scholarly journals Integer Decomposition Property of Free Sums of Convex Polytopes

2016 ◽  
Vol 20 (3) ◽  
pp. 601-607 ◽  
Author(s):  
Takayuki Hibi ◽  
Akihiro Higashitani
Keyword(s):  
Convex Polytopes ◽  
Decomposition Property ◽  
Integer Decomposition Property

scholarly journals Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals

2010 ◽  
Vol 106 (1) ◽  
pp. 88 ◽  
Author(s):  
Luis A. Dupont ◽  
Rafael H. Villarreal
Keyword(s):  
Monomial Ideal ◽  
Lattice Points ◽  
Edge Ideal ◽  
Comparability Graph ◽  
Decomposition Property ◽  
Edge Ideals ◽  
Finite Poset ◽  
Integer Decomposition Property ◽  
Torsion Free ◽  
Min Cut

The normality of a monomial ideal is expressed in terms of lattice points of blocking polyhedra and the integer decomposition property. For edge ideals of clutters this property characterizes normality. Let $G$ be the comparability graph of a finite poset. If $\mathrm{cl}(G)$ is the clutter of maximal cliques of $G$, we prove that $\mathrm{cl}(G)$ satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. Then we prove that edge ideals of complete admissible uniform clutters are normally torsion free.

scholarly journals Convex-normal (Pairs of) Polytopes

2017 ◽  
Vol 60 (3) ◽  
pp. 510-521
Author(s):  
Christian Haase ◽  
Jan Hofmann
Keyword(s):  
Lattice Polytopes ◽  
Decomposition Property ◽  
Integer Decomposition Property

AbstractIn 2012, Gubeladze (Adv. Math. 2012) introduced the notion ofk-convex-normal polytopes to show that integral polytopes all of whose edges are longer than 4d(d+ 1) have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no diòerence betweenk- and (k+ 1)-convex-normality (fork≥ 3) and improve the bound to 2d(d+ 1). In the second part we extend the definition to pairs of polytopes. Given two rational polytopesPandQ, where the normal fan ofPis a reûnement of the normal fan ofQ, if every edge ePofPis at least d times as long as the corresponding face (edge or vertex) eQofQ, then.

scholarly journals Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices

2018 ◽  
Vol 100 ◽  
pp. 122-142 ◽  
Author(s):  
Benjamin Braun ◽  
Robert Davis ◽  
Liam Solus
Keyword(s):  
Decomposition Property ◽  
Integer Decomposition Property

scholarly journals Integer Decomposition Property for Cayley Sums of Order and Stable Set Polytopes

2020 ◽  
Vol 69 (4) ◽  
pp. 765-778 ◽  
Author(s):  
Takayuki Hibi ◽  
Hidefumi Ohsugi ◽  
Akiyoshi Tsuchiya
Keyword(s):  
Stable Set ◽  
Decomposition Property ◽  
Integer Decomposition Property

scholarly journals Lattice Polytopes from Schur and Symmetric Grothendieck Polynomials

10.37236/9621 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Margaret Bayer ◽  
Bennet Goeckner ◽  
Su Ji Hong ◽  
Tyrrell McAllister ◽  
McCabe Olsen ◽  
...  
Keyword(s):  
Complete Characterization ◽  
Algebraic Combinatorics ◽  
Schur Polynomials ◽  
Ehrhart Theory ◽  
Lattice Polytopes ◽  
Newton Polytopes ◽  
Decomposition Property ◽  
Integer Decomposition Property ◽  
Grothendieck Polynomials

Given a family of lattice polytopes, two common questions in Ehrhart Theory are determining when a polytope has the integer decomposition property and determining when a polytope is reflexive. While these properties are of independent interest, the confluence of these properties is a source of active investigation due to conjectures regarding the unimodality of the $h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we prove that these polytopes have the integer decomposition property by using the fact that both families of polynomials have saturated Newton polytope. Furthermore, in both cases, we provide a complete characterization of when these polytopes are reflexive. We conclude with some explicit formulas and unimodality implications of the $h^\ast$-vector in the case of Schur polynomials.

scholarly journals Integer Decomposition Property of Dilated Polytopes

10.37236/4204 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
David A. Cox ◽  
Christian Haase ◽  
Takayuki Hibi ◽  
Akihiro Higashitani
Keyword(s):  
Convex Polytope ◽  
Fundamental Question ◽  
Decomposition Property ◽  
Integral Convex Polytope ◽  
Integer Decomposition Property

An integral convex polytope $\mathcal{P} \subset \mathbb{R}^N$ possesses the integer decomposition property if, for any integer $k > 0$ and for any $\alpha \in k \mathcal{P} \cap \mathbb{Z}^{N}$, there exist $\alpha_{1}, \ldots, \alpha_k \in \mathcal{P} \cap \mathbb{Z}^{N}$ such that $\alpha = \alpha_{1} + \cdots + \alpha_k$. A fundamental question is to determine the integers $k > 0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. In the present paper, combinatorial invariants related to the integer decomposition property of dilated polytopes will be proposed and studied.

The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’Aini Aris ◽  
Azlina Jumadi
Keyword(s):  
Convex Polytopes ◽  
Polynomial Systems ◽  
Matrix Formulation ◽  
Bivariate Polynomials ◽  
Homogeneous Coordinates ◽  
The Matrix ◽  
Projective Vector ◽  
Coordinate Rings ◽  
Resultant Matrix ◽  
The Given

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

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