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 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.

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 $34$.

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

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 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

scholarly journals GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

2019 ◽  
Vol 7 ◽  
Author(s):  
SPENCER BACKMAN ◽  
MATTHEW BAKER ◽  
CHI HO YUEN
Keyword(s):  
Spanning Trees ◽  
Finite Abelian Group ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Lattice Points ◽  
Combinatorial Interpretation ◽  
Ehrhart Polynomial ◽  
Ehrhart Theory ◽  
Regular Matroid ◽  
Definition Of

Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$ . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) $\operatorname{Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$ ). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph $\text{Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential way and are inherently ‘nonmatroidal’. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid $M$ and bases of $M$ , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of $M$ admits a canonical simply transitive action on the set ${\mathcal{G}}(M)$ of circuit–cocircuit reversal classes of $M$ , and then define a family of combinatorial bijections $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ between ${\mathcal{G}}(M)$ and bases of $M$ . (Here $\unicode[STIX]{x1D70E}$ (respectively $\unicode[STIX]{x1D70E}^{\ast }$ ) is an acyclic signature of the set of circuits (respectively cocircuits) of $M$ .) We then give a geometric interpretation of each such map $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ in terms of zonotopal subdivisions which is used to verify that $\unicode[STIX]{x1D6FD}$ is indeed a bijection. Finally, we give a combinatorial interpretation of lattice points in the zonotope $Z$ ; by passing to dilations we obtain a new derivation of Stanley’s formula linking the Ehrhart polynomial of $Z$ to the Tutte polynomial of $M$ .

scholarly journals Smooth Fano Polytopes Whose Ehrhart Polynomial Has a Root with Large Real Part

2012 ◽  
Vol 47 (3) ◽  
pp. 624-628 ◽  
Author(s):  
Hidefumi Ohsugi ◽  
Kazuki Shibata
Keyword(s):  
Ehrhart Polynomial

scholarly journals Norm Bounds for Ehrhart Polynomial Roots

2008 ◽  
pp. 1-3
Author(s):  
Benjamin Braun
Keyword(s):  
Ehrhart Polynomial ◽  
Polynomial Roots
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