Factorizations of the same length in abelian monoids
Evelia R. García Barroso, Ignacio García-Marco, Irene Márquez-Corbella
FFactorizations of the same length in affinesemigroups
Evelia R. Garc´ıa Barroso, Ignacio Garc´ıa-Marco, Irene M´arquez-Corbella ∗†‡
July 14, 2020
Abstract
Let
S ⊆ Z n be a pointed affine semigroup. In this paper we developa general strategy to study the set of elements in S having at least twofactorizations of the same length, namely the ideal L S . To this end, wework with the toric ideal of a certain semigroup associated to S . Ourwork can be seen as a new approach generalizing [9], which only studiesthe case of numerical semigroups. When S is a numerical semigroupwe give three results: (1) we compute explicitly a set of generators ofthe ideal L S when S is minimally generated by an almost arithmeticsequence; (2) we address the question of when L S is a principal ideal;(3) we classify the computational problem of determining the largestinteger not in L S as an N P -hard problem.
Keywords:
Affine semigroup; Toric ideal; Non-unique factorization;Ap`ery set
Mathematics Subject Classification (2010) An affine semigroup is a finitely generated submonoid of Z m . Affine semi-groups are a powerful interface between Combinatorics and Algebraic Geom-etry since they constitute a combinatorial tool for studying Toric Geometry ∗ Departamento de Matem´aticas, Estad´ıstica e I.O., Universidad de La Laguna, Aptdo.Correos 460, 38200, La Laguna, Tenerife, Spain. † This work was partially supported by the Spanish MICINN PID2019-105896GB-I00,the Spanish MICINN PID2019-104844GB-I00 and MASCA (ULL Research Project) ‡ E-mail: [email protected], [email protected], [email protected] a r X i v : . [ m a t h . A C ] J u l see, e.g. [26, 8, 11]). Given a finite set A = { a , . . . , a n } ⊆ Z m , we denoteby (cid:104)A(cid:105) := { λ a + . . . + λ n a n | λ , . . . , λ n ∈ N } , the affine semigroup generated by A . An affine semigroup S ⊂ Z m is pointed if the only invertible element is the neutral element of the monoid or, equiv-alently, if S ∩ ( −S ) = { } . An affine semigroup has a unique minimal (withrespect to the inclusion) set of generators if and only if it is pointed (see [11,Proposition 1.2.23] or [7, Section 1] for this and other properties of pointedaffine semigroups). We will refer to this set of generators as the minimalset of generators of the semigroup. Unless stated otherwise, when we write S = (cid:104) a , . . . , a n (cid:105) ⊆ Z m for a pointed affine semigroup, we are assuming that A = { a , . . . , a n } is its minimal set of generators.Now, consider a pointed affine semigroup S = (cid:104) a , . . . , a n (cid:105) ⊆ Z m and anelement b ∈ S . Since b ∈ S , there exists an n -tuple λ = ( λ , . . . , λ n ) ∈ N n such that b = λ a + . . . λ n a n . In this case we say that λ = ( λ , . . . , λ n ) is a factorization of b in S of length (cid:96) ( λ ) := λ + . . . + λ n . Now we define the set L S of elements in S having (at least) two factorizations of the same length,i.e., L S def = { b ∈ S | b has two different factorizations of the same length } . In this paper we investigate this set L S . In the particular setting that S is a numerical semigroup this problem was addressed in [9]. Numericalsemigroups are an interesting family of pointed affine semigroups. Moreprecisely, a numerical semigroup is a submonoid of N with finite complementover N (for a thorough study of numerical semigroups we refer the readerto [3, 24]). In [9], the authors prove that given S = (cid:104) a , . . . , a n (cid:105) ⊆ N anumerical semigroup, then L S = ∅ if and only if n = 2, and describe L S when n = 3.This paper goes further into the study of factorization properties of affinemonoids by means of its corresponding toric ideal. For a general referencein the theory of non-unique factorizations domains and monoids, we referto [17]. For a recent account of the progress of factorization invariants inaffine semigroups, we refer the reader to the recent papers [16, 18] and thereferences therein. 2 utline of the article Section § B = { b , . . . , b s } ⊆ S , that is, the setAp S ( B ) = { x ∈ S | x − b i / ∈ S , ∀ i ∈ { , . . . , s }} . Although we use Ap´ery sets as a tool in the other sections, we believe thatthe results in this section are interesting in their own. In Theorem 2.2, wepresent how to compute an Ap´ery set by means of the (toric) ideal of thesemigroup and a factorization of the elements of B . This result provides analternative to [21, Theorem 8]. Then, in Theorem 2.7, we characterize whenthis Ap´ery set is finite, it turns out that Ap S ( B ) is finite if and only if theideal generated by B and the zero element form a pointed affine semigroup.We also prove that this is also equivalent to the fact that the cone definedby B coincides with the one defined by S .The main results of the paper are in Section §
