arXiv subjects

Matt Larson

Publications and source records attributed to Matt Larson.

34 records · Page 2Linked to original sources

arXiv2025

Lefschetz properties of local face modules

Local face modules are modules over face rings whose Hilbert function is the local $h$-vector of a triangulation of a simplex. We study when Lefschetz properties hold for local face modules. We prove new inequalities for local $h$-vectors of vertex-induced triangulations by proving Lefschetz properties for local face modules of these triangulations. We show that, even for regular triangulations, Lefschetz properties can fail for local face modules in positive characteristic.

math.CO↗

arXiv2025

Straightening laws for Chow rings of matroids

We give elementary and non-inductive proofs of three fundamental theorems about Chow rings of matroids: the standard monomial basis, Poincare duality, and the dragon-Hall-Rado formula. Our approach, which also works for augmented Chow rings of matroids, is based on a straightening law. This approach also gives a decomposition of the Chow ring of a matroid into pieces indexed by flats.

math.CO↗

arXiv2025

Rank functions and invariants of delta-matroids

In this note, we give a rank function axiomatization for delta-matroids and study the corresponding rank generating function. We relate an evaluation of the rank generating function to the number of independent sets of the delta-matroid, and we prove a log-concavity result for that evaluation using the theory of Lorentzian polynomials.

math.CO↗

arXiv2024

K-theoretic positivity for matroids

Hilbert polynomials have positivity properties under favorable conditions. We establish a similar "K-theoretic positivity" for matroids. As an application, for a multiplicity-free subvariety of a product of projective spaces such that the projection onto one of the factors has birational image, we show that a transformation of its K-polynomial is Lorentzian. This partially answers a conjecture of Castillo, Cid-Ruiz, Mohammadi, and Montano. As another application, we show that the h*-vector of a simplicially positive divisor on a matroid is a Macaulay vector, affirmatively answering a question of Speyer for a new infinite family of matroids.

math.AG↗

arXiv2024

K-classes of delta-matroids and equivariant localization

Delta-matroids are "type B" generalizations of matroids in the same way that maximal orthogonal Grassmannians are generalizations of Grassmannians. A delta-matroid analogue of the Tutte polynomial of a matroid is the interlace polynomial. We give a geometric interpretation for the interlace polynomial via the K-theory of maximal orthogonal Grassmannians. To do so, we develop a new Hirzebruch-Riemann-Roch-type formula for the type B permutohedral variety.

math.CO↗

arXiv2024

Signed permutohedra, delta-matroids, and beyond

We establish a connection between the algebraic geometry of the type B permutohedral toric variety and the combinatorics of delta-matroids. Using this connection, we compute the volume and lattice point counts of type B generalized permutohedra. Applying tropical Hodge theory to a new framework of "tautological classes of delta-matroids," modeled after certain vector bundles associated to realizable delta-matroids, we establish the log-concavity of a Tutte-like invariant for a broad family of delta-matroids that includes all realizable delta-matroids. Our results include new log-concavity statements for all (ordinary) matroids as special cases.

math.AG↗

arXiv2024

$K$-rings of wonderful varieties and matroids

We study the $K$-ring of the wonderful variety of a hyperplane arrangement and give a combinatorial presentation that depends only on the underlying matroid. We use this combinatorial presentation to define the $K$-ring of an arbitrary loopless matroid. We construct an exceptional isomorphism, with integer coefficients, to the Chow ring of the matroid that satisfies a Hirzebruch--Riemann--Roch-type formula, generalizing a recent construction of Berget, Eur, Spink, and Tseng for the permutohedral variety (the wonderful variety of a Boolean arrangement). As an application, we give combinatorial formulas for Euler characteristics of arbitrary line bundles on wonderful varieties. We give analogous constructions and results for augmented wonderful varieties, and for Deligne--Mumford--Knudsen moduli spaces of stable rational curves with marked points.

math.AG↗

arXiv2024

The Bergman fan of a polymatroid

We introduce the Bergman fan of a polymatroid and prove that the Chow ring of the Bergman fan is isomorphic to the Chow ring of the polymatroid. Using the Bergman fan, we establish the Kähler package for the Chow ring of the polymatroid, recovering and strengthening a result of Pagaria-Pezzoli.

