Table Of ContentMINIMAL PRESENTATIONS OF SHIFTED NUMERICAL MONOIDS
REBECCA CONAWAY, FELIX GOTTI, JESSE HORTON, CHRISTOPHER O’NEILL,
ROBERTO PELAYO, MESA WILLIAMS, AND BRIAN WISSMAN
7
1 Abstract. A numerical monoid is an additive submonoid of the non-negative in-
0 tegers. Given a numerical monoid S, consider the family of “shifted” monoids M
n
2 obtained by adding n to each generator of S. In this paper, we examine minimal
n relations among the generators of Mn when n is sufficiently large, culminating in a
a description that is periodic in the shift parameter n. We explore several applications
J
to computation, combinatorial commutative algebra, and factorization theory.
0
3
]
C 1. Introduction
A
A minimal presentation of a numerical monoid M (that is, an additive submonoid
.
h
of the natural numbers) encapsulates the minimal relations among generators of M.
t
a
Such minimal presentations arise in the study of toric ideals, where they correspond
m
to minimal generating sets for kernels of monomial maps [7], and algebraic statistics,
[
where they correspond to Markov bases [9]. Additionally, many arithmetic invariants
1
of interest in combinatorial commutative algebra and factorization theory can be eas-
v
5 ily recovered (both theoretically and computationally) from a minimal presentation,
5 making them a particularly useful tool in computational algebra [10, 14].
5
8 In this paper, we examine families of numerical monoids obtained by “shifting” a
0 chosen generating set. In particular, given positive integers r < ··· < r , consider
1 k
.
1 numerical monoids of the form
0
7 M = (cid:104)n,n+r ,...,n+r (cid:105),
n 1 k
1
: indexed by a shift parameter n. Our main result is Theorem 4.9, which describes how
v
Xi minimal presentations of Mn vary with large n. More specifically, we give an explicit
bijection between the minimal presentations of M and those of M when n > r2.
r n n+rk k
a Following Theorem 4.9, we characterize the behavior of several arithmetic invari-
ants determined by minimal presentations (e.g. Betti numbers and catenary degree),
resulting in periodic or periodic-linear descriptions in each case. Some of these char-
acterizations are new, while others strengthen existing results in the literature [6, 15].
Our approach unifies these results (old and new) as consequences of a deeper structural
phenomenon that occurs among the minimal relations of M for large n, and improves
n
each lower bound on n that was previously given; see Remark 5.4 for a thorough
discussion of the benefits of our approach and resulting improvements.
Date: January 31, 2017.
1
2 CONWAY, GOTTI, HORTON, O’NEILL, PELAYO, WILLIAMS, AND WISSMAN
One of the primary consequences of Theorem 4.9 lies in the realm of computation.
While minimal presentations (and many of the arithmetic invariants they determine)
are generally more difficult to compute for monoids with large generators, our results
give a way to more efficiently perform these computations in some cases by instead
computing a minimal presentation for a numerical monoid with smaller generators in
thesameshiftedfamily. WediscussthespecificsinRemark5.1,includingaforthcoming
implementation in the popular GAP package numericalsgps [8].
2. Background
Inthissection,weprovidethenecessarydefinitionsrelatedtothefactorizationtheory
of numerical monoids. In what follows, let N denote the set of non-negative integers.
Definition 2.1. Anumerical monoid M isanadditivesubmonoidofN. Whenwewrite
M = (cid:104)m ,...,m (cid:105), we assume m < ··· < m , and the chosen generators m ,...,m
1 t 1 t 1 t
are called irreducible elements or atoms. We say M is primitive if gcd(m ,...,m ) = 1.
1 t
Definition 2.2. Fix a numerical monoid M = (cid:104)m ,...,m (cid:105) and a ∈ M. A factoriza-
1 t
tion of a is an expression
a = z m +···+z m
1 1 t t
of a as a sum of irreducible elements of M, which we often represent with the tuple
z = (z ,...,z ) ∈ Nt. The length of a factorization z of a is the total number
1 t
|z| = z +···+z
1 t
of irreducible elements appearing in z, and the support of z is the set
supp(z) = {m : z > 0}
i i
of distinct irreducible elements appearing in z.
