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CHRISTOPHER O’NEILL
My research lies in combinatorial commutative algebra of rings and monoids. I am also interested
in discrete geometry and its connections to computation, algorithms, and the development and use
of software in mathematical research.
Research overview. Non-unique factorization theory aims to classify integral domains and commutative cancellative monoids where uniqueness of factorization of non-unit elements as products
of irreducibles fails. Since its inception in the study of algebraic number fields, non-unique factorization theory has remained closely connected to additive group theory and combinatorial number
theory. Combinatorial factorization invariants play a major role in the study of non-unique factorization, as they give concrete methods of quantifying the abundance and variety of factorizations.
Finitely generated monoids have played an increasingly important role in the study of non-unique
factorization theory. Many landmark results in factorization theory center around demonstrating
extremal behavior of certain factorization invariants, and the setting of finitely generated monoids
is broad enough to encompass a large variety of such behavior [ACHP07, CK15]. At the same
time, finitely generated monoids benefit from algorithms and implementations for explicit computation of invariant values, which have been instrumental in their study [DGM15, Gar15]. As such,
finitely generated monoids provide a wealth of explicit examples, and obtaining a more thorough
understanding of their factorization structure has implications on the general setting.
My recent work involving finitely generated monoids has produced
(i) a definitive link with combinatorial commutative algebra, yielding a new framework through
which to examine factorization structure [One15], and
(ii) more efficient algorithms for computing invariant values [BOP14b, DGM15].
In Sections 2-5, several factorization invariants from the literature are discussed and methods for
future work concern the effective computation of factorization invariants for finitely generated
monoids are presented, following an overview of the my recent contributions in Section 1.
Undergraduate research. Suitably chosen research problems on factorization theory in monoids
are ideal for undergraduate projects, as they offer a high probability of producing publishable output while remaining accessible to undergraduate students. While significant contributions to open
problems in commutative algebra often require extensive background in the subject, factorization
theory can place algebraic objects in combinatorial settings, making them more concrete to work
with explicitly. In fact, several of the motivating results presented below are products of undergraduate research projects I have coadvised, either in the REU setting [CCMMP14, HKMOP14,
KOP15, OPTW15] or independently [BOP14a, BOP14b].
Many of the problems presented below are excellent starting places for undergraduate research
projects, and I intend to include undergraduates on several such projects. As an example, initial
investigations for Project 1 will involve utilizing mathematics software packages to compute concrete
examples, providing a natural starting place for students to acquire familiarity and build intuition.
Project 4 will also benefit from computation in its early stages, and a first semester in abstract
algebra is a sufficient prerequisite to begin work with block monoids. Additionally, the survey
article [OPe14] contains several open problems accessible to undergraduates.
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1. Invariants of non-unique factorization
A factorization of an element m in an integral domain R is an expression of m as a product of
irreducible elements; R is factorial if every nonzero nonunit admits a unique factorization (up to
multiplication by a unit). First examined in the context of rings of integers of algebraic number
fields [Car60], non-unique factorization theory aims to classify and quantify the non-uniqueness of
factorizations that arise in integral domains that fail to be factorial.
Since the factorial condition on a domain R does not depend on the additive structure of R, one
can study factorizations in R by focusing on the structure of R∗ = R \ {0}, its multiplicative (cancellative, commutative) monoid. Often, this is done by either considering R∗ itself, or by restricting
to a suitable submonoid of R∗ . For instance, when R is the monoid algebra over a cancellative commutative monoid M , the factorization properties of R are often studied by examining the monoid
M . Since the emergence of this idea [And97], non-unique factorization theory is frequently studied
in monoids in lieu of integral domains [GH06], resulting in applications to combinatorial number
theory [Hal91, Man73], algebraic geometry [Kun70, Ros99], and elsewhere.
