arXiv subjects

Christopher O'Neill

Publications and source records attributed to Christopher O'Neill.

At least 37 records · Page 2Linked to original sources

arXiv2023

Convexity in (colored) affine semigroups

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 semigroups, where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg's theorem and colorful Helly's theorem for semigroups, as well as a version of colorful Caratheodory's theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.

math.AC↗

arXiv2023

On faces of the Kunz cone and the numerical semigroups within them

A numerical semigroup is a cofinite subset of the non-negative integers that is closed under addition and contains 0. Each numerical semigroup $S$ with fixed smallest positive element $m$ corresponds to an integer point in a rational polyhedral cone $\mathcal C_m$, called the Kunz cone. Moreover, numerical semigroups corresponding to points in the same face $F \subseteq \mathcal C_m$ are known to share many properties, such as the number of minimal generators. In this work, we classify which faces of $\mathcal C_m$ contain points corresponding to numerical semigroups. Additionally, we obtain sharp bounds on the number of minimal generators of $S$ in terms of the dimension of the face of $\mathcal C_m$ containing the point corresponding to $S$.

math.CO↗

arXiv2023

Enumerating numerical sets associated to a numerical semigroup

A numerical set $T$ is a subset of $\mathbb N_0$ that contains $0$ and has finite complement. The atom monoid of $T$ is the set of $x \in \mathbb N_0$ such that $x+T \subseteq T$. Marzuola and Miller introduced the anti-atom problem: how many numerical sets have a given atom monoid? This is equivalent to asking for the number of integer partitions with a given set of hook lengths. We introduce the void poset of a numerical semigroup $S$ and show that numerical sets with atom monoid $S$ are in bijection with certain order ideals of this poset. We use this characterization to answer the anti-atom problem when $S$ has small type.

math.CO↗

arXiv2023

Numerical semigroups, polyhedra, and posets III: minimal presentations and face dimension

This paper is the third in a series of manuscripts that examine the combinatorics of the Kunz polyhedron $P_m$, whose positive integer points are in bijection with numerical semigroups (cofinite subsemigroups of $\mathbb Z_{\ge 0}$) whose smallest positive element is $m$. The faces of $P_m$ are indexed by a family of finite posets (called Kunz posets) obtained from the divisibility posets of the numerical semigroups lying on a given face. In this paper, we characterize to what extent the minimal presentation of a numerical semigroup can be recovered from its Kunz poset. In doing so, we prove that all numerical semigroups lying on the interior of a given face of $P_m$ have identical minimal presentation cardinality, and we provide a combinatorial method of obtaining the dimension of a face from its corresponding Kunz poset.

math.CO↗

arXiv2022

When is a numerical semigroup a quotient?

A natural operation on numerical semigroups is taking a quotient by a positive integer. If $\mathcal S$ is a quotient of a numerical semigroup with $k$ generators, we call $\mathcal S$ a $k$-quotient. We give a necessary condition for a given numerical semigroup $\mathcal S$ to be a $k$-quotient, and present, for each $k \ge 3$, the first known family of numerical semigroups that cannot be written as a $k$-quotient. We also examine the probability that a randomly selected numerical semigroup with $k$ generators is a $k$-quotient.

math.AC↗

arXiv2022

Graver bases of shifted numerical semigroups with 3 generators

A numerical semigroup $M$ is a subset of the non-negative integers that is closed under addition. A factorization of $n \in M$ is an expression of $n$ as a sum of generators of $M$, and the Graver basis of $M$ is a collection $Gr(M_t)$ of trades between the generators of $M$ that allows for efficient movement between factorizations. Given positive integers $r_1, \ldots, r_k$, consider the family $M_t = \langle t + r_1, \ldots, t + r_k\rangle$ of "shifted" numerical semigroups whose generators are obtained by translating $r_1, \ldots, r_k$ by an integer parameter $t$. In this paper, we characterize the Graver basis $Gr(M_t)$ of $M_t$ for sufficiently large $t$ in the case $k = 3$, in the form of a recursive construction of $Gr(M_t)$ from that of smaller values of $t$. As a consequence of our result, the number of trades in $Gr(M_t)$, when viewed as a function of $t$, is eventually quasilinear. We also obtain a sharp lower bound on the start of quasilinear behavior.

math.AC↗

arXiv2022

Numerical semigroups, polyhedra, and posets I: the group cone

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 introduced by Kunz, has integer points in bijection with numerical semigroups of fixed multiplicity, and another, introduced by Hellus and Waldi, has integer points corresponding to oversemigroups of numerical semigroups with two generators. In this paper, we provide a combinatorial framework from which to study both families of polyhedra. We introduce a new family of polyhedra called group cones, each constructed from some finite abelian group, from which both of the aforementioned families of polyhedra are directly determined but that are more natural to study from a standpoint of polyhedral geometry. We prove that the faces of group cones are naturally indexed by a family of finite posets, and illustrate how this combinatorial data relates to semigroups living in the corresponding faces of the other two families of polyhedra.

math.CO↗

arXiv2021

Length density and numerical semigroups

Length density is a recently introduced factorization invariant, assigned to each element $n$ of a cancellative commutative atomic semigroup $S$, that measures how far the set of factorization lengths of $n$ is from being a full interval. We examine length density of elements of numerical semigroups (that is, additive subsemigroups of the non-negative integers).

math.AC↗

arXiv2021

Numerical semigroups, polyhedra, and posets II: locating certain families of semigroups

