Searcharxiv⌕ Search

arXiv subjects

Jesus A. De Loera

Publications and source records attributed to Jesus A. De Loera.

15 recordsLinked to original sources
arXiv2026

Benchmarks in Leipzig

Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the three-day workshop Benchmarks in Leipzig with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100~questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs and their predecessors, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive. In September 2026, we added a fourth stage in which the next generation of models attempted all 100 questions once more, after which only 1 question remains unsolved.

math.HO↗
arXiv2026

Ehrhart Functions of Weighted Lattice Points

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 weighted Ehrhart functions and their generating functions and associated rings. These will be called $q$-weighted, $r$-weighted, and $s$-weighted Ehrhart functions, respectively. The key questions we investigate are \emph{When are the weighted Ehrhart series rational functions and which classical Ehrhart theory properties are preserved? And, when are the abstract formal power series the Hilbert series of Ehrhart rings of some polytope?} We prove generalizations about weighted Ehrhart $h^*$-coefficients of $q$-weighted Ehrhart series, and show $q$- and $s$-weighted Ehrhart reciprocity theorems. Then, we show the $q$- and $r$-weighted Ehrhart rings are the (classical) Ehrhart rings of weight lifting polytopes.

math.CO↗
arXiv2023

Convexity in (colored) affine semigroups

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 semigroups, where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg's theorem and colorful Helly's theorem for semigroups, as well as a version of colorful Caratheodory's theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.

math.AC↗
arXiv2020

The lattice of cycles of an undirected graph

We study bases of the lattice generated by the cycles of an undirected graph, defined as the integer linear combinations of the 0/1-incidence vectors of cycles. We prove structural results for this lattice, including explicit formulas for its dimension and determinant, and we present efficient algorithms to construct lattice bases, using only cycles as generators, in quadratic time. By algebraic considerations, we relate these results to the more general setting with coefficients from an arbitrary Abelian group. Our results generalize classical results for the vector space of cycles of a graph over the binary field to the case of an arbitrary field.

math.CO↗
arXiv2019

Data-driven Algorithm Selection and Parameter Tuning: Two Case studies in Optimization and Signal Processing

Machine learning algorithms typically rely on optimization subroutines and are well-known to provide very effective outcomes for many types of problems. Here, we flip the reliance and ask the reverse question: can machine learning algorithms lead to more effective outcomes for optimization problems? Our goal is to train machine learning methods to automatically improve the performance of optimization and signal processing algorithms. As a proof of concept, we use our approach to improve two popular data processing subroutines in data science: stochastic gradient descent and greedy methods in compressed sensing. We provide experimental results that demonstrate the answer is ``yes'', machine learning algorithms do lead to more effective outcomes for optimization problems, and show the future potential for this research direction.

cs.LG↗
arXiv2018

Random Monomial Ideals

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 random monomial ideals. We prove theorems about the probability distributions, expectations and thresholds for events involving monomial ideals with given Hilbert function, Krull dimension, first graded Betti numbers, and present several experimentally-backed conjectures about regularity, projective dimension, strong genericity, and Cohen-Macaulayness of random monomial ideals.

math.AC↗
arXiv2017

Sparse solutions of linear Diophantine equations

We present structural results on solutions to the Diophantine system $A{\boldsymbol y} = {\boldsymbol b}$, ${\boldsymbol y} \in \mathbb Z^t_{\ge 0}$ with the smallest number of non-zero entries. Our tools are algebraic and number theoretic in nature and include Siegel's Lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.

math.OC↗
arXiv2015

A Quantitative Doignon-Bell-Scarf Theorem

The famous Doignon-Bell-Scarf Theorem is a Helly-type result about the existence of integer solutions on systems of linear inequalities. The purpose of this paper is to present the following quantitative generalization: Given an integer $k$, we prove that there exists a constant $c(n,k)$, depending only on the dimension $n$ and $k$, such that if a polyhedron ${x: Ax \leq b}$ contains exactly k integer solutions, then there exists a subset of the rows, of cardinality no more than $c(n,k)$, defining a polyhedron that contains exactly the same $k$ integer points. In this case $c(n,0) = 2^n$ is the original case of Doignon-Bell-Scarf for infeasible systems of inequalities. We work on both upper and lower bounds for the constant $c(n,k)$ and discuss some consequences, including a Clarkson-style algorithm to find the $l$-th best solution of an integer program with respect to the ordering induced by the objective function.

math.MG↗
arXiv2015

Parametric Polyhedra with at least $k$ Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

Given an integral $d \times n$ matrix $A$, 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 lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset $\mbox{ Sg}_{\geq k}(A)$ of all vectors $b \in \mbox{ Sg}(A)$ such that $P_A(b) \cap {\mathbb Z}^n $ has at least $k$ solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ are fixed natural numbers, one can compute in polynomial time an encoding of $\mbox{ Sg}_{\geq k}(A)$ as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives.

math.CO↗
arXiv2015

On Augmentation Algorithms for Linear and Integer-Linear Programming: From Edmonds-Karp to Bland and Beyond

Motivated by Bland's linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we discuss three closely-related natural augmentation rules for linear and integer-linear optimization. In several nice situations, we show that polynomially-many augmentation steps suffice to reach an optimum. In particular, when using "discrete steepest-descent augmentations" (i.e., directions with the best ratio of cost improvement per unit 1-norm length), we show that the number of augmentation steps is bounded by the number of elements in the Graver basis of the problem matrix, giving the first ever strongly polynomial-time algorithm for $N$-fold integer-linear optimization. Our results also improve on what is known for such algorithms in the context of linear optimization (e.g., generalizing the bounds of Kitahara and Mizuno for the number of steps in the simplex method) and are closely related to research on the diameters of polytopes and the search for a strongly polynomial-time simplex or augmentation algorithm.

math.OC↗
arXiv2012

Not all simplicial polytopes are weakly vertex-decomposable

In 1980 Provan and Billera defined the notion of weak $k$-decomposability for pure simplicial complexes. They showed the diameter of a weakly $k$-decomposable simplicial complex $Δ$ is bounded above by a polynomial function of the number of $k$-faces in $Δ$ and its dimension. For weakly 0-decomposable complexes, this bound is linear in the number of vertices and the dimension. In this paper we exhibit the first examples of non-weakly 0-decomposable simplicial polytopes.

math.CO↗
arXiv2009

Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.

math.OC↗
arXiv2008

A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope

We present a multivariate generating function for all n x n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B_n of n x n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B_n and any of its faces.

math.CO↗
arXiv2001

Algebraic Unimodular Counting

We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Gröbner bases, and rational generating functions as in Barvinok's algorithm. We report polyhedral and computational results for two special cases: counting contingency tables and Kostant's partition function.

math.CO↗