Definition 2.3. Fix a numerical monoid M = (cid:104)m ,...,m (cid:105) and a ∈ M. The factor-
1 t
ization homomorphism of M is the map π : Nt → M given by
π(z ,...,z ) = z m +···+z m .
1 t 1 1 t t
The set of factorizations of a is the set
Z (a) = π−1(a) = {z ∈ Nt : π(z) = a} ⊂ Nt.
M
When there can be no confusion, we often omit the subscript and simply write Z(a).
We conclude this section with Theorem 2.4, which appeared as [1, Theorem 4.3]
for minimally generated, primitive numerical monoids. The statement below follows
immediately from the proof of the original statement given in [1].
MINIMAL PRESENTATIONS OF SHIFTED NUMERICAL MONOIDS 3
Theorem 2.4. Fix m < ··· < m , and suppose M = (cid:104)m ,...,m (cid:105) is not necessarily
1 t 1 t
primitive or minimally generated by m ,...,m . The function m : M → N sending
1 t
each a ∈ M to its smallest factorization length satisfies
m(a+m ) = m(a)+1
t
for all a > m m .
t−1 t
Notation. Through the remainder of this paper, fix r < ··· < r and n ∈ N, and let
1 k
S = (cid:104)r ,...,r (cid:105) and M = (cid:104)n,n+r ,...,n+r (cid:105)
1 k n 1 k
denote additive submonoids of N. Unless otherwise stated, we assume M is primitive
n
and minimally generated as written, but we do not make either assumption for S.
Note that choosing n as the first generator of M ensures that every numerical monoid
n
falls into exactly one shifted family.
3. Sufficiently shifted numerical monoids
In this section, we give the Theorem 3.4 and Corollary 3.5, which identify the core
obstructiontoTheorem4.9forsmalln(inthesenseofRemark4.10). Thisresultcomes
in the form of a description of the factorizations of Betti elements (Definition 3.1),
whose factorizations encapsulate the minimal relations among atoms.
Definition 3.1. Fix a numerical monoid M and a ∈ M. The factorization graph of a,
denoted ∇ , has vertex set Z(a), and two vertices z,z(cid:48) ∈ Z(a) are connected by an edge
a
whenever they have at least one irreducible in common. We say a is a Betti element
of M if its factorization graph ∇ is disconnected, and write Betti(M) for the set of
a
Betti elements of M.
Example 3.2. The Betti elements of M = (cid:104)6,9,20(cid:105) are 18 and 60, since
Z(18) = {(3,0,0),(0,2,0)} and
Z(60) = {(10,0,0),(7,2,0),(4,4,0),(1,6,0),(0,0,3)}
both yield disconnected factorization graphs. Here, the factorizations of 18 represent
the minimal relation between 6 and 9, namely that in any factorization of an element
a ∈ M, onecanreplacethreecopiesof6withtwocopiesof9toyieldanewfactorization
of a. Similarly, 60 is the first element that can be factored using all three irreducibles,
and thus gives the minimal ways to exchange copies of 20 for copies of 6 and 9.
In contrast, the element 126 ∈ M is not a Betti element of M, even though
(1,0,6),(0,14,0) ∈ Z(126) have no irreducibles in common. Indeed, this relation can
be obtained by twice exchanging three 20’s for ten 6’s, yielding (21,0,0), and then
repeatedly exchanging three 6’s for two 9’s until (0,14,0) is obtained. This is repre-
sented by a path through the factorization graph ∇ connecting (1,0,6) to (0,14,0)
126
that passes through 9 vertices, including (21,0,0).
4 CONWAY, GOTTI, HORTON, O’NEILL, PELAYO, WILLIAMS, AND WISSMAN
Before stating and proving Theorem 3.4 and Corollary 3.5, we prove Lemma 3.3,
whichidentifiesthelocationsintheproofofTheorem3.4thatrequirentobesufficiently
large. Note that Theorem 3.4 is the source of the bound given in nearly every “eventual
behavior” result in this paper; see Remark 4.10 for more detail.
Lemma 3.3. Fix a ∈ S and s = (s ,...,s ) ∈ Z(a).