The non-uniqueness of factorizations in a given monoid M can be quantified using combinatoriallyflavored factorization invariants, which assign to each element of M (or to M as a whole) a
value measuring its failure to admit unique factorization. For example, the catenary degree invariant assigns to each monoid element m ∈ M a non-negative integer c(m) measuring the distance between its factorizations, computed using a discrete graph whose vertices are factorizations
[CCMMP14, CGL09]. The catenary degree c(M ) of the monoid is then defined as the supremum of
the catenary degrees achieved by its elements [BGG11], providing a global (monoid-wide) measure of
the factorization structure. Many other standard invariants, such as delta sets [BCKR06, CGP14],
elasticity [BOP14a, CHM06, CGGR01], and ω-primality [AC10, OPe14], have been well studied
for several classes of monoids and will each be formally introduced in later sections.
Let ZM (m) denote the set of factorizations of the monoid element m ∈ M . The set
Z(M ) = {ZM (m) : m ∈ M },
of factorizations of the monoid M is an example of a perfect factorization invariant since it encapsulates enough information to uniquely determine M up to isomorphism [OPe15]. However,
such complete information comes at a cost: extracting information from it (or even simply writing
it down) is a nontrivial task. In fact, the set of factorizations of a sufficiently large finitely generated monoid element alone contains enough information to recover the monoid structure of M .
Many invariants derived from the set of factorizations, such as the elasticity and catenary degree
invariants, are more manageable and easier to work with, but necessitate a loss of information.
Further traction is often gained by examining factorization invariants for particular classes of
monoids arising in the literature. The factorization structure of numerical monoids [GR09] (cofinite
additive submonoids of Z≥0 ) has been of much recent interest [BCKR06, CK15, OPe15], due in
part to its role in related areas of mathematics. For example, factorizations of a given numerical
monoid element coincide with non-negative integer solutions to a linear equation, which is central to
discrete optimization and integer programming [CGLR06, DeL05, DDK13]. Additionally, numerical
monoids arise naturally in the Frobenius coin-exchange problem, which asks for the largest nonnegative integer value that cannot be evenly changed using a collection of relatively prime coin
values n1 , . . . , nk [BR07, Chapter 1]. In this context, each coin value represents an irreducible
element of the numerical monoid M = hn1 , . . . , nk i ⊂ Z≥0 , and factorizations of a monoid element
m ∈ M correspond to distinct ways of making change for m.
Recent work examining the factorization structure of numerical monoids has uncovered explicit
characterizations of the eventual behavior of numerous factorization invariants [OPe15]. For example, the catenary degree invariant c : M → Z≥0 (defined in Section 5) for any numerical monoid
M = hn1 , . . . , nk i satisfies c(m + n1 · · · nk ) = c(m) for m 0 (that is, the function c is eventually
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Figure 1. A plot showing the catenary degree (left) and ω-primality (right) invariants for the numerical monoid h7, 13, 17i ⊂ Z≥0 , produced using SAGE [SAGE].
periodic), and the ω-primality invariant ω : M → Z>0 (defined in Section 4) satisfies
ω(m) =
1
n1 m
+ a0 (m)
for some n1 -periodic function a0 and m 0 [CCMMP14, OPe13]. The above function is an example
of a quasipolynomial, that is, a polynomial with periodic coefficients. Figure 1 depicts the eventual
behavior of both the catenary degree and ω-primality functions for a particular numerical monoid.
Many of the aforementioned eventual behavior results rely on the fact that numerical monoids
(i) possess only finitely many irreducible elements and (ii) can be totally ordered. Recently, I have
successfully generalized many of these results from numerical monoids to affine monoids (finitely
generated submonoids of Zk≥0 ), using techniques from combinatorial commutative algebra [One15].
As affine monoids need not possess a total ordering, these extensions are stated using an appropriate
multivariate quasipolynomial analog.
Definition 1.1. A function f : Zk≥0 → R is eventually quasipolynomial if (i) f restricts to a
polynomial on each of finitely many cones (that is, sets of the form
C(β; α1 , . . . , αr ) = β + α1 Z≥0 + · · · + αr Z≥0
for β and linearly independent α1 , . . . , αr ∈ Zk≥0 ), and (ii) the union of these cones equals Zk≥0 .
Hilbert’s theorem, a cornerstone of commutative algebra, states that the Hilbert function of
any finitely generated, positively multigraded module is eventually quasipolynomial [Sta96, Fie00].
