scholarly journals On Counterexamples to a Conjecture of Wills and Ehrhart Polynomials whose Roots have Equal Real Parts

10.37236/3757 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Matthias Henze
Keyword(s):  
Lattice Point ◽  
Convex Bodies ◽  
Discrete Analog ◽  
Side Length ◽  
Ehrhart Polynomials ◽  
Lattice Polytopes ◽  
Lattice Polytope ◽  
Positive Side ◽  
Reflexive Polytopes

As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a $0$-symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side length two. We discuss several counterexamples to this conjecture and, on the positive side, we identify a family of lattice polytopes that fulfill the claimed inequalities. This family is related to the recently introduced class of $l$-reflexive polytopes.

scholarly journals Interlacing Ehrhart polynomials of reflexive polytopes

2017 ◽  
Vol 23 (4) ◽  
pp. 2977-2998 ◽  
Author(s):  
Akihiro Higashitani ◽  
Mario Kummer ◽  
Mateusz Michałek
Keyword(s):  
Ehrhart Polynomials ◽  
Reflexive Polytopes

Minimal width and diameter of lattice‐point‐free convex bodies

Mathematika ◽  
1981 ◽  
Vol 28 (2) ◽  
pp. 255-264 ◽  
Author(s):  
P. McMullen ◽  
J. M. Wills
Keyword(s):  
Lattice Point ◽  
Convex Bodies ◽  
Minimal Width

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.

Note on lattice-point-free convex bodies

1998 ◽  
Vol 126 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Poh Wah Awyong ◽  
Martin Henk ◽  
Paul R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Bodies

scholarly journals Lattice Point Inequalities for Centered Convex Bodies

2016 ◽  
Vol 30 (2) ◽  
pp. 1148-1158
Author(s):  
Sören Lennart Berg ◽  
Martin Henk
Keyword(s):  
Lattice Point ◽  
Convex Bodies

scholarly journals An Ehrhart Series Formula For Reflexive Polytopes

10.37236/1153 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Benjamin Braun
Keyword(s):  
Ehrhart Polynomial ◽  
Lattice Polytopes ◽  
Reflexive Polytopes ◽  
Ehrhart Series ◽  
Multiplicative Relationship ◽  
Free Sum

It is well known that for $P$ and $Q$ lattice polytopes, the Ehrhart polynomial of $P\times Q$ satisfies $L_{P\times Q}(t)=L_P(t)L_Q(t)$. We show that there is a similar multiplicative relationship between the Ehrhart series for $P$, for $Q$, and for the free sum $P\oplus Q$ that holds when $P$ is reflexive and $Q$ contains $0$ in its interior.

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 $34$.

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