64 citations · 418 across the 52 of their papers we have counts for
58 papers
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 $…
From Ham-Sandwich to Centerpoints: Semialgebraic Algorithms for Cutting Polytopal Measures
Marie-Charlotte Brandenburg, Jesús A. De Loera, Chiara Meroni
We design exact algorithms for the ham-sandwich and centerpoint theorems for polytopal measures. Our key observation is that the cap-volume function of such a measure, i.e., the vo…
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…
Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones
Jesús A. De Loera, Brittney Marsters, Luze Xu +1
We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigr…
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…
Irreducible Markov Chains on spaces of graphs with fixed degree-color sequences
Félix Almendra-Hernández, Jesús A. De Loera, Sonja Petrović
We study a colored generalization of the famous simple-switch Markov chain for sampling the set of graphs with a fixed degree sequence. Here we consider the space of graphs with co…
Markov bases: a 25 year update
Félix Almendra-Hernández, Jesús A. De Loera, Sonja Petrović
In this paper, we evaluate the challenges and best practices associated with the Markov bases approach to sampling from conditional distributions. We provide insights and clarifica…
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…
Convexity in (colored) affine semigroups
Jesus A. De Loera, Christopher O'Neill, Chengyang Wang
In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Caratheodory. Additionally, we develop a new theory of colored affine se…
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…
Turning Mathematics Problems into Games: Reinforcement Learning and Gröbner bases together solve Integer Feasibility Problems
Yue Wu, Jesús A. De Loera
Can agents be trained to answer difficult mathematical questions by playing a game? We consider the integer feasibility problem, a challenge of deciding whether a system of linear…
Geometric Policy Iteration for Markov Decision Processes
Yue Wu, Jesús A. De Loera
Recently discovered polyhedral structures of the value function for finite state-action discounted Markov decision processes (MDP) shed light on understanding the success of reinfo…
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…
Random Monomial Ideals
Jesus A. De Loera, Sonja Petrovic, Lily Silverstein +2
Inspired by the study of random graphs and simplicial complexes, and motivated by the need to understand average behavior of ideals, we propose and study probabilistic models of ra…
Quantitative combinatorial geometry for continuous parameters
J. A. De Loera, R. N. La Haye, D. Rolnick +1
We prove variations of Carathéodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among…
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…