64 citations · 418 across the 52 of their papers we have counts for
31 papers · 1 filter
On Unavoidable Faces of High-Dimensional Polytopes
Jesús A. De Loera, Ethan X. Fang, Shengtao Guo +2
Kalai's cube--simplex conjecture asserts that for all positive integers , there is an integer such that every polytope of dimension at least has eit…
Sharp Bounds on the Independence Number of Simplicial Spheres
Jesús A. De Loera, Ethan X. Fang, Junwei Lu +2
We study the maximum size of an independent set in the graph of a simplicial sphere. Let denote this maximum over all simplicial -spheres on vertices, and let $…
Persistent Subdivisions of Coxeter Permutahedra
Timothy Blanton, Jesús A. De Loera, Melissa Sherman-Bennett
We investigate the realizations of Coxeter permutahedra which are also Coxeter matroid polytopes; these are polytopes of the form where is a…
There are matroid toric ideals without quadratic Gröbner bases
Jesús A. De Loera, Luis Ferroni, Santiago Morales +1
Our paper shows that if a matroid contains the Fano plane or its dual as a minor, then its toric ideal does not have any quadratic Gröbner basis. More than 25 years ago, Hibi, Herz…
Critical moments of slices and slabs of the cube (and other polyhedral norms)
Marie-Charlotte Brandenburg, Jesús A. De Loera, Yu Luo +1
In this article, we present a unified algebraic-combinatorial framework for computing explicit, piecewise rational, and combinatorially indexed parametric formulas for volumes and…
Ehrhart Theory over Abelian Group Rings
Robert Davis, Jesús A. De Loera, Alexey Garber +3
We introduce a unified framework for Ehrhart theory in which lattice point enumerators take coefficients in an Abelian group ring, encoding substantially richer algebraic data than…
An arithmetic measure of width for convex bodies
Jesús A. De Loera, Brittney Marsters, Christopher O'Neill
We introduce the arithmetic width of a convex body, defined as the number of distinct values a linear functional attains on the lattice points within the body. Arithmetic width ref…
Ehrhart Functions of Weighted Lattice Points
Jesus A. De Loera, Carlos E. Valencia, Rafael H. Villarreal +1
This paper studies three different ways to assign weights to the lattice points of a convex polytope and discusses the algebraic and combinatorial properties of the resulting weigh…
Sums of Weighted Lattice Points of Polytopes
Jesús A. De Loera, Laura Escobar, Nathan Kaplan +1
We study the problem of counting lattice points of a polytope that are weighted by an Ehrhart quasi-polynomial of a family of parametric polytopes. As applications one can compute…
The Best Ways to Slice a Polytope
Marie-Charlotte Brandenburg, Jesús A. De Loera, Chiara Meroni
We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane a…
Weighted Ehrhart Theory: Extending Stanley's nonnegativity theorem
Esme Bajo, Robert Davis, Jesús A. De Loera +4
We generalize R. P. Stanley's celebrated theorem that the -polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under contai…
Ramsey Numbers through the Lenses of Polynomial Ideals and Nullstellensätze
Jesús A. De Loera, William J. Wesley
In this article we study the Ramsey numbers through Hilbert's Nullstellensatz and Alon's Combinatorial Nullstellensatz. We give polynomial encodings whose solutions corres…
The Polyhedral Geometry of Pivot Rules and Monotone Paths
Alexander E. Black, Jesús A. De Loera, Niklas Lütjeharms +1
Motivated by the analysis of the performance of the simplex method we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules w…
The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg
Jesus A. De Loera, Xavier Goaoc, Frédéric Meunier +1
We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg from combinatorial geometry. We…
The Diameters of Network-flow Polytopes satisfy the Hirsch Conjecture
S. Borgwardt, J. A. De Loera, E. Finhold
We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network…
The Hierarchy of Circuit Diameters and Transportation Polytopes
Steffen Borgwardt, Jesús A. De Loera, Elisabeth Finhold +1
The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations researc…
Three Ehrhart Quasi-polynomials
Velleda Baldoni, Nicole Berline, Jesús A. De Loera +2
Let be a semi-rational parametric polytope, where is a real multi-parameter. We study intermediate sums of polynomial functions on ,…
Edges vs Circuits: a Hierarchy of Diameters in Polyhedra
Steffen Borgwardt, Jesús A. De Loera, Elisabeth Finhold
The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation i…
Parametric Polyhedra with at least Lattice Points: Their Semigroup Structure and the k-Frobenius Problem
Iskander Aliev, Jesus A. De Loera, Quentin Louveaux
Given an integral matrix , the well-studied affine semigroup $\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\}$ can be stratified by the number of latt…
A Rainbow Ramsey Analogue of Rado's Theorem
Jesús A. De Loera, Reuben N. La Haye, Amanda Montejano +2
We present a Rainbow Ramsey version of the well-known Ramsey-type theorem of Richard Rado. We use techniques from the Geometry of Numbers. We also disprove two conjectures proposed…
Intermediate Sums on Polyhedra II: Bidegree and Poisson Formula
Velleda Baldoni, Nicole Berline, Jesús A. De Loera +2
We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-poly…
Weak Orientability of Matroids and Polynomial Equations
J. A. De Loera, J. Lee, S. Margulies +1
This paper studies systems of polynomial equations that provide information about orientability of matroids. First, we study systems of linear equations over GF(2), originally allu…
Combinatorics and Geometry of Transportation Polytopes: An Update
Jesús A. De Loera, Edward D. Kim
A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise n…
Not all simplicial polytopes are weakly vertex-decomposable
Jesus A. De Loera, Steven Klee
In 1980 Provan and Billera defined the notion of weak -decomposability for pure simplicial complexes. They showed the diameter of a weakly -decomposable simplicial complex $Δ…