1 k
(a) If |s| ≥ r and s = 0, then there is a shorter factorization s(cid:48) ∈ Z(a) with s(cid:48) > 0.
k k k
(b) If a > r r and s has minimum factorization length, then s > 0.
k−1 k k
(c) If a > r2, then |s| ≥ r .
k k
Proof. [1, Lemma 4.1] and Theorem 2.4. (cid:3)
Theorem 3.4. Suppose n > r2, and let z and z(cid:48) be factorizations of a Betti element
k
β ∈ M in different connected components of ∇ . If |z| > |z(cid:48)|, then z > 0 and z(cid:48) > 0.
n β 0 k
Proof. We begin by observing that
k k
(cid:88) (cid:88)
β −|z|n = z n+ z (n+r )−|z|n = z r ,
0 i i i i
i=1 i=1
which yields an explicit bijection between the factorizations of β ∈ M of length (cid:96) and
n
the factorizations of β−(cid:96)n ∈ S of length at most (cid:96). Let s = (z ,...,z ) ∈ Z (β−|z|n)
1 k S
and s(cid:48) = (z(cid:48),...,z(cid:48)) ∈ Z (β −|z(cid:48)|n) denote the factorizations in S corresponding to z
1 k S
and z(cid:48), respectively. Notice that since |z| > |z(cid:48)|, we have
β −|z(cid:48)|n ≥ n+β −|z|n ≥ n > r2,
k
so |s(cid:48)| ≥ r by Lemma 3.3(c).
k
Next, we claim some factorization in the same connected component of ∇ as z(cid:48) has
β
positive last component. Certainly if z(cid:48) > 0 the claim is proved. Otherwise, since
k
β−|z(cid:48)|n ≥ r r , applying Lemma 3.3(b) produces a factorization s(cid:48)(cid:48) ∈ Z (β−|z(cid:48)|n)
k−1 k S
with s(cid:48)(cid:48) > 0 obtained from s(cid:48) by replacing all but one atom with a minimum length
k
factorization. The corresponding factorization z(cid:48)(cid:48) = (|z(cid:48)|−|s(cid:48)(cid:48)|,s(cid:48)(cid:48),...,s(cid:48)(cid:48)) ∈ Z(β) of β
1 k
under the above bijection is connected to z(cid:48) in ∇ and has z(cid:48)(cid:48) > 0 and z(cid:48)(cid:48) > 0.
β 0 k
Now, since z and z(cid:48) lie in different connected components of ∇ , the above claim
β
implies z = 0. This means |s| ≤ r , since otherwise Lemma 3.3(a) would produce a
k k
factorization connected to z in ∇ with positive last coordinate. As such,
β
|z| > |z(cid:48)| ≥ |s(cid:48)| ≥ r ≥ |s|,
k
whichyieldsz = |z|−|s| > 0. Lastly,weconcludez(cid:48) > 0,asotherwisethefactorization
0 k
z(cid:48)(cid:48) constructed above would be connected to both z and z(cid:48) in ∇ since it has positive
β
first coordinate. (cid:3)
Corollary 3.5. Suppose n > r2 and that M is primitive, and let d = gcd(r ,...,r ).
k n 1 k
Any two factorizations z,z(cid:48) ∈ Z(β) of a Betti element β ∈ Betti(M ) lying in different
n
(cid:12) (cid:12)
connected components of ∇ satisfy (cid:12)|z|−|z(cid:48)|(cid:12) ∈ {0,d}.
β
MINIMAL PRESENTATIONS OF SHIFTED NUMERICAL MONOIDS 5
Proof. By [2, Proposition 2.9], |z|−|z(cid:48)| ∈ dZ, so suppose by way of contradiction that
|z| − |z(cid:48)| ≥ 2d. Since β − |z(cid:48)|n − n ∈ S, there exists a factorization z(cid:48)(cid:48) ∈ Z(β) with
|z(cid:48)(cid:48)| = |z(cid:48)| + d. By Theorem 3.4, both z(cid:48)(cid:48) and z(cid:48)(cid:48) must be positive, meaning z and z(cid:48)
0 k
are both connected to z(cid:48)(cid:48) in ∇ . (cid:3)
β
4. Minimal presentations
Let π : Nk+1 → M denote the factorization homomorphism of M , that is,
n n n
k
(cid:88)
π (z) = z n+ z (n+r ),
n 0 i i
i=1
andletkerπ denotetheequivalencerelationonNk+1 givenby(z,z(cid:48)) ∈ kerπ whenever
n n
π (z) = π (z(cid:48)), (that is, when z and z(cid:48) are factorizations for the same element in M ).
