scholarly journals Computing the Ehrhart polynomial of a convex lattice polytope

1994 ◽  
Vol 12 (1) ◽  
pp. 35-48 ◽  
Author(s):  
A. I. Barvinok
Keyword(s):  
Ehrhart Polynomial ◽  
Lattice Polytope ◽  
Convex Lattice

scholarly journals Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials

2019 ◽  
Vol 10 (1) ◽  
pp. 27-63 ◽  
Author(s):  
Loïc Foissy
Keyword(s):  
Hopf Algebra ◽  
Point Of View ◽  
Duality Principle ◽  
Algebraic Proof ◽  
Ehrhart Polynomial ◽  
Algebraic Point ◽  
Ehrhart Polynomials ◽  
Lattice Polytope ◽  
Algebra Morphism

Abstract To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism with values in \mathbb{Q}[X] . We deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber’s formula. We also give non-commutative versions of these results, where polynomials are replaced by packed words. We obtain, in particular, a non-commutative duality principle.

The Ehrhart Polynomial of a Lattice Polytope

1997 ◽  
Vol 145 (3) ◽  
pp. 503 ◽  
Author(s):  
Ricardo Diaz ◽  
Sinai Robins
Keyword(s):  
Ehrhart Polynomial ◽  
Lattice Polytope

scholarly journals Ehrhart Polynomial Roots of Reflexive Polytopes

10.37236/7780 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Gábor Hegedüs ◽  
Akihiro Higashitani ◽  
Alexander Kasprzyk
Keyword(s):  
Recent Work ◽  
Ehrhart Polynomial ◽  
Polynomial Roots ◽  
Lattice Polytope ◽  
Reflexive Polytopes ◽  
Half Strip ◽  
Reflexive Polytope

Recent work has focused on the roots $z\in\mathbb{C}$ of the Ehrhart polynomial of a lattice polytope $P$. The case when $\Re{z}=-1/2$ is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when $\mathrm{dim}(P)\leq 7$. We also consider the "half-strip condition", where all roots $z$ satisfy $-\mathrm{dim}(P)/2\leq\Re{z}\leq \mathrm{dim}(P)/2-1$, and show that this holds for any reflexive polytope with $\mathrm{dim}(P)\leq 5$. We give an example of a $10$-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi–Shibata in dimension $34$.

Erratum: The Ehrhart Polynomial of a Lattice Polytope

1997 ◽  
Vol 146 (1) ◽  
pp. 237
Author(s):  
Ricardo Diaz ◽  
Sinai Robins
Keyword(s):  
Ehrhart Polynomial ◽  
Lattice Polytope

scholarly journals THE BOUNDARY VOLUME OF A LATTICE POLYTOPE

2011 ◽  
Vol 85 (1) ◽  
pp. 84-104
Author(s):  
GÁBOR HEGEDÜS ◽  
ALEXANDER M. KASPRZYK
Keyword(s):  
Lattice Points ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lattice Polytope ◽  
Birkhoff Polytope ◽  
Convex Lattice ◽  
Smooth Polytope ◽  
Necessary And Sufficient

AbstractFor a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(∂P) is derived in terms of the number of boundary lattice points on the first ⌊d/2⌋ dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulas for the f-vector of a smooth polytope in dimensions three, four, and five. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.

scholarly journals On the Ehrhart Polynomial of Minimal Matroids

2021 ◽  
Author(s):  
Luis Ferroni
Keyword(s):  
Ehrhart Polynomial ◽  
Matroid Polytopes ◽  
Connected Matroid ◽  
Fixed Rank

AbstractWe provide a formula for the Ehrhart polynomial of the connected matroid of size n and rank k with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $$h^*$$ h ∗ -real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $$h^*$$ h ∗ -real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.

scholarly journals Norm Bounds for Ehrhart Polynomial Roots

2007 ◽  
Author(s):  
Benjamin Braun
Keyword(s):  
Ehrhart Polynomial ◽  
Polynomial Roots

scholarly journals Lattice polytopes, Hecke operators, and the Ehrhart polynomial

2007 ◽  
Vol 13 (2) ◽  
pp. 253-276 ◽  
Author(s):  
Paul E. Gunnells ◽  
Fernando Rodriguez Villegas
Keyword(s):  
Hecke Operators ◽  
Ehrhart Polynomial ◽  
Lattice Polytopes

Reconstruction of Q-Convex Lattice Sets

2007 ◽  
pp. 31-54
Author(s):  
S. Brunetti ◽  
A. Daurat
Keyword(s):  
Convex Lattice
Sign in / Sign up

Export Citation Format

Share Document