## 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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