Associated Graded Rings for Numerical Monoid Rings

Numerical monoids, i.e., the additive submonoids of non-negative integers with finite complement, are central objects of study at the crossroads of combinatorics, additive number theory, and commutative algebra. A well-known invariant, attached to a numerical monoid, is the Frobenius number. It is the largest integer not in the monoid. Despite many partial results, no general formula is known for the Frobenius number in terms of given generators of the monoid. We introduce a new invariant of a numerical monoid, called the second Frobenius number. It carries important structural information on the monoid. A commutative algebra perspective explains the relationship between the original and the second Frobenius numbers: both are the threshold points for stabilization of Hilbert functions – for the monoid ring itself in the classical setting and for the associated graded algebra in our setting. Our main theoretical result is an explicit upper bound for the second Frobenius number in terms of given generators of the monoid. We also develop several algorithms for computing the second Frobenius number and present many computational results, strongly suggesting that the Hilbert functions may be stabilizing much faster than our theoretical bound.


Acknowledgments
I would like to thank my wife for encouraging me to undertake a master's degree and supporting me through the process. I honestly could not have done it without her. I would like to thank my advisor Dr. Joseph Gubeladze who kindly and humbly worked with me on this thesis. He always took the time to teach me and to answer the many questions I had. The generosity of his time is deeply appreciated. I would also like to thank David Erickson for his help solving some of the algorithmic challenges we faced.
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Introduction
Ferdinand Georg Frobenius was a German mathematician born February 14, 1877 in a suburb of Berlin. He worked on diverse fields such as elliptic functions, differential equations, number theory, and group theory. He has numerous mathematical ideas named after him but one in particular will be the focus of this paper.
In the early part of the 20th century, Frobenius proposed the Diophantine Frobenius Problem which would motivate the study of numerical monoids. The problem asks what is the largest positive integer (called the Frobenius number ) that is not representable as a nonnegative integer linear combination of relatively prime positive integers? For example, if we take the subset of the natural numbers {5, 7, 9} and consider all possible combinations of these numbers if we can only add them (including repetitions), then we get the following set {5, 7, 9, 10, 12, 14, →} where the symbol → means that every integer greater than 14 is in this set. We can see the largest natural number not in this set is 13.
What Frobenius ended up doing by proposing this question was to motivate the study of the gaps in the natural numbers. This study is a fascinating tour de force of linear algebra, number theory, and abstract algebra; see [RG09;Ram05;CGO20;ADG20]. The Frobenius number problem has a ring theoretical interpretation which we will explain after introducing several algebraic concepts.
In this paper we denote the nonnegative integers by Z + and N to denote the natural numbers {1, 2, 3, . . .}.
A numerical monoid M is a subset of the nonnegative integers that contains the additive identity 0, is closed under addition and has a finite complement (possibly empty) in the nonnegative integers. Every numerical monoid M can be written (non uniquely in general) as Z + m 1 + · · · + Z + m e := {a 1 m 1 + · · · + a e m e | a 1 , . . . , a e ∈ Z + } for some relatively prime natural numbers m 1 , . . . , m e , called generators of M . In the case k is algebraically closed, this surjection via the Hilbert Nullstellensatz induces an embedding of the monomial variety, corresponding to k[M ], in the affine space of dimension e. Correspondingly, e is called the embedding dimension, which also explains our notation.
in terms of the generators of M (Theorem 4.7). In the second part of the work we implement several algorithms for computing F ′ (M ) and use it to develop computational data on many examples of numerical monoids. These computations provide strong evidence that F ′ (M ) is considerably smaller than the theoretical upper bound in Theorem 4.7.
We expect that the second Frobenius number is at least as interesting as the original Frobenius number and it provides a fertile ground for exploration of numerical monoids from this novel perspective.
Chapter 1 Semigroups and monoids