n n n
The equivalence relation kerπ is a congruence since it is also closed under translation,
n
that is, (z +u,z(cid:48) +u) ∈ kerπ whenever (z,z(cid:48)) ∈ kerπ and u ∈ Nk+1.
n n
Definition 4.1. FixanumericalmonoidM = (cid:104)m ,...,m (cid:105)andletπ : Nt → M denote
1 t
the factorization homomorphism of M. A presentation for M is a set of relations
ρ ⊂ kerπ such that kerπ is the unique minimal (w.r.t. containment) congruence on Nt
containing ρ. Equivalently, this is true if between any two factorizations (z,z(cid:48)) ∈ kerπ,
there exists a chain a ,a ,...,a with a = z, a = z(cid:48), and
0 1 r 0 r
(a ,a ) = (b ,b(cid:48))+(u ,u ) ∈ kerπ
i−1 i i i i i
for some (b ,b(cid:48)) ∈ ρ and u ∈ Nt for each i ≤ r. We say ρ is minimal if it is minimal
i i i
with respect to containment among all presentations of M.
Minimal presentations are one of the fundamental tools with which to study the
factorization structure of finitely generated monoids. Each minimal presentation of a
monoid M can be viewed as a particular choice of minimal relations that are suffi-
cient for relating any two factorizations of the elements of M. For a more thorough
introduction, we refer the reader to [13, Chapter 9] and [14, Chapter 7].
Example 4.2. The minimal presentations of M = (cid:104)6,9,20(cid:105) from Example 3.2 are
{((3,0,0),(0,2,0)),((10,0,0),(0,0,3))}, {((3,0,0),(0,2,0)),((7,2,0),(0,0,3))},
{((3,0,0),(0,2,0)),(( 4,4,0),(0,0,3))}, {((3,0,0),(0,2,0)),((1,6,0),(0,0,3))},
each of which has exactly one relation for each Betti element. As per the discussion in
Example 3.2, each minimal presentation provides enough relations among the minimal
generators of M to relate any two factorizations of elements of M.
6 CONWAY, GOTTI, HORTON, O’NEILL, PELAYO, WILLIAMS, AND WISSMAN
Example 4.3. Let S = (cid:104)6,9,20(cid:105), and consider the following minimal presentations.
M : (( 0, 0, 8, 0),(3, 2, 0, 3)), (( 0, 1, 6, 0),(4, 0, 0, 3)), (( 0, 3, 0, 0),(1, 0, 2, 0)),
450
((20, 5, 0, 0),(0, 0, 0,24)), ((25, 1, 0, 0),(0, 0, 4,21)), ((26, 0, 0, 0),(0, 2, 2,21))
M : (( 0, 0, 8, 0),(3, 2, 0, 3)), (( 0, 1, 6, 0),(4, 0, 0, 3)), (( 0, 3, 0, 0),(1, 0, 2, 0)),
470
((21, 5, 0, 0),(0, 0, 0,25)), ((26, 1, 0, 0),(0, 0, 4,22)), ((27, 0, 0, 0),(0, 2, 2,22))
M : (( 0, 0, 8, 0),(3, 2, 0, 3)), (( 0, 1, 6, 0),(4, 0, 0, 3)), (( 0, 3, 0, 0),(1, 0, 2, 0)),
490
((22, 5, 0, 0),(0, 0, 0,26)), ((27, 1, 0, 0),(0, 0, 4,23)), ((28, 0, 0, 0),(0, 2, 2,23))
Each first-row relation (z,z(cid:48)) satisfies |z| = |z(cid:48)|, and each second-row relation (z,z(cid:48))
satisfies |z| = |z(cid:48)| + 1, z > 0 and z(cid:48) > 0. Theorem 3.4 ensures that every relation
0 3
above satisfies one of these two criteria.
