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David A. Cox

Publications and source records attributed to David A. Cox.

At least 19 recordsLinked to original sources
arXiv2026

Symbolic Integration of Differential Forms: From Abel to Zeilberger

This paper focuses on symbolic integration of differential forms, with a particular emphasis on historical and modern developments, from Abel's addition theorems for Abelian integrals to Zeilberger's creative telescoping for parameterized integrals. It explores closed rational $p$-forms and provides algorithmic approaches for their integration, extending classical results like Hermite reduction and Liouville's theorem. The integration of closed differential forms with parameters is further examined through telescopers, offering a unified framework for handling both algebraic and transcendental cases.

math.CA↗
arXiv2024

Notes on Three Formulas of Abel

These notes explore three amazing formulas proved by Abel in his 1826 Paris memoir on what we now call Abelian integrals. We discuss the first two formulas from the point of view of symbolic computation and explain their connection to residues and partial fractions. The third formula arises from the first two and is related to the genus and lattice points in the Newton polygon.

math.AG↗
arXiv2023

Subresultants and the Shape Lemma

In nice cases, a zero-dimensional complete intersection ideal over a field of characteristic zero has a Shape Lemma. There are also cases where the ideal is generated by the resultant and first subresultant polynomials of the generators. This paper explores the relation between these representations and studies when the resultant generates the elimination ideal. We also prove a Poisson formula for resultants arising from the hidden variable method.

math.AC↗
arXiv2020

Stickelberger and the Eigenvalue Theorem

The Eigenvalue Theorem shows that solving a zero-dimensional polynomial system can be recast as an eigenvalue problem. This paper explores the relation between the Eigenvalue Theorem and the work of Ludwig Stickelberger (1850-1936).

math.AC↗
arXiv2020

On the base point locus of surface parametrizations: formulas and consequences

This paper shows that the multiplicity of the base points locus of a projective rational surface parametrization can be expressed as the degree of the content of a univariate resultant. As a consequence, we get a new proof of the degree formula relating the degree of the surface, the degree of the parametrization, the base points multiplicity, and the degree of the rational map induced by the parametrization. In addition, we extend both formulas to the case of dominant rational maps of the projective plane and describe how the base point loci of a parametrization and its reparametrizations are related. As an application of these results, we explore how the degree of a surface reparametrization is affected by the presence of base points.

math.AG↗
arXiv2020

Moment Maps, Strict Linear Precision, and Maximum Likelihood Degree One

We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into polytopes with strict linear precision, and in the process we use results from and find remarkable connections between Symplectic Geometry, Geometric Modeling, Algebraic Statistics, and Algebraic Geometry.

math.AG↗
arXiv2016

Transfinite diameter on complex algebraic varieties

We use methods from computational algebraic geometry to study Chebyshev constants and the transfinite diameter of a pure $m$-dimensional affine algebraic variety in $\mathbb{C}^n$ ($m\leq n$). The main result is a generalization of Zaharjuta's integral formula for the Fekete-Leja transfinite diameter.

math.AG↗
arXiv2015

On closed graphs II

A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we explore various aspects of closed graphs, including the number of closed labelings and clustering coefficients.

math.CO↗
arXiv2014

On Closed Graphs I

A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal, claw-free, and has a property we call narrow, which holds when every vertex is distance at most one from all longest shortest paths of the graph.

math.CO↗
arXiv2014

Strata of rational space curves

The mu-invariant mu = (μ_1,μ_2,μ_3) of a rational space curve gives important information about the curve. In this paper, we describe the structure of all parameterizations that have the same mu-type, what we call a mu-stratum, and as well the closure of strata. Many of our results are based on papers by the second author that appeared in the commutative algebra literature. We also present new results not in the earlier papers, including an explicit formula for the codimension of the locus of non-proper parametrizations within each mu-stratum and a decomposition of the smallest mu-stratum based on which two-dimensional rational normal scroll the curve lies on.

math.AG↗
arXiv2014

Erratum to "The Homogeneous Coordinate Ring of a Toric Variety", along with the original paper

This submission consists of two papers: 1) an erratum that corrects an error in the proof of Proposition 4.3 in my paper "The Homogeneous Coordinate Ring of a Toric Variety", and 2) the original (unchanged) version of the paper, published in 1995. The original paper introduced the homogeneous coordinate ring of a toric variety (now called the total coordinate ring or Cox ring) and gave a quotient construction. The paper also studied sheaves on a toric variety, and in Section 4 described its automorphism group. The error in the proof of Proposition 4.3 resulted from the faulty assumption that a certain set of graded endomorphisms forms a ring; rather, it is a monoid under composition. The erratum notes this error and gives a correct proof of the proposition.

alg-geom↗
arXiv2013

Integer decomposition property of dilated polytopes

Let $\mathcal{P} \subset \mathbb{R}^N$ be an integral convex polytope of dimension $d$ and write $k \mathcal{P}$, where $k = 1, 2, \ldots$, for dilations of $\mathcal{P}$. We say that $\mathcal{P}$ possesses the integer decomposition property if, for any integer $k = 1, 2, \ldots$ and for any $α\in k \mathcal{P} \cap \mathbb{Z}^N$, there exist $α_{1}, \ldots, α_k$ belonging to $\mathcal{P} \cap \mathbb{Z}^N$ such that $α= α_1 + \cdots + α_k$. A fundamental question is to determine the integers $k > 0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. In the present paper, combinatorial invariants related to the integer decomposition property of dilated polytopes will be proposed and studied.

math.CO↗
arXiv2012

The Galois theory of the lemniscate

This article studies the Galois groups that arise from division points of the lemniscate. We compute these Galois groups two ways: first, by class field theory, and second, by proving the irreducibility of lemnatomic polynomials, which are analogs of cyclotomic polynomials. We also discuss Abel's theorem on the lemniscate and explain how lemnatomic polynomials relate to Chebyshev polynomials.

math.NT↗
arXiv2008

Primitive Collections and Toric Varieties

This paper studies Batyrev's notion of primitive collection. We use primitive collections to characterize the nef cone of a quasi-projective toric variety whose fan has convex support, a result stated without proof by Batyrev in the smooth projective case. When the fan is non-simplicial, we modify the definition of primitive collection and explain how our definition relates to primitive collections of simplicial subdivisons. The paper ends with some open problems.

math.AG↗
arXiv2006

Bezoutians and Tate Resolutions

This paper gives an explicit construction of the Tate resolution of sheaves arising from the d-fold Veronese embedding of P^n. Our description involves the Bezoutian of n+1 homogenous forms of degree d in n+1 variables. We give applications to duality theorems, including Koszul duality.

math.AC↗
arXiv2001

Equations of Parametric Surfaces via Syzygies

The revised version has two additional references and a shorter proof of Proposition 5.7. This version also makes numerous small changes and has an appendix containing a proof of the degree formula for a parametrized surface.

math.AG↗
arXiv2000

Virtual Fundamental Classes of Zero Loci

Let V be a convex vector bundle over a smooth projective manifold X, and let Y be the subset of X which is the zero locus of a regular section of V. This mostly expository paper discusses a conjecture which relates the virtual fundamental classes of X and Y. Using an argument due to Gathmann, we prove a special case of the conjecture. The paper concludes with a discussion of how our conjecture relates to the mirror theorems in the literature.

math.AG↗