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Inequalities and Ehrhart \(\delta \)-vectors. (English) Zbl 1181.52024

Fix a lattice \(L\) and let \(P\) be a \(d\)-dimensional lattice polytope in \(L \otimes_{\mathbb Z} {\mathbb R}\), i.e., the convex hull of finitely many points in \(L\). The lattice-point counting function
\[ i(P,t) := \# \left( tP \cap L \right) \]
is a polynomial in the positive integer variable \(r\), by Ehrhart’s fundamental theorem [E. Ehrhart, C. R. Acad. Sci., Paris 254, 616–618 (1962; Zbl 0100.27601)]. Hence the generating function (the Ehrhart series) of \(i(P,r)\) is a rational function of the form \[ \sum_{ t \geq 0 } i(P,t) \, x^t = {{ \delta_d x^d + \delta_{ d-1 } x^{ d-1 } + \cdots + \delta_0 } \over { (1-x)^{ d+1 } }} \, , \] and Stanley [R. P. Stanley, Ann. Discrete Math. 6, 333–342 (1980; Zbl 0812.52012)] proved that the \(\delta_k\)’s are nonnegative integers. Let \(s\) be the degree of \(\delta_d x^d + \delta_{ d-1 } x^{ d-1 } + \cdots + \delta_0\) and set \(l = d+1-s\) and \(\delta_k = 0\) for \(k<0\) or \(k>s\).
The paper under review gives several new inequalities for the \(\delta_k\)’s, partly to improve on older results and partly to give purely combinatorial proofs for known results. For example, the new set of inequalities \[ \delta_{ 2-l } + \cdots + \delta_0 + \delta_1 \leq \delta_j + \delta_{ j-1 } + \cdots + \delta_{ j-l+1 } \qquad (j = 2, 3, \dots, d-1) \] generalizes T. Hibi’s [Adv. Math. 105, No. 2, 162–165 (1994; Zbl 0807.52011)] inequalities \(1 \leq \delta_1 \leq \delta_k\) (\(k = 2, 3, \dots, d-1\)) which hold only when \(\delta_d \neq 0\) [M. Henk and M. Tagami, “Lower bounds on the coefficients of Ehrhart polynomials”, Eur. J. Comb. 30, No. 1, 70–83 (2009; Zbl 1158.52014)].
To mention one more sample theorem from the paper: suppose the boundary of \(P\) admits a regular unimodular lattice triangulation, then \(\delta_{ j+1 } \geq \delta_{ d-j }\) and
\[ \delta_0 + \delta_1 + \cdots + \delta_{ j+1 } \leq \delta_d + \delta_{ d-1 } + \cdots + \delta_{ d-j } + \left( {{ \delta_1 - \delta_d + j + 1 } \atop { j+1 }} \right)\,, \]
for \(j = 0, 1, \dots, \lfloor {d \over 2} \rfloor - 1\). These inequalities extend a theorem of C. Athanasiadis [Electron. J. Comb. 11, No. 2, Research paper R6, 13 p. (2004; Zbl 1068.52016)].
The proofs are based on a simple but powerful decomposition of the polynomial
\[ \left( 1 + t + \cdots + t^{ l-1 } \right) \left( \delta_d x^d + \delta_{ d-1 } x^{ d-1 } + \cdots + \delta_0 \right) \]
into two palindromic polynomials.

MSC:

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
11H06 Lattices and convex bodies (number-theoretic aspects)
11P21 Lattice points in specified regions
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References:

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