Hilbert’s theorem is often applied to problems in enumerative combinatorics by constructing a
graded module whose Hilbert function coincides with a counting function of interest [Ehr62, Mac71,
DLMO09]. For example, constructions of this type yield an alternative proof that the number of
proper k-colorings of a finite undirected graph G is a polynomial in k whose degree is the number
of vertices of G [Sta12].
My recent work characterizing the eventual behavior of factorization invariants for affine monoids
applies Hilbert’s theorem in this manner [One15]. More specifically, given M ⊂ Zk≥0 affine and one
of several factorization invariants defined on the level of monoid elements, a family of multigraded
modules is constructed whose Hilbert functions determine the invariant value of every element of M .
Applying Hilbert’s theorem to the modules in this family yields a description of the factorization
invariant in terms of multivariate eventually quasipolynomial functions. When M is a numerical
monoid (that is, when k = 1), this specializes to a known result for each invariant.
As an example, the aforementioned ω-primality invariant ω : M → Z>0 over any affine monoid
M is shown to coincide with an eventual quasipolynomial in which the degree of each polynomial
restriction is at most 1. If M = hn1 , . . . , nk i is a numerical monoid, then more can be said. In
particular, the polynomial restrictions occur on n1 1-dimensional cones, each generated by n1 ,
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CHRISTOPHER O’NEILL
whose union has finite complement in M . The restriction of the ω-function to each 1-dimensional
cone is linear, and the leading term of each linear restriction is n11 . The remaining cones are all
0-dimensional, one for each element of M outside of the 1-dimensional cones.
In addition to generalizing several core numerical monoid results to affine monoids, my work
has produced a definitive link between combinatorial commutative algebra and the factorization
invariants of interest in this setting. The result is a new framework through which to examine the
factorization structure of affine monoids, which should help to ease the transition from the recent
focus on numerical monoids to this more general setting. Projects proposed below are motivated
both by this newfound connection, and by fundamental questions arising in factorization theory.
2. Factorization length and elasticity
The length of a factorization f of a monoid element m ∈ M is the number of irreducibles in f ;
L(m) denotes the set of factorization lengths of m (its length set). Many factorization invariants
are derived from factorization lengths, the simplest and most natural of which are maximum and
minimum factorization length, which assign to each monoid element the values M(m) = max L(m)
and m(m) = min L(m), respectively.
My work discussed in Section 1 characterizes the eventual behavior of maximum and minimum
factorization length for affine monoids [One15], generalizing an earlier result for numerical monoids
from joint work with Barron and Pelayo [BOP14a] and specializing foundational asymptotic results [GH92, CGGR02]. As a consequence of Theorem 2.1 below, the maximum and minimum
factorization length functions on any affine monoid M are uniquely determined by their values on
a bounded region, and an explicit region is given when M is a numerical monoid.
Theorem 2.1 ([One15]). The max and min factorization length functions M, m : M → Z>0 are
eventually quasilinear for any affine monoid M ⊂ Zk≥0 . Moreover, if M = hn1 , . . . , nk i ⊂ Z≥0 is a
numerical monoid, then
M(m) =
1
n1 m
+ a0 (m)
and
m(m) =
1
nk m
+ b0 (m)
for m ≥ nk−1 nk , where a0 and b0 are n1 -periodic and nk -periodic functions, respectively.
Maximum and minimum factorization length arise in discrete optimization problems, where
factorizations in a given affine monoid coincide with integer solutions to a given set of linear
equations [CGLR06, DeL05, DDK13]. In this setting, a monoid element’s factorizations that achieve
the maximum (resp. minimum) factorization length are precisely those solutions with maximal
(resp. minimal) `1 -norm. Theorem 2.1 implies that the extremal `1 -norms of the factorizations
of sufficiently large affine monoid elements m ∈ M are parametrized by the extremal `1 -norms
of their divisors. Moreover, the proof of Theorem 2.1 for numerical monoids characterizes which
factorizations achieve such values, yielding an explicit parametrization ideal for direct application
to discrete optimization problems.
Project 1. Characterize the cones arising in the quasipolynomials in Theorem 2.1.
The connection to Hilbert functions discussed in Section 1 brings additional tools from combinatorial commutative algebra to the table for investigating Project 1. For instance, recent developments involving the so-called multigraded Castelnuovo-Mumford regularity [MS04, MS05] bound
the translation factors of cones arising in multigraded Hilbert function restrictions.
