author={Matthias Beck and Jes{\'u}s A. De Loera and Mike Develin and Julian Pfeifle and Richard P. Stanley},
  journal={arXiv: Combinatorics},
The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coeffi- cients of Ehrhart polynomials and relate them to known inequalities. We also investigate the roots of Ehrhart polynomials. We prove that for fixed d, there exists a bounded region of C containing all roots of Ehrhart polynomials of d-polytopes, and that all real roots of these polynomials lie in (−d, ⌊d/2⌋). In contrast, we… 

Figures from this paper

Ehrhart polynomials of cyclic polytopes
  • Fu Liu
  • Mathematics
    J. Comb. Theory, Ser. A
  • 2005
Periods of Ehrhart Coefficients of Rational Polytopes
This paper constructs families of polytopes in which the periods of the coefficient functions take on various prescribed values, as partial progress on this problem is made.
On Positivity of Ehrhart Polynomials
  • Fu Liu
  • Mathematics
    Association for Women in Mathematics Series
  • 2019
Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients
Notes on the Roots of Ehrhart Polynomials
The Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n2, where n is the dimension and this improves on the previously best known bound n and complements a recent result of Braun.
Contributions to the Theory of Ehrhart
In this thesis, we study the Ehrhart polynomials of different polytopes. In the 1960's Eugene Ehrhart discovered that for any rational d-polytope P, the number of lattice points, i(P,m), in the mth
M. Beck, J. De Loera, M. Develin, J. Pfeifle and R. Stanley found that the roots of the Ehrhart polynomial of a d-dimensional lattice polytope are bounded above in norm by 1 + (d + 1)!. We provide an
Gale duality bounds for roots of polynomials with nonnegative coefficients
Ehrhart polynomials and successive minima
We investigate the Ehrhart polynomial for the class of 0-sym- metric convex lattice polytopes in Euclidean n-space R n . It turns out that the roots of the Ehrhart polynomial and Minkowski's


Star-shaped complexes and Ehrhart polynomials
We study Ehrhart polynomials of star-shaped triangulations of balls by means of Cohen-Macaulay rings and canonical modules. A polyhedral complex IF in RN is a finite set of convex polytopes in RN
The Coloring Ideal and Coloring Complex of a Graph
Let G be a simple graph on d vertices. We define a monomial ideal K in the Stanley-Reisner ring A of the order complex of the Boolean algebra on d atoms. The monomials in K are in one-to-one
Lattice points in lattice polytopes
IfK is the underlying point-set of a simplicial complex of dimension at mostd whose vertices are lattice points, and ifG(K) is the number of lattice points inK, then the lattice point
Geometry of Polynomials
During the years since the first edition of this well-known monograph appeared, the subject (the geometry of the zeros of a complex polynomial) has continued to display the same outstanding vitality
Lectures on Polytopes
Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward
On the real roots of the Bernoulli polynomials and the Hurwitz zeta-function
The behaviour of the real roots of the Bernoulli polynomials Bm(a) for large m is investigated.
On the zeros of certain polynomials
We prove that certain naturally arising polynomials have all of their roots on a vertical line.
Hilbert Polynomials in Combinatorics
We prove that several polynomials naturally arising in combinatorics are Hilbert polynomials of standard graded commutative k-algebras.