Theorem 4.9 characterizes the relationship between successive minimal presenta-
tions. In particular, the same equal-length relations appear in all three given minimal
presentations, and each remaining relation for M is obtained from a relation for
n+20
M by adding (e ,e ).
n 0 k
Proposition 4.4 defines the map Φ used to construct the bijection between minimal
n
presentations in Theorem 4.9, and Proposition 4.6 gives several key properties of Φ .
n
In particular, it is shown that Φ preserves symmetric and translation closure, and
n
preserves monotone chain connectivity (Definition 4.5).
Proposition 4.4. The map Φ : kerπ → kerπ given by
n n n+r
k
(z +(cid:96)e ,z(cid:48) +(cid:96)e ) if |z| > |z(cid:48)|
0 k
Φ (z,z(cid:48)) = (z +(cid:96)e ,z(cid:48) +(cid:96)e ) if |z| < |z(cid:48)|
n k 0
(z,z(cid:48)) if |z| = |z(cid:48)|
(cid:12) (cid:12)
for (z,z(cid:48)) ∈ kerπ and (cid:96) = (cid:12)|z|−|z(cid:48)|(cid:12) is well defined.
n
Proof. Fix (z,z(cid:48)) ∈ kerπ with z = (z ,...,z ) and z(cid:48) = (z(cid:48),...,z(cid:48)). By symmetry, we
n 0 k 0 k
can assume that (cid:96) = |z|−|z(cid:48)| ≥ 0. Now, we simply use π (z) = π (z(cid:48)) to verify that
n n
k k
(cid:88) (cid:88)
π (z +de ) = (|z|+(cid:96))(n+r )+ z r = (|z(cid:48)|+2(cid:96))(n+r )−(cid:96)n+ z(cid:48)r
n+rk 0 k i i k i i
i=1 i=1
k
(cid:88)
= (|z(cid:48)|+(cid:96))(n+r )+(cid:96)r + z(cid:48)r = π (z(cid:48) +(cid:96)e ),
k k i i n+rk k
i=1
as desired. (cid:3)
Definition 4.5. A chain z = a ,a ,...,a = z(cid:48) between factorizations z,z(cid:48) in a nu-
0 1 r
merical monoid is monotone if |a |,|a |,...,|a | is a monotone sequence.
0 1 r
Proposition 4.6. Fix n, ρ ⊂ kerπ , and (z,z(cid:48)) ∈ ρ, and let (w,w(cid:48)) = Φ (z,z(cid:48)).
n n
(a) The map Φ is injective.
n
(b) The map Φ preserves length differences: |z|−|z(cid:48)| = |w|−|w(cid:48)|.
n
MINIMAL PRESENTATIONS OF SHIFTED NUMERICAL MONOIDS 7
(c) The map Φ preserves the reflexive, symmetric, and translation closure operations:
n
if ρ is reflexive, symmetric, or closed under translation, then so is Φ (ρ).
n
(d) The map Φ preserves monotone chain connectivity: if ρ is translation-closed and
n
there exists a monotone ρ-chain from z to z(cid:48), then there exists a monotone Φ (ρ)-
n
chain from w to w(cid:48).
Proof. It is easy to check that |z| − |z(cid:48)| = |w| − |w(cid:48)|, from which injectivity follows.
Both Φ(z,z) = (z,z) and Φ(z(cid:48),z) = (w(cid:48),w) follow directly from definitions as well.
Next, fixing u ∈ Nk+1 and assuming by symmetry that (cid:96) = |z|−|z(cid:48)| ≥ 0, we have
Φ (z+u,z(cid:48)+u) = (z+u+(cid:96)e ,z(cid:48)+u+(cid:96)e ) = (z+(cid:96)e ,z(cid:48)+(cid:96)e )+(u,u) = Φ (z,z(cid:48))+Φ (u,u).
n 0 k 0 k n n
It remains to prove the final claim.