3. Delta sets
One of the first invariants studied in factorization theory [CS90, CS92], the delta set of a monoid
element m ∈ M is defined as the set of successive differences of elements of its length set L(m) and
provides a coarse measure of the “gaps” between the factorization lengths of m. More precisely,
if we write the length set L(m) = {l1 < · · · < lr } of m in increasing order and without repetition,
we define the delta set of m to be the set ∆M (m) = {li − li−1 : 2 ≤ i ≤ r}. This invariant detects
several different kinds of non-unique factorization behavior. For instance, if ∆(m) = ∅, then every
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Figure 2. A SAGE [SAGE] plot showing the delta sets of elements in the numerical
monoid h17, 33, 53, 71i. In this plot, a point at (n, d) indicates that d ∈ ∆(n).
factorization of m has the same length, and if ∆(m) = {1}, then the factorization lengths of m are
consecutive. Generally speaking, the delta set ∆(m) measures how sparse the length set of m is,
and its larger elements indicate “missing factorizations” in M .
S
Delta sets in a given monoid M are often studied using the union ∆(M ) = m∈M ∆M (m) of the
delta sets of its elements, a set which globally measures how sparse the factorization lengths of M
are. For instance, ∆(M ) = ∅ if and only if M is half-factorial, meaning any two factorizations of
the same element have identical length.
It is known that gcd(∆(M )) = min(∆(M )) for any cancellative, commutative monoid M with
nonempty delta set [GH06]. Additionally, under certain conditions (such as if M is finitely generated), the delta set ∆(M ) is known to be finite. However, no further restrictions are known
regarding which finite sets occur. In particular, it is not known if there is a finite set D satisfying
gcd(D) = min(D) which does not occur as the delta set of some monoid. This motivates one of the
most persistent open questions concerning delta sets, namely the delta set realization problem:
Problem 3.1. Given a finite set D ⊂ Z≥0 satisfying gcd(D) = min(D), does there exist a monoid
M satisfying ∆(M ) = D?
Further restictions are known for certain classes of monoids, but no such restrictions are known
for numerical monoids. In fact, recent results of Colton and Kaplan provide strong evidence that
the conditions stated in Problem 3.1 may in fact be the only conditions governing numerical monoid
delta sets [CK15]. In particular, they construct an infinite family of numerical monoids whose delta
sets have arbitrarily large gaps, taking the form {d, td} for any given d and t.
The difficulty of Problem 3.1 lies in proving that a particular value does not appear in the delta
set of a given monoid M , as doing so requires a high level of control over the relations between
factorizations throughout M . For example, the numerical monoid M = h17, 33, 53, 71i ⊂ Z≥0 has
delta set ∆(M ) = {2, 4, 6}, but the value 6 only occurs in the delta sets of the monoid elements
266, 283 and 300; see Figure 2. Such examples demonstrate the subtlety in producing monoids like
those constructed by Colton and Kaplan in [CK15].
Numerical monoids are a natural setting in which to investigate Problem 3.1. Much of the recent
work involving factorization invariants in numerical monoids benefits from established computer
software packages [DGM15, Gar15, SAGE], and the delta set invariant is no exception. In particular,
two recently developed algorithms for numerical monoid delta set computation include counterexamples to several conjectures, each claiming that some particular finite set does not occur as the
delta set of any numerical monoid [CK15]. The first algorithm (joint work with Garcı́a-Sánchez)
utilizes techniques from combinatorial commutative algebra to compute the delta set algorithm
of any affine monoid. The key ingredient is Theorem 3.2 below, which generalizes the eventual
periodicity of numerical monoid delta sets [CHK09].
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Theorem 3.2 ([One15]). Given an affine monoid M ⊂ Zk≥0 and fixing d ∈ ∆(M ), the function
M → {0, 1}, defined by sending m 7→ 1 whenever d ∈ ∆(m), is eventually quasiconstant. As a
consequence, if M = hn1 , . . . , nk i ⊂ Z≥0 is a numerical monoid, then for all m 0 we have
∆(m + n1 nk ) = ∆(m).
