arXiv subjects

Jesús A. De Loera

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

At least 19 recordsLinked to original sources

arXiv2026

On Lattice Diameter Segments: Algorithms and Structure

The lattice diameter of a bounded set $S \subset \mathbb{R}^d$ counts the maximal number of lattice points in a segment whose endpoints are lattice points in $S$. Such a segment is called a lattice diameter segment of $S$. This basic invariant in the geometry of numbers has several interesting applications. Our first type of contribution is algorithmic: We describe a polynomial-time algorithm that computes lattice diameter segments of polytopes in fixed dimension, with a simpler algorithm for polygons. In contrast, we show hardness of computing the lattice diameters of polytopes of variable dimension and for semi-algebraic sets. Our second type of contribution concerns the structure of lattice diameter segments: Generalizing the classical Ehrhart function, we prove that the function that counts distinct lattice diameter segments in dilations of a lattice polytope is eventually a quasi-polynomial in the dilation factor. Finally, we prove a Borsuk-type theorem on the number of parts needed to partition a set of lattice points such that each part has strictly smaller lattice diameter.

math.CO↗

arXiv2026

On Unavoidable Faces of High-Dimensional Polytopes

Kalai's cube--simplex conjecture asserts that for all positive integers $\ell,k$, there is an integer $f(\ell,k)$ such that every polytope of dimension at least $f(\ell,k)$ has either a simplex $\ell$-face or a cube $k$-face; let $f_s(\ell,k)$ denote the threshold restricted to simple polytopes. Finiteness of $f(\ell,k)$ is known only for $\ell,k \leq 2$. In addition, Kalai proved that $f_s(2,k) \leq 2k^2$. Here we prove that $f_s(\ell,k)$ is finite for all $\ell \geq 2$ and $k \geq 3$, the first such result beyond $\ell = 2$, with $f_s(2,k) \leq 2k^2-1$ and $f_s(\ell,k) \leq \tfrac{1}{2}k^2\ell\,2^k$ for $\ell \geq 3$. In the opposite direction, we obtain the lower bounds $f(\ell,k) \geq (5\lfloor \ell/2 \rfloor + (\ell \bmod 2) - 1)(k-1)+1$ and $f_s(\ell,k) \geq \max\{4,\,2(\ell-1)\}(k-1)+1$. A companion question asks for the minimum possible size of a 3-face within a higher-dimensional polytope. Meisinger, Kleinschmidt and Kalai proved that every rational $d$-polytope with $d \geq 9$ has a $3$-face with fewer than $78$ vertices or fewer than $78$ facets. Here we improve their bound: every convex polytope of dimension at least $15$ has a $3$-face with at most $13$ facets. One step of our proof requires an explicit exact rational certificate or identity on flag numbers. This certificate is computed using linear programming.

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arXiv2026

Sharp Bounds on the Independence Number of Simplicial Spheres

We study the maximum size of an independent set in the graph of a simplicial sphere. Let $β(d,n)$ denote this maximum over all simplicial $(d-1)$-spheres on $n$ vertices, and let $α(d,n)$ denote the maximum restricted to flag $(d-1)$-spheres. For every fixed $d\geq4$, we prove $β(d,n)=n-Θ(n^{1/\lfloor d/2\rfloor})$. For flag spheres, we show $α(d,n)\geq n-4\sqrt n+O(1)$ for all $d\geq4$ and determine the correct asymptotic order $α(d,n)=n-Θ(\sqrt n)$ for dimensions $d=4,5$. We also investigate the independence sets of Bier spheres and show that, in contrast to our other results, for this very large family of spheres, the independence number cannot be larger than $\left\lfloor\frac{n}{2}\right\rfloor.$

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arXiv2026

Ehrhart Theory over Abelian Group Rings

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 classical counts. We prove that fundamental results of Ehrhart theory extend to this setting through a generalized Brion theorem, including rational generating functions, reciprocity phenomena, connections to volume, and vertex-cone decompositions. We further show how to derive $q$-enumerative and weighted theories from this setting, recasting several major refinements of Ehrhart theory as consequences of a single algebraic mechanism. We also show how our framework combines with equivariant Ehrhart theory.

math.CO↗

arXiv2026

From Ham-Sandwich to Centerpoints: Semialgebraic Algorithms for Cutting Polytopal Measures