math.CO↗

arXiv2023

Kazhdan-Lusztig polynomials of braid matroids

We provide a combinatorial interpretation of the Kazhdan--Lusztig polynomial of the matroid arising from the braid arrangement of type $\mathrm{A}_{n-1}$, which gives an interpretation of the intersection cohomology Betti numbers of the reciprocal plane of the braid arrangement. Moreover, we prove an equivariant version of this result. The key combinatorial object is a class of matroids arising from series-parallel networks. As a consequence, we prove a conjecture of Elias, Proudfoot, and Wakefield on the top coefficient of Kazhdan--Lusztig polynomials of braid matroids, and we provide explicit generating functions for their Kazhdan--Lusztig and $Z$-polynomials.

math.CO↗

arXiv2023

Stellahedral geometry of matroids

We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety, and show that valuative, homological, and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro-Smirnov-Vaintrob on Postnikov-Shapiro algebras, and calculate the Chern-Schwartz-MacPherson classes of matroid Schubert cells. The central construction is the "augmented tautological classes of matroids," modeled after certain vector bundles on the stellahedral toric variety.

math.AG↗

arXiv2023

Intersection theory of polymatroids

Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented wonderful varieties of subspace arrangements, we generalize several algebro-geometric techniques developed in recent years to study matroids. We show that intersection numbers in the augmented Chow ring of a polymatroid are determined by a matching property known as the Hall--Rado condition, which is new even in the case of matroids.

math.AG↗

arXiv2023

Resolutions of local face modules, functoriality, and vanishing of local $h$-vectors

We study the local face modules of triangulations of simplices, i.e., the modules over face rings whose Hilbert functions are local $h$-vectors. In particular, we give resolutions of these modules by subcomplexes of Koszul complexes as well as functorial maps between modules induced by inclusions of faces. As applications, we prove a new monotonicity result for local $h$-vectors and new results on the structure of faces in triangulations with vanishing local $h$-vectors.

math.CO↗

arXiv2022

Expanding Accurate Person Recognition to New Altitudes and Ranges: The BRIAR Dataset

Face recognition technology has advanced significantly in recent years due largely to the availability of large and increasingly complex training datasets for use in deep learning models. These datasets, however, typically comprise images scraped from news sites or social media platforms and, therefore, have limited utility in more advanced security, forensics, and military applications. These applications require lower resolution, longer ranges, and elevated viewpoints. To meet these critical needs, we collected and curated the first and second subsets of a large multi-modal biometric dataset designed for use in the research and development (R&D) of biometric recognition technologies under extremely challenging conditions. Thus far, the dataset includes more than 350,000 still images and over 1,300 hours of video footage of approximately 1,000 subjects. To collect this data, we used Nikon DSLR cameras, a variety of commercial surveillance cameras, specialized long-rage R&D cameras, and Group 1 and Group 2 UAV platforms. The goal is to support the development of algorithms capable of accurately recognizing people at ranges up to 1,000 m and from high angles of elevation. These advances will include improvements to the state of the art in face recognition and will support new research in the area of whole-body recognition using methods based on gait and anthropometry. This paper describes methods used to collect and curate the dataset, and the dataset's characteristics at the current stage.

cs.CV↗

arXiv2021

The Arakelov-Zhang pairing and Julia sets

The Arakelov-Zhang pairing $\langleψ,ϕ\rangle$ is a measure of the "dynamical distance" between two rational maps $ψ$ and $ϕ$ defined over a number field $K$. It is defined in terms of local integrals on Berkovich space at each completion of $K$. We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height $h_ϕ$ and the standard Weil height $h$, and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction.

math.NT↗

arXiv2020

Inverse problems for minimal complements and maximal supplements

Given a subset $W$ of an abelian group $G$, a subset $C$ is called an additive complement for $W$ if $W+C=G$; if, moreover, no proper subset of $C$ has this property, then we say that $C$ is a minimal complement for $W$. It is natural to ask which subsets $C$ can arise as minimal complements for some $W$. We show that in a finite abelian group $G$, every non-empty subset $C$ of size $|C| \leq 2^{2/3}|G|^{1/3}/((3e \log |G|)^{2/3}$ is a minimal complement for some $W$. As a corollary, we deduce that every finite non-empty subset of an infinite abelian group is a minimal complement. We also derive several analogous results for ``dual'' problems about maximal supplements.

math.CO↗

arXiv2019

Power maps in finite groups

In recent work, Pomerance and Shparlinski have obtained results on the number of cycles in the functional graph of the map $x \mapsto x^a$ in $\mathbb{F}_p^*$. We prove similar results for other families of finite groups. In particular, we obtain estimates for the number of cycles for cyclic groups, symmetric groups, dihedral groups and $SL_2(\mathbb{F}_q)$. We also show that the cyclic group of order $n$ minimizes the number of cycles among all nilpotent groups of order $n$ for a fixed exponent. Finally, we pose several problems.

math.CO↗