As with the proof of Theorem 2.1, the quasiconstant function appearing in Theorem 3.2 is
constructed using Hilbert’s theorem for multigraded modules. More specifically, given an affine
monoid M , a sequence of multigraded polynomial ideals I0 ⊂ I1 ⊂ I2 ⊂ · · · is constructed with
the property that the quotient Id /Id−1 has a nonzero element of graded degree m ∈ M if and only
if d ∈ ∆(m). Since each quotient Id /Id−1 is graded, this implies that Id−1 ( Id if and only if
j ∈ ∆(M ), a membership criterion for ∆(M ) that avoids searching individual monoid elements.
The minimal monoid elements m ∈ M for which d ∈ ∆(m) can be recovered directly from the
elements of the reduced Gröbner basis of each ideal Id . A deeper understanding of these minimal
monoid elements is critical for progress in Problem 3.1; indeed, a recent preprint from Garcı́a
Sánchez et.al. [GLM15] provides a solution to Project 2 below for numerical monoids with three
minimal generators, resulting in a complete answer to Problem 3.1 in this special case. Additionally,
an answer to Project 2 would improve the efficiency of Algorithm 3.3 by further restricting the set
of monoid elements whose delta sets must be computed.
Project 2. For affine M ⊂ Zk≥0 , characterize the minimal elements of M whose delta sets contain
each element of ∆(M ).
A second algorithm for computing numerical monoid delta sets, joint with Barron and Pelayo
[BOP14b], utilizes dynamic programming to compute length sets in finitely generated monoids
without the need to first compute factorizations. This allows the delta sets of large monoid elements
to be quickly computed, since the number of distinct factorization lengths in finitely generated
monoids has linear growth rate [FH06, GH92]. Algorithm 3.3 below, together with the eventual
periodicity of the delta set for numerical monoids, produces an algorithm to compute the delta set
of any numerical monoid.
Algorithm 3.3 ([BOP14b, DGM15]). The length set of any affine monoid element m ∈ M can be
dynamically computed without computing any monoid element factorizations.
In contrast to numerical monoids, the monoid B(G) of zero-sum sequences over any finite Abelian
group G (called a block monoid ) is known to have delta set ∆(B(G)) = {1, . . . , max ∆(B(G))}.
Although max ∆(B(G)) has been found for some Abelian groups G, it remains an open problem in
general [GZ15]. The use of computer software in computing max ∆(B(G)) has been infeasible until
recently, due to the large (exponential in |G|) number of generators that block monoids possess
[Pon04]. Indeed, the set of factorizations grows too quickly to be effectively computed for large
elements, and computing the ideals used in Theorem 3.2 is infeasable since the underlying ring has
one variable per monoid generator. Algorithm 3.3 avoids this issue, since the size of the length
set grows independently of the number of generators. Additionally, for block monoids, it is known
that max ∆(B(G)) occurs in the delta set of some element whose length set contains 2, yielding a
bounded set of elements whose delta sets must be computed in order to obtain ∆(B(G)) [GGS11].
Project 3. Optimize Algorithm 3.3 to compute max ∆(B(G)) for finite Abelian groups G.
4. Measuring primality
Developed by Geroldinger [Ger97], the ω-primality invariant assigns to each non-unit element
r in an integral domain R a value ω(r) ∈ Z>0 ∪ {∞}, defined as follows. Recall that r is prime
if whenever r divides a product u1 · · · un of n elements, then r must divide one of the ui . In this
case, we set ω(r) = 1. If ω(r) = 3, then whenever r divides a product u1 · · · un of n ≥ 3 elements,
there exist a collection of 3 elements ui , uj , and uk appearing in the product such that r divides
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ui uj uk . In this way, ω(r) = 1 if and only if r is prime in R, and the value ω(r) measures how far
the element r is from being prime.
Every prime element in an integral domain (or more generally, a cancellative commutative
monoid) is irreducible, but the converse need not hold. In fact, the existence of non-prime irreducible elements in a monoid M coincides with the existence of non-unique factorizations. As such,
the behavior of the ω-function over the whole monoid M measures how far from factorial M is. Much
of the initial work studying ω-primality focused on ω-values of irreducible elements [Ger97, GH08],
and more recently, examining ω-values of arbitrary non-unit elements has produced algorithms and
closed formulas in several settings [AC10, CPS14, OPe14].
