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 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 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 Mixed Ehrhart polynomials

10.37236/5815 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Christian Haase ◽  
Martina Juhnke-Kubitzke ◽  
Raman Sanyal ◽  
Thorsten Theobald
Keyword(s):  
Mixed Volume ◽  
Ehrhart Polynomial ◽  
Ehrhart Polynomials ◽  
Lattice Polytopes ◽  
Real Roots

For lattice polytopes $P_1,\ldots, P_k \subseteq \mathbb{R}^d$, Bihan (2016) introduced the discrete mixed volume $DMV(P_1,\dots,P_k)$ in analogy to the classical mixed volume.  In this note we study the associated mixed Ehrhart polynomial $ME_{P_1, \dots,P_k}(n) = DMV(nP_1, \dots, nP_k)$.  We provide a characterization of all mixed Ehrhart coefficients in terms of the classical multivariate Ehrhart polynomial. Bihan (2016) showed that the discrete mixed volume is always non-negative. Our investigations yield simpler proofs for certain special cases.We also introduce and study the associated mixed $h^*$-vector. We show that for large enough dilates $r  P_1, \ldots, rP_k$ the corresponding mixed $h^*$-polynomial has only real roots and as a consequence  the mixed $h^*$-vector becomes non-negative. 

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 q-analogues of Ehrhart polynomials

2015 ◽  
Vol 59 (2) ◽  
pp. 339-358 ◽  
Author(s):  
F. Chapoton
Keyword(s):  
General Theory ◽  
Classical Theory ◽  
Linear Form ◽  
Partially Ordered Sets ◽  
Weighted Sums ◽  
Ordered Sets ◽  
Ehrhart Polynomials ◽  
Lattice Polytopes ◽  
Ehrhart Series ◽  
Special Case

AbstractWe consider weighted sums over points of lattice polytopes, where the weight of a point v is the monomial qλ(v) for some linear form λ. We propose a q-analogue of the classical theory of Ehrhart series and Ehrhart polynomials, including Ehrhart reciprocity and involving evaluation at the q-integers. The main novelty is the proposal to consider q-Ehrhart polynomials. This general theory is then applied to the special case of order polytopes associated with partially ordered sets. Some more specific properties are described in the case of empty polytopes.

scholarly journals Symmetric Decompositions and the Veronese Construction

2021 ◽  
Author(s):  
Katharina Jochemko
Keyword(s):  
Generating Functions ◽  
Linear Inequalities ◽  
Lattice Polytopes ◽  
Lattice Polytope ◽  
Dimensional Lattice ◽  
Ehrhart Series ◽  
Dilation Factor ◽  
Symmetric Decomposition ◽  
Rational Generating Functions

Abstract We study rational generating functions of sequences $\{a_n\}_{n\geq 0}$ that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences $\{a_{rn}\}_{n\geq 0}$. We prove that if the numerator polynomial for $\{a_n\}_{n\geq 0}$ is of degree $s$ and its coefficients satisfy a set of natural linear inequalities, then the symmetric decomposition of the numerator for $\{a_{rn}\}_{n\geq 0}$ is real-rooted whenever $r\geq \max \{s,d+1-s\}$. Moreover, if the numerator polynomial for $\{a_n\}_{n\geq 0}$ is symmetric, then we show that the symmetric decomposition for $\{a_{rn}\}_{n\geq 0}$ is interlacing. We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the $h^\ast $-polynomial of every dilation of a $d$-dimensional lattice polytope of degree $s$ has a real-rooted symmetric decomposition whenever the dilation factor $r$ satisfies $r\geq \max \{s,d+1-s\}$. Moreover, if the polytope is Gorenstein, then this decomposition is interlacing.

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.

Polytopes and 𝐾-Theory

2004 ◽  
Vol 11 (4) ◽  
pp. 655-670
Author(s):  
W. Bruns ◽  
J. Gubeladze
Keyword(s):  
Lattice Polytopes ◽  
Unit Simplex

Abstract This is an overview of results from our experiment of merging two seemingly unrelated disciplines – higher algebraic 𝐾-theory of rings and the theory of lattice polytopes. The usual 𝐾-theory is the “theory of a unit simplex”. A conjecture is proposed on the structure of higher polyhedral 𝐾-groups for certain class of polytopes for which the coincidence of Quillen's and Volodin's theories is known.

scholarly journals NON-MILITARY FORMS OF ENDANGERING SECURITY OF MODERN COUNTRY

2019 ◽  
Vol 26 (6) ◽  
pp. 1749-1761
Author(s):  
Hatidza Berisha
Keyword(s):  
The State ◽  
Security Threats ◽  
Technological Revolution ◽  
New Energy ◽  
Free Sum

Modern challenges and threats to security, according to the criteria of space and resources used for the purpose of their realization, are in principle external and internal, armed and unarmed. Due to intense migration, technological revolution in communications, greater freedom and speed of movement, porosity of boundaries and free transfer of various technologies, security threats today are not only a free sum of acts of endangering. Though they have the same name and the same goal, modern threats, today, carry completely new energy and quality of endangerment. This is best seen on the case of terrorism. Contemporary terrorism and "liberal" terrorism in the seventies have a common name and goal, while their range, destructiveness, brutality, the number of victims and the effects that they produce differ greatly.The paper presents a review of safety, taking into account the different views from which this term is observed. At the same time, relations between the object - values and threats are considered. Possible reference security objects are defined, an approximation is made on two objects, country and individual, and using the state of Critical School of Security, the focus is placed on sources of threats, and forms of security threats. Terrorism has been dealt with thematically and its relationship with other non-military forms of threats to security has been explained.

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