Suppose ρ is translation-closed and there is a monotone decreasing ρ-chain from z
to z(cid:48). By induction on chain length, we can assume there is a single intermediate
factorization z(cid:48)(cid:48). Letting (cid:96) = |z|−|z(cid:48)(cid:48)| ≥ 0 and (cid:96)(cid:48) = |z(cid:48)(cid:48)|−|z(cid:48)| ≥ 0, we have
Φ (z +(cid:96)(cid:48)e ,z(cid:48)(cid:48) +(cid:96)(cid:48)e ) = (z +((cid:96)+(cid:96)(cid:48))e ,z(cid:48)(cid:48) +(cid:96)(cid:48)e +(cid:96)e )
n 0 0 0 0 k
and
Φ (z(cid:48)(cid:48) +(cid:96)e ,z(cid:48) +(cid:96)e ) = (z(cid:48)(cid:48) +(cid:96)e +(cid:96)(cid:48)e ,z(cid:48) +((cid:96)+(cid:96)(cid:48))e )
n k k k 0 k
which form a monotone decreasing Φ (ρ)-chain from w to w(cid:48). (cid:3)
n
The main obstruction to Theorem 4.9 for arbitrary n is that Φ needs only pre-
n
serve connectivity by monotone chains. Proposition 4.8 ensures that for n sufficiently
large, any pair of factorizations (z,z(cid:48)) ∈ kerπ is connected by a monotone chain, and
n
Example 4.7 demonstrates why this can fail for small n.
Example 4.7. Let S = (cid:104)3,14(cid:105). The element 1078 ∈ M has factorization set
74
Z (1078) = {(0,14,0),(11,0,3),(0,6,7)}.
M74
Notice that the only chains between (0,14,0) and (11,0,3) are a monotone chain di-
rectly between them and a non-monotone chain through (0,6,7). Since both relations
in the non-monotone chain are translations, no minimal presentation of M contains
74
the relation ((0,14,0),(11,0,3)). On the other hand, we have
Z (1274) = {(0,14,0),(11,0,3)},
M88
so every minimal presentation of M contains the relation ((0,14,0),(11,0,3)).
88
Proposition 4.8. Fix n > r2 and a minimal presentation ρ ⊆ kerπ . There exists a
k n
monotone ρ-chain between any (z,z(cid:48)) ∈ kerπ .
n
Proof. Without loss of generality, assume gcd(z,z(cid:48)) = 0. By way of contradiction,
assume there is no monotone ρ-chain from z to z(cid:48). Since Cong(ρ) = kerπ , there exists
n
a chain z = a ,a ,...,a = z(cid:48) of factorizations such that for each i < r, we have
0 1 r
(a ,a ) = (b ,b(cid:48))+(u ,u ), (b ,b(cid:48)) ∈ ρ,u ∈ Nk+1,
i i+1 i i i i i i i
8 CONWAY, GOTTI, HORTON, O’NEILL, PELAYO, WILLIAMS, AND WISSMAN
where b and b(cid:48) occur in distinct connected components of the graph ∇ of β = π (b ).
i i β n i
By Corollary 3.5, we have |a | − |a | ∈ {−1,0,1} for each i ≤ r. As such, the
i i−1
sequence |a |,|a |,...,|a | of factorization lengths has non-sequential repeated values.
0 1 r
Without loss of generality, we can replace z(cid:48) with the factorization whose length is
the first non-sequential repeated value in this sequence, and replace z with the last
factorization before z(cid:48) with |z| = |z(cid:48)|. As such, |a | = |a | = |a | = |a |, and
0 r 1 r−1
|a | = |a | whenever 1 < i < r.
i i−1
First, suppose |a | = |a |+1. Applying Theorem 3.4 to the pairs (b ,b(cid:48)) and (b ,b(cid:48)),
1 0 1 1 r r
we see that z > 0 and z(cid:48) > 0, which contradict the assumption that gcd(z,z(cid:48)) = 0.
k k
Likewise, if |a | = |a | − 1, then Theorem 3.4 implies that z > 0 and z(cid:48) > 0, which
1 0 0 0
again contradict the assumption that gcd(z,z(cid:48)) = 0. This completes the proof. (cid:3)
Together, Propositions 4.6 and 4.8 yield Theorem 4.9, the main result of this section.