Several recent papers have examined a rational polyhedron $P_m$ whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing $m$. A combinatorial description of the faces of $P_m$ was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of $P_m$ containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of $P_m$. In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of $P_m$.

math.CO↗

arXiv2021

The fundamental theorem of finite fields: a proof from first principles

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 theory to prove it. Yet the combinatorial and algebraic coding theory applications of finite fields can show up early on for students in STEM. To make the FTFF more accessible to students lacking exposure to Galois theory, we provide a proof from algebraic "first principles."

math.HO↗

arXiv2021

Factorization length distribution for affine semigroups IV: a geometric approach to weighted factorization lengths in three-generator numerical semigroups

For numerical semigroups with three generators, we study the asymptotic behavior of weighted factorization lengths, that is, linear functionals of the coefficients in the factorizations of semigroup elements. This work generalizes many previous results, provides more natural and intuitive proofs, and yields a completely explicit error bound.

math.CO↗

arXiv2020

Factorization length distribution for affine semigroups II: asymptotic behavior for numerical semigroups with arbitrarily many generators

For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length statistics. This involves a variety of tools that are not standard in the subject, such as algebraic combinatorics (Schur polynomials), probability theory (weak convergence of measures, characteristic functions), and harmonic analysis (Fourier transforms of distributions). We provide instructive examples which demonstrate the power and generality of our techniques. We also highlight unexpected consequences in the theory of homogeneous symmetric functions.

math.CO↗

arXiv2020

Factorization length distribution for affine semigroups III: modular equidistribution for numerical semigroups with arbitrarily many generators

For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrary modulus. In particular, we prove that the factorization lengths are equidistributed across all congruence classes that are not trivially ruled out by modular considerations.

math.CO↗

arXiv2020

On length densities

For a commutative cancellative monoid $M$, we introduce the notion of the length density of both a nonunit $x\in M$, denoted $\mathrm{LD}(x)$, and the entire monoid $M$, denoted $\mathrm{LD}(M)$. This invariant is related to three widely studied invariants in the theory of non-unit factorizations, $L(x)$, $\ell(x)$, and $ρ(x)$. We consider some general properties of $\mathrm{LD}(x)$ and $\mathrm{LD}(M)$ and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid $M$ with irrational length density, we show that if $M$ is finitely generated, then $\mathrm{LD}(M)$ is rational and there is a nonunit element $x\in M$ with $\mathrm{LD}(M)=\mathrm{LD}(x)$ (such a monoid is said to have accepted length density). While it is well-known that the much studied asymptotic versions of $L(x)$, $\ell (x)$ and $ρ(x)$ (denoted $\overline{L}(x)$, $\overline{\ell}(x)$, and $\overlineρ (x)$) always exist, we show the somewhat surprising result that $\overline{\mathrm{LD}}(x) = \lim_{n\rightarrow \infty} \mathrm{LD}(x^n)$ may not exist. We also give some finiteness conditions on $M$ that force the existence of $\overline{\mathrm{LD}}(x)$.

math.AC↗

arXiv2020

On minimal presentations of shifted affine semigroups with few generators

An affine semigroup is a finitely generated subsemigroup of $(\mathbb Z_{\ge 0}^d, +)$, and a numerical semigroup is an affine semigroup with $d = 1$. A growing body of recent work examines shifted families of numerical semigroups, that is, families of numerical semigroups of the form $M_n = \langle n + r_1, \ldots, n + r_k \rangle$ for fixed $r_1, \ldots, r_k$, with one semigroup for each value of the shift parameter $n$. It has been shown that within any shifted family of numerical semigroups, the size of any minimal presentation is bounded (in fact, this size is eventually periodic in $n$). In this paper, we consider shifted families of affine semigroups, and demonstrate that some, but not all, shifted families of 4-generated affine semigroups have arbitrarily large minimal presentations.

math.CO↗

arXiv2020

On parametrized families of numerical semigroups

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 $P_n = \langle f_1(n), \ldots, f_k(n) \rangle$ for polynomial functions $f_i$. We conjecture that for large $n$, the Betti numbers, Frobenius number, genus, and type of $P_n$ each coincide with a quasipolynomial. This conjecture has already been proven in general for Frobenius numbers, and for the remaining quantities in the special case when $P_n = \langle n, n + r_2, \ldots, n + r_k \rangle$. Our main result is to prove our conjecture in the case where each $f_i$ is linear. In the process, we develop the notion of weighted factorization length, and generalize several known results for standard factorization lengths and delta sets to this weighted setting.

math.AC↗

arXiv2020

Discovery learning in an interdisciplinary course on finite fields and applications

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 lectures and active learning. Under the discussed classroom model, there are two lecture days and two discovery-based discussion days each week. Discussions center around group activities that build intuition for abstract concepts while avoiding excessive technical machinery. Teaching this course presents some unique challenges, as much of the content typically appears in a second course in abstract algebra, yet the students exhibit a wide range of mathematical preparation.

math.HO↗

arXiv2020

Weighted Means of B-Splines, Positivity of Divided Differences, and Complete Homogeneous Symmetric Polynomials

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 symmetric polynomials of even degree $2p$, the positivity of which is a classical result by D. B. Hunter. We extend Hunter's result to complete homogeneous symmetric polynomials of fractional degree, which are defined via Jacobi's bialternant formula. We show in particular that these polynomials have positive real part for real degrees $μ$ with $|μ-2p|< 1/2$. We also prove a positivity criterion for linear combinations of the classical complete homogeneous symmetric polynomials and a sufficient criterion for the positivity of linear combinations of products of such polynomials.

math.CO↗