Christian Bey

Notes on the Roots of Ehrhart Polynomials

We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n2, where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n2. For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes. We further give a characterisation of the roots of Ehrhart polyomials in the three-dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part -1/2. These polytopes belong to the class of reflexive polytopes.

An upper bound on the sum of squares of degrees in a hypergraph

We give an upper bound on the sum of squares of l-degrees in a k-uniform hypergraph in terms of l, k and the number of vertices and edges of the hypergraph, where a l-degree is the number of edges of the hypergraph containing a fixed l-element subset of the vertices. For ordinary graphs this bound coincides with one given by de Caen. We show that our bound implies the quadratic LYM-inequality for 2-level antichains of subsets of a finite set.

Old and New Results for the Weighted t-Intersection Problem via AK-Methods

Let [n]: = {1, ..., n}, 2[n] be the power set of [n] and s ∈ [n]. A family F ⊆ 2[n] is called t-intersecting in [s] if

Polynomial Lym Inequalities

\left| {{X_1} \cap {X_2} \cap \left[ s \right]} \right| \geqslant t\,for\,all\,{X_1},{X_2}\, \in \,F.

The Erdős-Ko-Rado bound for the function lattice

Let ω: 2[n] → ℝ+ be a given weight function and

Notes on lattice points of zonotopes and lattice-face polytopes

{M_s}\left( {n,t;\omega } \right):\, = \max \left\{ {\omega \left( F \right)} \right.:F\,is\,t - \operatorname{int} er\sec ting\,in\left. {\,\left[ s \right]} \right\}.