Theorem 4.1 below describes the eventual behavior of ω-primality for affine monoids [One15],
generalizing the existing result for numerical monoids [OPe13, GMV14a]. The proof uses Hilbert’s
theorem in a similar manner to that used in Theorem 2.1, except that the final step in recovering
the ω-function requires taking the element-wise maximum of finitely many (eventually quasilinear)
Hilbert functions. Although this still implies the ω-function is eventually quasilinear, the resulting
characterization leaves the most room for refinement among those appearing in [One15].
Theorem 4.1 ([One15]). The ω-primality function ω : M → Z>0 is eventually quasilinear for any
affine monoid M ⊂ Zk≥0 . Moreover, if M = hn1 , . . . , nk i ⊂ Z≥0 is numerical, then
ω(m) =
1
n1 m
+ a0 (m)
for some n1 -periodic function a0 and m 0.
The ω-values of irreducible elements in block monoids admit a particularly concise characterization [Ger97, GK10]. However, little is known about the ω-values of the remaining block monoid
elements. As noted earlier, block monoids are affine and satisfy a certain saturation condition,
making them a natural setting in which to refine Theorem 4.1.
Project 4. Characterize the ω-values of elements of B(G) for finitely Abelian groups G.
Much of the recent progress on ω-primality for numerical monoids has benefited from the development of algorithms [ACK11, Bla11, GMV14a]. Joint work with Barron and Pelayo [BOP14b]
has yielded an algorithm for computing ω-values in numerical monoids using similar dynamic programming techniques to those used in Algorithm 3.3 for delta sets. Algorithm 4.2 below works
by inductively computing the ω-value of every divisor of a numerical monoid element m prior to
computing ω(m). More specifically, Algorithm 4.2 first performs finitely many preliminary computations (the “base step”), and then computes each value ω(m) using the output from both the
preliminary computations and the computation for ω(m0 ) for each m0 ≤ m (the “inductive step”).
The advantages of Algorithm 4.2 are twofold. First, while existing algorithms quickly become
too slow on large monoid elements, the inductive nature of Algorithm 4.2 reduces the size of the
computation that must be performed for each element, thereby offering significant performance
improvements. Second, Algorithm 4.2 computes a large collection of ω-values at once instead of
treating teach ω-value as a separate computation. This is ideal for examining eventual behavior,
since many sequential values are often required to discern precise periodic behavior.
Algorithm 4.2 ([BOP14b]). Given a numerical monoid M , the ω-value of each m ∈ M can be
dynamically computed in a manner whch produces the ω-value of every monoid element m0 ≤ m.
Generalizing Algorithm 4.2 to affine monoids would allow for the effective use of software in
refining Theorem 4.1. Although the inductive step of Algorithm 4.2 immediately generalizes to
affine monoids, the base step may require infinitely many computations. In practice, only finitely
many preliminary computations are actually required, but it is difficult to pin down a finite set
which is sufficient to begin the inductive process. Preliminary computations indicate that the
saturation condition characterizing block monoids may yield a finite set suitable for initiating the
dynamic algorithm. In addition to providing the first step generalizing Algorithm 4.2 to affine
monoids, an answer to Project 5 below would allow the use of computer data on Project 4.
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Figure 3. A SAGE plot [SAGE] depicting the catenary degrees of elements of the
numerical monoid h17, 41, 43, 59, 61i.
Project 5. Develop a dynamic algorithm to compute ω-values for block monoids.
Joint work on Project 5 is already in its early stages with the other authors of [BOP14b].
5. The catenary degree
The catenary degree invariant, described briefly in Section 1, assigns to each element m ∈ M
a nonnegative integer c(m) derived from combinatorial properties of the set of factorizations of
m. Generally speaking, the value c(m) measures the distance between the factorizations of m,
analogous to how delta sets measure distance between factorization lengths.
