Computing the Ehrhart polynomial of a convex lattice polytope

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.

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The Ehrhart Polynomial of a Lattice Polytope

Annals of Mathematics ◽

10.2307/2951842 ◽

1997 ◽

Vol 145 (3) ◽

pp. 503 ◽

Cited By ~ 30

Author(s):

Ricardo Diaz ◽

Sinai Robins

Keyword(s):

Ehrhart Polynomial ◽

Lattice Polytope

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Ehrhart Polynomial Roots of Reflexive Polytopes

The Electronic Journal of Combinatorics ◽

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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Erratum: The Ehrhart Polynomial of a Lattice Polytope

Annals of Mathematics ◽

10.2307/2951835 ◽

1997 ◽

Vol 146 (1) ◽

pp. 237

Author(s):

Ricardo Diaz ◽

Sinai Robins

Keyword(s):

Ehrhart Polynomial ◽

Lattice Polytope

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THE BOUNDARY VOLUME OF A LATTICE POLYTOPE

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972711002577 ◽

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.

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On the Ehrhart Polynomial of Minimal Matroids

Discrete & Computational Geometry ◽

10.1007/s00454-021-00313-4 ◽

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.

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