Semigroups
Our study of numerical monoids starts with that of semigroups. A semigroup is a pair (S, +) with S a set and + an associative binary operation on S. Semigroups do not need inverses nor an identity element like a group does. We will assume all semigroups in this paper are commutative and we will omit the binary operation + and denote the semigroup just by S. Thus N is a semigroup under addition as is the set of positive even integers 2N.
A subsemigroup is a subset of a semigroup S that is closed under the operation of S.
Thus, 2N is a subsemigroup of N.
The intersection of any number of subsemigroups is a subsemigroup. To see this, one must show that the intersection of a set of subsemigroups of S is closed under the operation of S. Each element in the intersection is in every subsemigroup and each subsemigroup is closed under the operation of S by definition. Thus the intersection is closed which makes the intersection a subsemigroup.
Let S be a semigroup and let s 1 , . . . , s k be any collection of elements of S. The smallest subsemigroup that contains s 1 , . . . , s k is the intersection of all subsemigroups that contain s 1 , . . . , s k . We denote this set ⟨s 1 , . . . , s k ⟩ and we have k ⟨s 1 , . . . , s k ⟩ = a i s i | a i ∈ Z + and at least one Remember it is not necessary for the additive identity to be an element of a semigroup.
If ⟨s 1 , . . . , s k ⟩ = S for some k ∈ N then we say that S is generated by s 1 , . . . , s k and S is finitely generated. When s ∈ ⟨s 1 , . . . , s k ⟩ is written as s = k i=1 a i s i with a i ∈ N, then we call this sum a representation of s ∈ S.

Monoids
A monoid M is a semigroup with a neutral element 0. A subset N of a monoid M is a submonoid of M if it is a subsemigroup that contains the neutral element. Note that the set {0} is a submonoid of M and is called the trivial submonoid.
As with the case with semigroups, the intersection of any number of submonoids is a submonoid. Thus, for any subset N of a monoid M , the smallest submonoid containing N is the intersection of all submonoids of M that contain N . This is also denoted by ⟨N ⟩.
Whether this is a subsemigroup or a submonoid will be clear from context (just check if it ̸ contains 0). Since ⟨N ⟩ is closed under the operation of M , all elements have the form Notice that since {0} and the empty set ∅ are subsets of every submonoid. We will now discuss an important submonoid called a numerical monoid.

Chapter 2
Numerical monoids and the Frobenius number 2.1 Numerical monoids In this paper we are only interested in the submonoids of the nonnegative integers. These submonoids give a rich source of material to study. We therefore come to an important definition.
Definition 2.1. A numerical monoid is a submonoid of Z + with finite complement in Z + .
The set {0, 6, 7, 8, 9, →} is a numerical monoid, whereas the set of even nonnegative integers 2Z + is not because its complement in Z + is the infinite set of odd integers.
Since the complement of a numerical monoid M in Z + is finite, there must be a m 0 ∈ M , such that for all m > m 0 , m + 1 ∈ M . In other words, the sequence of numerical monoid elements includes all consecutive integers after some point. In this light, the following is a proposition that can be used as the definition of a numerical monoid. Let n ≥ (m − 1)(m + 1) and by the division algorithm there exists unique q, r ∈ Z such that n = qm + r, where 0 ≤ r < m. Notice that (m − 1)(m + 1) = (m − 1)m + (m − 1). So n = qm + r ≥ (m − 1)m + (m − 1). Then q ≥ m − 1 and m > r implies that m − 1 ≥ r.
Therefore we have written n as a nonnegative integer combination of m and m + 1 which means that n ∈ M . Furthermore, any n ≥ (m − 1)(m + 1) ∈ M . Thus the complement of M in Z + must be finite. Now we will focus on numerical monoids, given by generators. Proof. The subgroup of Z, generated by ⟨A⟩ is the same as the subgroup generated by A.
According to Proposition 2.2, the latter is Z if and only if ⟨A⟩ is a numerical monoid which, in turn, is equivalent to gcd(A) = 1. Thus for m ∈ 2Z + , ϕ(m) = ϕ(2x) = 2x = x. Therefore 2 Now some terminology about what it means to add two sets together. Let A, B ⊂ Z + .
Then Notice that the generators of M are absent from this set. Proof. Let m ∈ M * . We want to show that there exist m 1 , . . . , m k ∈ M * \ (M * + M * ) such that m = a 1 m 1 + · · · + a k m k for some a i ∈ Z + . If m ∈ / M * \ (M * + M * ) then there exist x, y ∈ M * such that m = x + y. Repeat this procedure with x and y and so on.
The process must terminate because of the Well Ordering Principle: after each step we get