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 volume cut off by a halfspace, is piecewise rational on a natural decomposition of the space of oriented hyperplanes. This lets us recast prescribed-proportion cutting problems as semialgebraic feasibility problems. For fixed ambient dimension, this yields polynomial-time algorithms to decide the existence of cuts, describe the full solution set, and sample or enumerate solutions. We extend this framework to the center transversal theorem, showing that spaces of deep affine flats are semialgebraic, which holds for centerpoints. We further show that the set of centerpoints of a convex polytope coincides with its floating body at level $1/(d+1)$, a useful semialgebraic description.

cs.CG↗

arXiv2026

Critical moments of slices and slabs of the cube (and other polyhedral norms)

In this article, we present a unified algebraic-combinatorial framework for computing explicit, piecewise rational, and combinatorially indexed parametric formulas for volumes and higher moments of slices and slabs of polyhedral norm balls. Our main method builds on prior work concerning a combinatorial decomposition of the parameter space of all slices of a polytope. We extend this framework to slabs, and find a polynomial-time algorithm in fixed dimension. We also exhibit computational methods to obtain moments of arbitrary order for all slices or slabs of any polyhedral norm ball, and an algebraic framework for analyzing their critical points. In addition, we present an experimental study of the $d$-dimensional unit cube. Our analysis recovers and reinterprets the known volume formulas for slabs and slices of the two- and three-dimensional cubes, first obtained by König and Koldobsky. Moreover, our method identifies a new complete family of fourteen rational functions giving the volumes of slices and slabs of the four-dimensional cube. We further compute explicit higher moments of slices and slabs in dimensions two and three, and derive explicit formulas for moments of arbitrary order for slices of the two-dimensional cube, describing their critical points. For the four-dimensional cube, we further use these formulas to identify candidate global maxima and minima for slice and slab volumes; the candidates are verified to be critical points by exact symbolic computation and are checked numerically to give the global extrema.

math.CO↗

arXiv2026

Persistent Subdivisions of Coxeter Permutahedra

We investigate the realizations of Coxeter permutahedra which are also Coxeter matroid polytopes; these are polytopes of the form $\mathrm{conv}(W \cdot \mathbf{a})$ where $W$ is a finite Coxeter group acting on $\mathbb{R}^n$ and $\mathbf{a}$ is generic. Our main focus is how the geometric properties of $\mathrm{conv}(W \cdot \mathbf{a})$ change as $\mathbf{a}$ changes, with particular attention to persistent simplices, triangulations, and subdivisions.

math.CO↗

arXiv2026

There are matroid toric ideals without quadratic Gröbner bases

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, Herzog, and Sturmfels established a direct connection between the existence of quadratic Gröbner bases and regular unimodular flag triangulations. Our paper solves a famous question posed by Herzog and Hibi on a polyhedral reformulation for the existence of quadratic Gröbner bases: we show that the base polytopes of the Fano plane and its dual do not have regular unimodular flag triangulations which implies the main result on Gröbner bases. Our proof relies on several novel tools: a lemma that connects the $1$-skeleton of a lattice polytope to the lattice points in its dilations, an encoding with Boolean formulas and SAT solvers, and symmetry-breaking arguments.

math.CO↗

arXiv2026

Irreducible Markov Chains on spaces of graphs with fixed degree-color sequences

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 colored vertices, in which we fix the degree sequence and another statistic arising from the vertex coloring, and prove that the set can be connected with simple color-preserving switches or moves. These moves form a basis for defining an irreducible Markov chain necessary for testing statistical model fit to block-partitioned network data. Our methods further generalize well-known algebraic results from the 1990s: namely, that the corresponding moves can be used to construct a regular triangulation for a generalization of the second hypersimplex. On the other hand, in contrast to the monochromatic case, we show that for \emph{simple} graphs, the 1-norm of the moves necessary to connect the space increases with the number of colors.

cs.DM↗

arXiv2025

Ramsey Numbers through the Lenses of Polynomial Ideals and Nullstellensätze

In this article we study the Ramsey numbers $R(r,s)$ through Hilbert's Nullstellensatz and Alon's Combinatorial Nullstellensatz. We give polynomial encodings whose solutions correspond to Ramsey graphs of order $n$, those that do not contain a copy of $K_r$ or $\bar{K}_s$. When these systems have no solution and $n \ge R(r,s)$, we construct Nullstellensatz certificates whose degrees are equal to the restricted online Ramsey numbers introduced by Conlon, Fox, Grinshpun and He. Moreover, we show that these results generalize to other numbers in Ramsey theory, including Rado, van der Waerden, and Hales-Jewett numbers. Finally, we introduce a family of numbers that relate to the coefficients of a certain "Ramsey polynomial" that gives lower bounds for Ramsey numbers.

