34 citations · 239 across the 41 of their papers we have counts for
51 papers
Numerical semigroups via projections and via quotients
Tristram Bogart, Christopher O'Neill, Kevin Woods
We examine two natural operations to create numerical semigroups. We say that a numerical semigroup is -normalescent if it is the projection of the set of integer…
When is a numerical semigroup a quotient?
Tristram Bogart, Christopher O'Neill, Kevin Woods
A natural operation on numerical semigroups is taking a quotient by a positive integer. If is a quotient of a numerical semigroup with generators, we call $\mathca…
Graver bases of shifted numerical semigroups with 3 generators
James Howard, Christopher O'Neill
A numerical semigroup is a subset of the non-negative integers that is closed under addition. A factorization of is an expression of as a sum of generators of …
Convexity in (colored) affine semigroups
Jesus A. De Loera, Christopher O'Neill, Chengyang Wang
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 se…
Enumerating numerical sets associated to a numerical semigroup
April Chen, Nathan Kaplan, Liam Lawson +2
A numerical set is a subset of that contains and has finite complement. The atom monoid of is the set of such that .…
On the cardinality of minimal presentations of numerical semigroups
Ceyhun Elmacioglu, Kieran Hilmer, Christopher O'Neill +2
In this paper, we consider the following question: "given the multiplicity and embedding dimension of a numerical semigroup , what can be said about the cardinality …
Length density and numerical semigroups
Cole Brower, Scott Chapman, Travis Kulhanek +4
Length density is a recently introduced factorization invariant, assigned to each element of a cancellative commutative atomic semigroup , that measures how far the set of f…
Factorization length distribution for affine semigroups IV: a geometric approach to weighted factorization lengths in three-generator numerical semigroups
Stephan Ramon Garcia, Christopher O'Neill, Gabe Udell
For numerical semigroups with three generators, we study the asymptotic behavior of weighted factorization lengths, that is, linear functionals of the coefficients in the factoriza…
The fundamental theorem of finite fields: a proof from first principles
Anastasia Chavez, Christopher O'Neill
A mathematics student's first introduction to the fundamental theorem of finite fields (FTFF) often occurs in an advanced abstract algebra course and invokes the power of Galois th…
Numerical semigroups, polyhedra, and posets III: minimal presentations and face dimension
Tara Gomes, Christopher O'Neill, Eduardo Torres Davila
This paper is the third in a series of manuscripts that examine the combinatorics of the Kunz polyhedron , whose positive integer points are in bijection with numerical semigr…
On length densities
Scott T. Chapman, Christopher O'Neill, Vadim Ponomarenko
For a commutative cancellative monoid , we introduce the notion of the length density of both a nonunit , denoted , and the entire monoid , denoted $\…
On minimal presentations of shifted affine semigroups with few generators
Christopher O'Neill, Isabel White
An affine semigroup is a finitely generated subsemigroup of , and a numerical semigroup is an affine semigroup with . A growing body of recent work…
Factorization length distribution for affine semigroups III: modular equidistribution for numerical semigroups with arbitrarily many generators
Stephan Ramon Garcia, Mohamed Omar, Christopher O'Neill +1
For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrar…
Weighted Means of B-Splines, Positivity of Divided Differences, and Complete Homogeneous Symmetric Polynomials
Albrecht Boettcher, Stephan Ramon Garcia, Mohamed Omar +1
We employ the fact certain divided differences can be written as weighted means of B-splines and hence are positive. These divided differences include the complete homogeneous symm…
Numerical semigroups, polyhedra, and posets II: locating certain families of semigroups
Jackson Autry, Abigail Ezell, Tara Gomes +4
Several recent papers have examined a rational polyhedron whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-…
Numerical semigroups, polyhedra, and posets I: the group cone
Nathan Kaplan, Christopher O'Neill
Several recent papers have explored families of rational polyhedra whose integer points are in bijection with certain families of numerical semigroups. One such family, first intro…
Factorization length distribution for affine semigroups II: asymptotic behavior for numerical semigroups with arbitrarily many generators
Stephan Ramon Garcia, Mohamed Omar, Christopher O'Neill +1
For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational r…
On parametrized families of numerical semigroups
Franklin Kerstetter, Christopher O'Neill
A numerical semigroup is an additive subsemigroup of the non-negative integers. In this paper, we consider parametrized families of numerical semigroups of the form $P_n = \langle…
Elasticity in Apery sets
Jackson Autry, Tara Gomes, Christopher O'Neill +1
A numerical semigroup is an additive subsemigroup of the non-negative integers, containing zero, with finite complement. Its multiplicity is its smallest nonzero element. T…
Wilf's conjecture in fixed multiplicity
Winfried Bruns, Pedro Garcia-Sanchez, Christopher O'Neill +1
We give an algorithm to determine whether Wilf's conjecture holds for all numerical semigroups with a given multiplicity , and use it to prove Wilf's conjecture holds whenever $…
Beyond Coins, Stamps, and Chicken McNuggets: an Invitation to Numerical Semigroups
Scott Chapman, Rebecca Garcia, Christopher O'Neill
We give a self contained introduction to numerical semigroups, and present several open problems centered on their factorization properties.
Sequentially embeddable graphs
Jackson Autry, Christopher O'Neill
We call a (not necessarily planar) embedding of a graph in the plane \emph{sequential} if its vertices lie in and the line segments between adjacent vertices cont…
Discovery learning in an interdisciplinary course on finite fields and applications
Christopher O'Neill, Lily Silverstein
The authors describe their approach to teaching a course on finite fields and combinatorial applications, including block designs and error-correcting codes, using a hybrid of lect…
A sequence of quasipolynomials arising from random numerical semigroups
Calvin Leng, Christopher O'Neill
A numerical semigroup is a subset of the non-negative integers that is closed under addition. For a randomly generated numerical semigroup, the expected number of minimum generator…