Decomposition length
Let M be a numerical monoid generated by a set {m 1 , . . . , m e } and m 1 < · · · < m e . For any element m ∈ M and a representation m = a 1 m 1 + · · · + a e m e , the e-tuple (a 1 , . . . , a e ) is a decomposition of m ∈ M .
Next we introduce the maximal decomposition lengths. For any m ∈ M \ {0}, a repre- Correspondingly, the maximal decomposition length, or simply the length, of an element For the number of elements in M that have a particular length k ∈ N we write The following lemma is a special case -the rank one case -of the Gordan Lemma, In particular, M = ⟨m 0 , m 1 , . . . , m m−1 ⟩.

Frobenius number
The Frobenius number of a numerical monoid M is an active research area [Ram05]. The main problem is to express F(M ), or at least an optimal upper bound in terms of a given generating set of M . Below we give a short synopses of some of the highlights in the field.
For m 1 , . . . , m e ∈ N with gcd(m 1 , . . . , m e ) = 1 and m 1 < · · · < m e , we denote  (d) For the following arithmetic sequence we have (e) For the following geometric sequence we have Thus T is the ℓ 1 norm of the generators. One of the most famous open problems in the field, Arnold's conjecture, says that F(M ) increases like T 1+1/(e−1) [Isk11,p. 526]. Arnold also conjectured that for "average behavior", F(M ) is [Isk11,p. 526]).

Chapter 3
Commutative algebra of numerical monoid rings All our rings are assumed to be commutative and unital. The symbol k will always denote a field and k * = k \ {0}. Also, all our monoids are commutative and ring homomorphisms are assumed to respect the units.

Algebras
Let k be a field and A a ring. Then, A is called a k-algebra if A contains a isomorphic copy of k as a subring. For simplicity of notation, we will identify k with its isomorphic copy in A. Every k-algebra is also a k-vector space. For two k algebras A and B, a ring homomorphism f : A → B is a k-algebra homomorphism if it is also a k-linear map.
The basic example of a k-algebra is the multivariate polynomial ring k[X 1 , . . . , X n ]. The quotient of a k-algebra by an ideal is also a k-algebra.
A k-algebra A is called finitely generated if there is a finite family of elements {a 1 , . . . , a n } ⊂ A, called generators of A, such that A is the smallest sub-algebra of A, containing the a i 's.
In this case we will write A = k[a 1 , . . . , a n ]. The assignment X i → a i , i = 1, . . . n, gives rise to a surjective k-algebra homomorphism Hence, the Isomorphism Theorem for rings implies that every finitely generated k-algebra is isomorphic to a quotient of the form k[X 1 , . . . , X n ]/I for some natural number n ∈ N and an ideal I ⊂ k[X 1 , . . . , X n ].

