All Items 9

The story of ideals, varieties and algorithms

Cox, David A. (Department of Mathematics and Statistics, Amherst College)

This expository article tells the story of how the book "Ideals, Varieties and Algorithms" came to be written.

The story of ideals, varieties and algorithms
On closed graphs, II

Cox, David A. (Department of Mathematics and Statistics, Amherst College)

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.

On closed graphs, II
Integer decomposition property of dilated polytopes

Cox, David A. (Department of Mathematics and Statistics, Amherst College)

An integral convex polytope P in R^N possesses the integer decomposition property if, for any integer k > 0 and for any integer point a in kP, there exist integer points a_1,...,a_k in P such that a = a_1 + ... + a_k. A fundamental question is to determine the integers k > 0 for which the dilated polytope kP 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.

Integer decomposition property of dilated polytopes
On closed graphs I

Cox, David A. (Department of Mathematics and Statistics, Amherst College)

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.

On closed graphs I
Mirror symmetry and polar duality of polytopes

Cox, David A. (Department of Mathematics and Statistics, Amherst College)

This expository article explores the connection between the polar duality from polyhedral geometry and mirror symmetry from mathematical physics and algebraic geometry. Topics discussed include duality of polytopes and cones as well as the famous quintic threefold and the toric variety of a reflexive polytope.

Mirror symmetry and polar duality of polytopes
The homogeneous coordinate ring of a toric variety

Cox, David A. (Department of Mathematics and Statistics, Amherst College)

The original version of a paper, published in 1995. The 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 homogeneous coordinate ring of a toric variety
Strata of rational space curves

Cox, David A. (Department of Mathematics and Statistics, Amherst College)

The μ-invariant image 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 μ-type, what we call a μ-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, including an explicit formula for the codimension of the locus of non-proper parametrizations within each μ-stratum and a decomposition of the smallest μ-stratum based on which two-dimensional rational normal scroll the curve lies on.

Strata of rational space curves
The homogeneous coordinate ring of a toric variety [Erratum]

Cox, David A. (Department of Mathematics and Statistics, Amherst College)

An erratum that corrects an error in the proof of Proposition 4.3 in the paper "The Homogeneous Coordinate Ring of a Toric Variety." 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.

The homogeneous coordinate ring of a toric variety [Erratum]
The Galois theory of the lemniscate

Cox, David A. (Department of Mathematics and Statistics, Amherst College)

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.

The Galois theory of the lemniscate