Theorem 4.9. For any n > r2, the image of any minimal presentation ρ of M under
k n
the map Φ : kerπ → kerπ is a minimal presentation of M . In particular,
n n n+r n+r
k k
Φ induces a one-to-one correspondence between the minimal presentations of M and
n n
the minimal presentations of M .
n+r
k
Proof. We begin by showing that any minimal presentation ρ ⊂ kerπ of M satisfies
n n
Cong(Φ (ρ)) = kerπ ,
n n+r
k
that is, the image of ρ under Φ is a presentation for M . Fix (w,w(cid:48)) ∈ kerπ ,
n n+r n+r
k k
and let m = π (w). By Proposition 4.8, there exists a monotone chain from w to
n+r
k
w(cid:48), which we can assume is monotone decreasing by Proposition 4.6(c). We can also
assume each step in this chain has the form (b,b(cid:48)) + (u,u) for some u ∈ Nk+1 and
b,b(cid:48) ∈ Z(β) lying in different connected components of ∇ . By Proposition 4.6(c), it
β
suffices to prove each (b,b(cid:48)) lies in the image of Φ , so it is enough to assume w and w(cid:48)
n
lie in different connected components of ∇ .
m
First, if |w| = |w(cid:48)|, then Φ (w,w(cid:48)) = (w,w(cid:48)) by Proposition 4.4. Otherwise, Corol-
n
lary 3.5 implies |w| = |w(cid:48)|+d, where d = gcd(r ,...,r ), and w ,w(cid:48) ≥ d follows from
1 k 0 k
the proof of Theorem 3.4. This means Φ (w −de ,w(cid:48) −de ) = (w,w(cid:48)), which proves
n 0 k
Φ (ρ) generates kerπ .
n n+r
k
Now, by Propositions 4.6(d) and 4.8, factorizations z,z(cid:48) ∈ Z (β) lie in distinct
Mn
connected components of ∇ if and only if (w,w(cid:48)) = Φ (z,z(cid:48)) lie in different connected
β n
components of the factorization graph of π (w), so the image Φ (ρ) is indeed min-
n+r n
k
imal as a presentation of M . Additionally, the above argument implies that any
n+r
k
minimal presentation ρ(cid:48) of M is contained in the image of Φ , so its preimage
n+r n
k
Φ−1(ρ(cid:48)) is a minimal presentation for M . This completes the proof. (cid:3)
n n
Remark 4.10. Resuming notation from Theorem 4.9, the bound n > r2 first appears
k
in Theorem 3.4, and this is the only result explicitly using the bound. In particular,
each subsequent result requiring n > r2 (including Corollary 3.5, Proposition 4.8,
k
MINIMAL PRESENTATIONS OF SHIFTED NUMERICAL MONOIDS 9
Theorem 4.9, Corollary 4.12, and several results in Section 5) only uses this bound to
(possibly indirectly) apply Theorem 3.4. As such, any improvement on the bound in
Theorem 3.4 immediately improves Theorem 4.9.
Example 4.11. For fixed S and n sufficiently large, the size of a minimal presentation
for M need not be fixed within a given r -period. For example, if S = (cid:104)6,9,20(cid:105), the
n k
size of a minimal presentation of M ranges from 4 (for n = 420) to 8 (for n = 417).
n
We conclude the section with Corollary 4.12, which uses Proposition 4.8 to charac-
terize the minimal relations for M whose factorizations have equal length.
n
Corollary 4.12. Fix n > r2 and a minimal presentation ρ ⊂ kerπ for M . Then
k n n
τ = {((z ,...,z ),(z(cid:48),...,z(cid:48))) : (z,z(cid:48)) ∈ ρ and |z| = |z(cid:48)|} ⊂ Nk ×Nk
1 k 1 k
is a presentation for S.