Monoid rings
Let M be a monoid. The monoid algebra k[M ] is the k-vector space over the basis M , where the multiplicative structure is defined by where λ i , µ j ∈ k, m i , n j ∈ M , and the monoid operation is written multiplicatively. The reader is referred to [BG09,Chapter 2] for generalities on monoid rings.
The monoid ring k[M ] is a k-algebra, where k embeds to k[M ] via λ → λ · 1 for the neutral element 1 ∈ M (changed from the additive notation 0 ∈ M ).
The algebra k[M ] also contains an isomorphic copy of M via the embedding m → 1 · m, where 1 ∈ k.
Notice, 1 ∈ k and 1 ∈ M are the same unit element of k[M ].
We will write elements of k[M ] as linear combinations i λ i m i , with the understanding that 1 ∈ M gets identified with 1 ∈ k (and the monoid operation is written multiplicatively).
The defining universal property of the monoid ring k[M ] is that, for any k-algebra A  k[X 1 , . . . , X n ], where the identification of the two k-algebras is through the k-algebra iso-morphism, defined by (a 1 , . . . , a n ) → X a 1 1 · · · X an n .
Example 3.2. For a numerical monoid monoid M , the monoid algebra k[M ] can be though of as the subalgebra of the univariate polynomial ring k[X], consisting of the polynomials, whose reduced forms only involve monomials of the form λX m , where λ ∈ k and m ∈ M .

Embedding dimension and multiplicity
Assume M ⊂ Z + is a numerical monoid, generated by coprime numbers m 1 , . . . , m e ∈ N.
Then we have the k-algebra homomorphisms: For the corresponding affine varieties (assuming k is algebraically closed), via Hilbert Nullstellensatz [AM69, Chapter 7], f induces an algebraic embedding of the monomial curve This monomial curve is given by and an algebraic surjection from the Spec(k[M ]) to the affine line A 1 k , given by which is generically of the form 'n points to one point'. Correspondingly, e is called the embedding dimension of M with respect to the given generators and m 1 is called the multiplicity of M . Notice, according to Lemma 2.5, the multiplicity is independent of the choice of generators. This terminology also explains our use of e for the number of generators of M .

Graded rings
A graded k-algebra A is an algebra, admitting a direct sum representation where A i ⊂ A is a k-vector subspace, respecting the multiplicative structure as follows: We call A n the degree n-homogeneous component of A. For an element a ∈ A n \ {0}, we write deg(a) = n.
A k-algebra can carry several different graded structures. For instance, we can make the polynomial ring k[X 1 , . . . , X n ] into a graded algebra in infinitely many different ways.
Example 3.3. Every family of natural numbers c 1 , . . . c n ∈ N defines a graded structure via putting deg(X i ) = c i . In more detail, under this grading we have When c 1 = c 2 = · · · = c n = 1, we get the standard grading of the polynomial ring.
is the Hilbert function of A (for this grading).
Example 3.4. For the standard grading of A = k[X 1 , . . . , X n ], we have We will need the following consequence of the general dimension theory of graded rings √ [AM69, Chapter 11], where 0 denotes the ideal of all nilpotent elements -the nil-radical: Lemma 3.5. Let A = k ⊕ A 1 ⊕ A 2 ⊕ · · · be a finitely generated graded k-algebra such that √ ∼ A/ 0 = k[X] as k-algebras, then A is a one-dimensional ring and, consequently, H A is eventually a constant function, i.e., H A (i) = H A (i + 1) for i > 0.
Remark. We point out that dimension theory in commutative algebra does not say how large i needs to be to guarantee the equality H A (i) = H A (i + 1).
For k-algebra A and an ideal I ⊂ A, one defines the associated graded algebra as follows: where: (a) I k is the k-th power of I, i.e. the ideal generated by all possible k-fold products a 1 · · · a k , where a 1 , . . . , a k ∈ I, (b) The multiplicative structure is determined by the pairings: The importance of this construction is that the affine scheme of gr I (A) is a flat deformation of that of A, called the Rees deformation [Eis95, Section 6.4] and it plays an important role in resolutions of singularities in algebraic geometry.

Chapter 4
The main theorem Throughout this section we assume M ⊂ Z + is a numerical monoid, generated by coprime numbers m 1 , . . . , m e ∈ N, satisfying m 1 < · · · < m e . We follow the notation introduced in (a) We have the graded structure where the homogeneous components are: Proof. The parts (a, b) follow from the definition of gr(k[M ]) and the equality I k ∩ M = ) is clear. That the multiplicative structures also agree follows from the part (b) and the observation that l(m k 1 ) = k for every k ∈ N.
We define the map f : gr(k[M ]) → k[Z + m 1 ] to be the k-algebra homomorphism, defined by ). In fact, the degree of m, viewed as an element of (For a similar description of gr(k[N ]) for the affine monoids of high rank, see [Gub].) Corollary 4.2.