3, where we develop ageneral strategy to study L S . In Proposition 3.2 we describe how to obtaina finite set of generators of the ideal L S by means of the toric ideal of asemigroup ˜ S that we associate to S . That is,˜ S = (cid:104) ( a , , ( a , , . . . , ( a n , (cid:105) ⊆ Z m +1 . (1)As a consequence, in Theorem 3.5, we describe S − L S as an Ap´ery setand, thus, the techniques developed in the previous section apply here. Inparticular, applying Theorem 2.7, we describe when L S ∪ { } is a pointedaffine semigroup or, equivalently, when S − L S is a finite set.In Section § § § L S when S is minimally generated by an almost arithmetic sequence. By almostarithmetic sequence we mean a set { m , . . . , m n , b } ⊆ Z + , where m <. . . < m n is an arithmetic sequence of positive integers and b is any positiveinteger. We split this study in three different cases: when b < m (inProposition 5.5), when b > m n (in Proposition 5.7) and when m < b < m n (in Proposition 5.10). The key idea to prove these results is to use [4,Theorem 2.2]. There, the authors describe a set of generators of the idealof some projective monomial curves, which, in this context coincide withthe toric ideal of ˜ S , and then we apply Proposition 3.4 with this set ofgenerators. 3n Section § L S is aprincipal ideal. We give a partial answer to this question by providing inCorollary 6.3 an infinite family of numerical semigroups such that L S is aprincipal ideal. This family consists of shiftings of numerical semigroupswith a unique Betti element (a family of semigroups studied in [15]), andgeneralizes three generated numerical semigroups. As an intermediate result,in Proposition 6.2 we describe a explicit set of generators of the toric idealof a family of numerical semigroups which turn to be semigroups with onlyone Betti minimal element (a family of semigroups studied in [14]).When S ⊆ N is a numerical semigroup and L S is not empty, then N −L S is a finite set. In Section § L S as an N P -hard problem. We derivethis result by mimicking the proof of the
N P -hardness of the Frobeniusproblem in [24] and some (easy) considerations. The same ideas also allow usto derive that, for a bounded value k ∈ Z + , computing the largest element ina numerical semigroup with at least k different factorizations (or k differentfactorizations of the same length) is N P -hard.We finish the paper with a conclusions Section § Let S = (cid:104) a , . . . , a n (cid:105) ⊆ Z m be a pointed affine semigroup and consider B = { b , . . . , b s } ⊆ S − { } a finite set of nonzero elements of S . We definethe Ap´ery set of S with respect to B asAp S ( B ) = { x ∈ S | x − b i / ∈ S , ∀ i ∈ { , . . . , s }} . (2)The goal of this section is to study Ap S ( B ) and the main results of thissection are Theorem 2.2 and Proposition 2.7. In the first we describe Ap´erysets in terms of the degrees of the elements of a certain basis of a K -vectorspace. In the second one we characterize when Ap´ery sets are finite. Theproblem of computing the Ap´ery set of an affine semigroup has been studiedin [23, 21]. In [21] the authors provide a method to compute the Ap´ery setof an affine semigroup based on Gr¨obner basis computations. Our Theorem2.2 is similar and is inspired by [21, Theorem 8], the main differences arethe following: In [21], the authors require an extra hypothesis implying thatthe Ap´ery set is finite that we do not assume. In Proposition 2.7 we provethat this extra hypothesis characterizes when Ap S ( B ) is finite. Anotherdifference is that our result does not need any choice of a monomial order.4 third difference is that Theorem 2.2 requires a factorization of b , . . . , b r ,while in [21] they do not require so. This is not a big limitation for us, sincewe are applying this result in Section 3 in a context where we already knowa factorization of the elements of B .To state and prove Theorem 2.2, we first introduce some basics on toricideals. Let K be any field, we denote by K [ x ] = K [ x , . . . , x n ] the ring ofpolynomials in the variables x , . . . , x n with coefficients in K . We write amonomial in K [ x ] as x α = x α · · · x α n n with α = ( α , . . . , α n ) ∈ N n . A pointed affine semigroup S = (cid:104) a , . . . , a n (cid:105) ∈ Z m induces a grading in K [ x ], which is given bydeg S ( x α ) = n (cid:88) i =1 α i a i for α = ( α , . . . , α n ) ∈ N n . We say that a polynomial f ∈ K [ x ] is S -homogeneous if all its monomi-als have the same S -degree. Moreover, an ideal is S -homogeneous if it isgenerated by S -homogeneous polynomials.Associated to S , we have the morphism of K -algebras: ϕ : K [ x ] −→ K [ t ± , . . . , t ± m ] x i (cid:55)−→ t a i = t a i · · · t a im m (3)where K [ t ± , . . . , t ± m ] denotes the Laurent polynomial ring in the variables t , . . . , t m . The toric ideal of S is I S = ker( ϕ ). Remark 2.1