Proof. Fix s,s(cid:48) ∈ Z (s) with |s| ≤ |s(cid:48)|. Let π : Nk → S denote the factorization
S
homomorphism of S, and let z = (|s(cid:48)| − |s|,s ,...,s ) and z(cid:48) = (0,s(cid:48),...,s(cid:48)). Then
1 k 1 k
(z,z(cid:48)) ∈ kerπ , so by Proposition 4.8 there exists a monotone ρ-chain from z to z(cid:48), but
n
since |z| = |z(cid:48)|, each factorization a in the chain must have length |z| as well. As such,
k
(cid:88)
π (a) = |z|n+ a r = |z|n+π(a ,...,a )
n i i 1 k
i=1
for each a in the chain, thus producing a τ-chain from s to s(cid:48). (cid:3)
Example 4.13. Resuming notation from Corollary 4.12, the presentation τ for S need
not be minimal. Indeed, if S = (cid:104)6,9,20(cid:105), then the minimal presentation ρ of M
450
given in Example 4.3 contains three equal-length relations, yielding the presentation
τ = {((0,8,0),(2,0,3)),((1,6,0),(0,0,3)),((3,0,0),(0,2,0))}
for S. The first of the above relations is redundant, as the latter two form a minimal
presentation τ(cid:48) for S. However, the only τ(cid:48)-chain between (0,8,0) and (2,0,3) is non-
monotone, which is why ρ must also contain the relation ((0,0,8,0),(3,2,0,3)).
5. Applications to factorization invariants
In this section, we explore several consequences of the results in Sections 3 and 4. We
begin with Remark 5.1, which discusses computational applications of Theorem 4.9,
and Corollary 5.2, which improves a recent result from commutative algebra (see Re-
mark 5.3 for more on this connection). Next, we characterize the behavior of several
arithmetical invariants of non-unique factorization over M for large n. The survey
n
article [12] gives an overview of several of the invariants discussed here.
10 CONWAY, GOTTI, HORTON, O’NEILL, PELAYO, WILLIAMS, AND WISSMAN
n M GAP Runtime [8] Remark 5.1
n
50 (cid:104)50,56,59,70(cid:105) 1 ms 1 ms
200 (cid:104)200,206,209,220(cid:105) 40 ms 40 ms
400 (cid:104)400,406,409,420(cid:105) 210 ms 210 ms
1000 (cid:104)1000,1006,1009,1020(cid:105) 3 sec 210 ms
5000 (cid:104)5000,5006,5009,5020(cid:105) 18 min 210 ms
10000 (cid:104)10000,10006,10009,10020(cid:105) 4.2 hr 210 ms
Table 1. Runtimecomparisonforcomputingaminimalpresentationfor
numerical monoids M with S = (cid:104)6,9,20(cid:105). All computations performed
n
using GAP and the package numericalsgps [8].
Remark 5.1. Minimal presentations are used frequently in computer software package
implementations, since many quantities of interest can then be quickly computed [10].
Additionally, minimal presentations have particular significance in commutative alge-
bra; see Remark 5.3. Most existing algorithms to compute a minimal presentation of a
given numerical monoid use Gro¨bner basis techniques, which become computationally
infeasible as the number and size of the generators grow large [7].
Theorem 4.9 yields a method of reducing this complexity in certain cases. In par-
ticular, if the generators of M = (cid:104)n ,...,n (cid:105) satisfy n > (n −n )2, then a minimal
1 k 1 k 1
presentation for M can be computed by first computing a minimal presentation for
M(cid:48) = (cid:104)n −R,...,n −R(cid:105), where R is some appropriately chosen multiple of n −n ,
1 k k 1
andthensuccessivelyapplyingthemapΦ fromProposition4.4untilaminimalpresen-
∗
tation for M is obtained. In cases where the generators of M(cid:48) are significantly smaller
than those of M, the resulting computation is much faster than directly computing a
minimal presentation for M.
Table 1 gives a sample of the improved runtimes that result from using Theorem 4.9.
AllruntimeswereobtainedinthecomputeralgebrasystemGAPandthenumericalsgps
package,astandardsettingfornumericalsemigroupcomputations. Animprovedimple-
mentation of the function MinimalPresentationOfNumericalSemigroup that utilizes
Theorem 4.9 is currently in development, and will be available with the next major
release of the numericalsgps package.
As a corollary of Theorem 4.9, we obtain an improved bound for a result conjectured
by Herzog and Srinivasan and later proved by Vu in [15].
Corollary 5.2. The function n (cid:55)→ |Betti(M )| is r -periodic for n > r2.
n k k
Proof. GivenanyminimalpresentationρforM , thesetBetti(M )consistsofprecisely
n n
the elements with factorizations appearing in ρ. Now, Theorem 3.4 and Corollary 3.5
imply that each relation in ρ involving factorizations of different lengths produces a
distinct Betti element, since the corresponding Betti element β must have exactly two