The second Frobenius number
In view of Corollary 4.2(c) we can introduce the following    Proof. This is equivalent to the claim that, for a maximal decomposition m = e i=1 a i m i and Lemma 4.6. Assume (a 1 , . . . , a e ) ∈ dec(M ) and Proof. This is equivalent to the claim that, for a maximal length decomposition e i=1 a i m i , Theorem 4.7. (b) It follows from [ONe17] that the function l(−) is eventually of the form in Theorem 4.7(c), but we also give an explicit lower bound from where this behavior shows up.
We will need two lemmas.
Proof. Assume to the contrary a j ≥ m 1 for some j ≥ 2. Then we can write where a ′ 1 = a 1 + m j and a ′ j = a j − m 1 . This contradicts the containment (a 1 , . . . , a e ) ∈ dec(M ) because a 1 + a j < a ′ 1 + a ′ j .
To state the second lemma, we first introduce the following objects: (a) For every k ∈ Z + , the affine hyperplane and affine half-space: (b) The two infinite right prisms: and Π = {(x 1 , x 2 , . . . , x e ) | x 2 , . . . , x e ≥ 0 and (e) The sequence of lattice polytopes (for some initial vlues of s the polytope P s may be empty).
For s ∈ Z + , the s-th element of M refers to the s-the element of M in the natural order.  for some t > s. In particular, Lemma 4.9(c) implies that, for every index t with P t ∩ G k = 0 and P t ⊂ G − k , one has t > s and, therefore, (4.1) implies H t ∩ Π = H t ∩ Π + . This, in turn, implies the following.
Claim. The set of hyperplanes H t , satisfying P t ∩ G k+1 = 0 and P t ⊂ G − k+1 is the parallel translate by e 1 of the set of hyperplanes H t ′ , satisfying P t ′ ∩ G k = 0 and P t ′ ⊂ G − k .
As above, we assume ke 1 ∈ H s . For every t ≥ s, the equality (4.2) implies For the equalities in Part (c), one needs to show that P t ∩ G k = 0 and P t ⊂ G − k for every index s < t ≤ s + m 1 − 1. Assume to the contrary this is not the case for some index t.
Then, P t ∩ G r = 0 and P t ⊂ G − r for some r > k. By Sublemma, H t is the parallel translate by (r − k)e 1 of some H t ′ with s ≤ t ′ < t. We arrive at the contradiction where the second equality is due to (4.3).

̸ ̸
Chapter 5 Generating the d k (M ) We keep the notation, introduced in Section 2.2.
Given a natural number n, in this section we describe an algorithm for computing the numbers d 1 (M ), d 2 (M ), . . . , d n (M ).

Algorithm 1.
Step 1. Declare a struct called node to encapsulate a monoid element's value and length, which is not necessarily its maximal decomposition length. This node will be used in a linked list which is generated in the next step.
Step 2. To compute the monoid with generating set {m 1 , . . . , m e }, create e for-loops where each loop ranges from 0 to nme . The nested loops will create duplicate elements with differm 1 ent coefficients and, not necessarily, different lengths (an element can have more than one representation with the same length).
It is important to create enough copies of each element less than nm e − m 1 so that element's maximum decomposition is found. An element less than nm e −m 1 can be generated with coefficients whose sum is less than or equal to n, but its maximum decomposition is not found.
During the loop, check if the element is less than nm e − m 1 before creating a node and inserting into the linked list.

Algorithm 2
Step 1. To reduce the computations, one can first extract Hilb(M ) from the generating set {m 1 , . . . , m e }. This can be done using Normaliz. The steps below are independent of this steps, though.