The catenary degree of a monoid element m is formally defined in two steps. First, the distance
d(f, f 0 ) between any two factorizations f and f 0 of m is defined in such a way that two factorizations
are closer together if they have more irreducibles in common. Second, the catenary degree of m
is defined as the minimum non-negative integer c(m) = r so that in the complete graph whose
vertices are the factorizations of m and whose edges are labeled by the distance function, any two
vertices are connected by a path whose edges are labeled at most c(m). Informally, an element m
has a larger catenary degree if its factorizations are “further apart”.
Like the delta set invariant, the catenary degree function has been shown to be eventually periodic
in numerical monoids [CCMMP14], and my recent work has generalized this result to Theorem 5.1
below for affine monoids [One15]. Figure 3 plots the catenary degree function c : M → Z≥0 for the
numerical monoid M = h17, 41, 43, 59, 61i, which satisfies c(m + 17) = c(m) for all m > 250.
Theorem 5.1 ([One15]). The catenary degree function c : M → Z≥0 is eventually quasiconstant
for any affine monoid M ⊂ Zk≥0 . As a consequence, if M = hn1 , . . . , nk i ⊂ Z≥0 is a numerical
monoid, then for all m 0 we have
c(m + n1 · · · nk ) = c(m).
The proof of Theorem 5.1 for numerical monoids appearing in [CCMMP14] is purely existential
and relies on the fact that a nonincreasing sequence of positive integers is eventually constant. As
such, no concrete bound on the start of catenary degree periodicity is known. Additionally, the
eventual period is only known to divide n1 · · · nk , but in all computed examples the actual period
is much smaller, usually dividing n1 nk . This lack of detail is in stark contrast to the analogous
results for the delta set and ω-primality invariants.
A dynamic algorithm for computing the catenary degree in numerical monoids, similar to Algorithms 3.3 and 4.2 for computing delta sets and ω-primality, would benefit furthering investigations
into this periodic behavior. The iterative nature of the proof of Theorem 5.1 for numerical monoids
given in [CCMMP14] provides strong evidence that such an algorithm exists.
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Project 6. Develop a dynamic algorithm in the spirit of Algorithms 3.3 and 4.2 to compute catenary degrees of numerical monoid elements.
The catenary degree c(M ) of a monoid M is defined as the supremum of the catenary degrees
achieved by elements of M [GH06]. For finitely generated monoids, this supremum is known to be
finite, and always occurs on a finite (computable) set of monoid elements (called Betti elements)
[CGLPR06, BGG11]. However, this value need not accurately reflect the factorization structure
of M . For some numerical monoids, the maximum catenary degree is the only nonzero catenary
degree achieved, yet for others, every non-negative integer between 2 and the maximum catenary
degree occurs as the catenary degree of some monoid element.
The catenary degree c(m) of a monoid element m ∈ M is closely related to the delta set ∆(m), although neither completely determines the other. On the other hand, the monoid-wide factorization
invariants ∆(M ) and c(M ), both of which are standard in the literature, describe the factorization
structure of M in drastically different levels of detail. Indeed, the former maintains a comprehensive
set of values, while the latter recalls only a single value. The set
C(M ) = {c(m) : m ∈ M }
of catenary degrees of M is a closer analog of the delta set ∆(M ), as it provides a comparably
detailed description of the factorization structure of M . Recently, joint work with Ponomarenko,
Tate, and Webb [OPTW15] demonstrates the following:
(i) the minimum nonzero value of C(M ) also occurs at a Betti element, and
(ii) some elements of C(M ) need not occur at Betti elements.
Currently, there is no known algorithm to compute the entire set of catenary degrees of a numerical monoid [Gar15]. However, the multigraded modules constructed in the proof of Theorem 5.1
are very similar to those used to prove Theorem 3.2 for delta sets. In particular, given an affine
monoid M , an ascending chain of multigraded modules N0 ⊂ N1 ⊂ N2 ⊂ · · · is constructed so
that the Hilbert functions of each pair of sequential modules (Nj−1 , Nj ) determine which monoid
elements have catenary degree j. As such, there should be an algorithm that computes any affine
monoid’s set of catenary degrees, analogous to the delta set algorithm resulting from Theorem 3.2.
Project 7. Develop an algorithm to compute the set of catenary degrees of any affine monoid by
computing the multigraded modules arising in the proof of Theorem 5.1.
10
CHRISTOPHER O’NEILL
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