathbb R^d\) is a lattice polytope, i.e., the convex hull of finitely many points in \(\mathbb Z^d\), then the counting function \(E(t) := \# ( tP \cap \mathbb Z^d )\) is a polynomial in the positive integer variable \(t\) [E. Ehrhart, C. R. Acad. Sci., Paris 254, 616–618 (1962; Zbl 0100.27601)]. When written in terms of the polynomial basis \({t \choose d}, {{t+1} \choose d}, \dots, {{t+d} \choose d}\), the coefficients of \(E(t)\) are nonnegative, due to a theorem of R. P. Stanley [Ann. Discrete Math. 6, 333–342 (1980; Zbl 0812.52012)]. It is natural to ask whether these coefficients – which form the \(h^*\)-vector of \(P\) – satisfies further constraints, the most natural among which is unimodality, i.e., for some \(k\), \[ h_0^* \leq \cdots \leq h_k^* \geq \cdots \geq h_d^* \, . \] Several of the known techniques to prove unimodality of a given class of \(h^*\)-vectors are algebraic in nature, due to the connection between lattice polytopes and Cohen-Macaulay semigroup algebras. The paper surveys these and many others unimodality results in this area of geometric combinatorics, and states a number of open problems.
For the entire collection see [Zbl 1348.05002].

MSC:

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
05A20 Combinatorial inequalities
52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
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