math.CO↗

arXiv2025

Discrete Curvatures and Convex Polytopes

We study Forman--Ricci and effective resistance curvatures on the skeleta of convex polytopes. Our guiding questions are: how frequently do polytopal graphs exhibit everywhere positive curvature, and what structural constraints does positivity impose? For Forman--Ricci curvature we derive an exact identity for the average edge curvature in terms of flag $f$-numbers and establish the existence of infinite families of Forman--Ricci-positive polytopes in every fixed dimension $d\ge 6$. We prove finiteness results in low dimension: there are only finitely many Forman--Ricci-positive $3$- and $4$-polytopes; for $d=5$ we show finiteness in the simplicial case, and conjecture its extension to $5$-polytopes more generally. For the resistance curvature $κ(v)$ we establish the existence of infinite families for all $d\ge 3$, and we provide a quantitative lower bound for $κ(v)$ in a simple $3$-polytope in terms of the lengths of the three $2$-faces incident to $v$. This bound leads to constructions of non-vertex-transitive, resistance-positive $3$-polytopes via $Δ$-operations, and a degree-based obstruction showing that if each neighbor of $v$ has degree at most $d_v-2$, then $κ(v)\le 0$. Our results suggest that positive curvature on polytopal skeletons is rare and constrained.

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arXiv2025

An arithmetic measure of width for convex bodies

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 refines lattice width by detecting gaps in the lattice point distribution and always provides a natural lower bound. We show that for large dilates of a convex body, the attained values form an arithmetic progression with only a bounded number of omissions near the extremes. For rational polytopes, we show that the arithmetic width grows eventually quasilinearly in the dilation parameter, with optimal directions reoccurring periodically. Lastly, we present algorithms to compute the arithmetic width. These results build new connections with discrete geometry, integer programming, and additive combinatorics.

math.CO↗

arXiv2025

On the Number of Vertices in a Hyperplane Section of a Polytope

We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of upper bound theorem) and discuss a new algorithm to find all sections. Second, we investigate the sequence of numbers of vertices produced by the different slices over all possible hyperplanes and analyze the gaps that arise in that sequence. We study these sequences for three-dimensional polytopes and for hypercubes. Our results were obtained with the help of large computational experiments, and we report on new data generated for hypercubes.

math.CO↗

arXiv2025

Semigroups of Integer Points in Convex Cones

We study the question whether the affine semigroup of integer points in a convex cone can be finitely generated up to symmetries of the cone. We establish general properties of finite generation up to symmetry, and then concentrate on the case of irrational polyhedral cones.

math.NT↗

arXiv2025

Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones

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 semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set. We show this is true for the cone of positive semidefinite matrices (PSD) and the second-order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, under the action of a finitely generated group. We also extend notions of total dual integrality, Gomory-Chvátal closure, and Carathéodory rank to integer points in arbitrary cones.

math.OC↗

arXiv2024

Optimization Tools for Computing Colorings of $[1,\cdots ,n]$ with Few Monochromatic Solutions on $3$-variable Linear Equations

A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $E$ there is a threshold value $R_k(E)$ (the Rado number of $E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \ge R_k(E)$, there exists at least one monochromatic solution. But one can further ask, how many monochromatic solutions is the minimum possible in terms of $n$? Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.

math.CO↗

arXiv2024

Weighted Ehrhart Theory: Extending Stanley's nonnegativity theorem

We generalize R. P. Stanley's celebrated theorem that the $h^\ast$-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the $h^\ast$-polynomial as a real-valued function for a larger family of weights. We then target the case when the weight function is the square of a single (arbitrary) linear form. We show stronger results for two-dimensional convex lattice polygons and give concrete examples showing tightness of the hypotheses. As an application, we construct a counterexample to a conjecture by Berg, Jochemko, and Silverstein on Ehrhart tensor polynomials.

math.CO↗

arXiv2024

Sums of Weighted Lattice Points of Polytopes

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 integrals and maximum values of such quasi-polynomials, as well as obtain new identities in representation theory. These topics have been of great interest to Michèle Vergne since the late 1980's. Our new contribution is a result that transforms weighted sums into unweighted sums, even when the weights are very general quasipolynomials. In some cases it leads to faster integration over a polytope. We can create new algebraic identities and conjectures in algebraic combinatorics and number theory.

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