The toric ideal I S has been thoroughly studied in the lit-erature (see, e.g., [26, 27]). For example, it is well known that I S is an S -homogeneous binomial ideal (it is generated by differences of monomials).Moreover, I S is a prime ideal of height ht( I S ) = n − rank( A ), where A isthe m × n matrix with columns a , . . . , a n . This ideal can be computed viaelimination of the variables t , . . . , t m in the following formula I S = (cid:104) x − t a , . . . , x n − t a n (cid:105) ∩ K [ x ] . It is easy to check that x α − x β ∈ I S if and only if deg S ( x α ) = deg S ( x β ),and as a consequence I S = (cid:104) x α − x β | deg S ( x α ) = deg S ( x β ) (cid:105) . (4)5oreover, since S is pointed, then there exists a positive N -grading suchthat I S is homogeneous with respect to this grading, and thus, a gradedversion of Nakayama’s lemma holds. As a consequence, all minimal sets ofbinomial generators of I S have the same number of elements and the same S -degrees.Consider now B = { b , . . . , b s } ⊆ S − { } . Since b i ∈ S , then onecan express b i = (cid:80) nj =1 β ij a j with β i = ( β i , . . . , β in ) ∈ N n . We consider x β i = x β i · · · x β in for all i ∈ { , . . . , s } . Theorem 2.2
Let S = (cid:104) a , . . . , a n (cid:105) be a pointed affine semigroup and let B = { b , . . . , b s } ⊆ S − { } . Set the monomial x β i = x β i · · · x β in ∈ K [ x ] where β i = ( β i , . . . , β in ) ∈ N n is a factorization of b i for all i ∈ { , . . . , s } .If we take D a monomial K -basis of K [ x ] / ( I S + (cid:10) x β , . . . , x β s (cid:11) ) , then themapping h : D −→ Ap S ( B ) x α (cid:55)−→ deg S ( x α ) = α a + . . . + α n a n is bijective. Proof.
We start with the morphism presented in Equation (3), we observethat ϕ is graded with respect to the grading deg S ( x i ) = a i and deg( t j ) =(0 , . . . , , ( j ) , , . . . , I S = ker( ϕ ) and it easy to checkthat Im( ϕ ) is the K -algebra K [ S ] def = K [ t a , . . . , t a n ] = (cid:76) b ∈S K t b . Thus, K [ x ] /I S (cid:39) K [ S ] and we denote by ˜ ϕ the corresponding graded isomorphimof K -algebras.Now we consider (cid:104) t b , . . . , t b s (cid:105) · K [ S ], the ideal in K [ S ] generated by t b , . . . , t b s , and the canonical projection map: π : K [ S ] −→ K [ S ] / ( (cid:104) t b , . . . , t b s (cid:105) · K [ S ]) t α (cid:55)−→ [ t α ]Since ϕ ( x β i ) = t b i , we have that ker( π ◦ ˜ ϕ ) = ( I S + (cid:104) x β , . . . , x β s (cid:105) ) /I S .Thus, by the third isomorphism theorem we have that there is a gradedisomorphism Ψ of K -algebrasΨ : K [ x ] / ( I S + (cid:104) x β , . . . , x β s (cid:105) ) −→ K [ S ] / ( (cid:104) t b , . . . , t b s (cid:105) · K [ S ]) . Moreover, K [ S ] / ( (cid:104) t b , . . . , t b s (cid:105)· K [ S ]) has a unique monomial basis, whichis { t b | b ∈ Ap S ( B ) } . Finally, we observe that the image of a monomialby Ψ is a monomial and hence, the image of any monomial basis D of6 [ x ] / ( I S + (cid:10) x β , . . . , x β s (cid:11) ) has to be { t b | b ∈ Ap S ( B ) } . The result followsfrom the fact that Ψ is graded and Ψ( x α ) = t deg S ( x α ) . (cid:5) Set J := I S + (cid:10) x β , . . . , x β s (cid:11) . To compute D , a monomial K -basis of K [ x ] /J , it suffices to choose any monomial ordering (cid:31) in K [ x ], and define D as the set of all the monomials not belonging to in (cid:31) ( J ), the initial idealof J with respect to (cid:31) . That is, D = { x α | x α / ∈ in (cid:31) ( J ) } . Notice that different monomial orders yield different K -bases. Nevertheless,Theorem 2.2 holds for any of these (and for any other monomial K -basis).Let us illustrate the previous result with an example. Example 2.3
Let S = (cid:104) a , . . . , a (cid:105) ⊆ Z with a = (0 , , a = (1 , , a =(1 , , a = (3 , , a = (4 ,
2) and consider the set B = { b , b , b } ⊆ S ,where b = (3 , , b = (4 , , b = (9 , . A computation with any softwarefor polynomial computations (e.g.,
Singular [12], CoCoA [1] or
Macaulay2 [19]) shows that I S = (cid:104) f , . . . , f (cid:105) with f = x − x x , f = x x − x x , f = x x − x x ,f = x − x x , f = x x − x x , f = x − x x . Let us compute a factorization β i of b i for i ∈ { , , } : b = 3 a , b = a + a , b = 3 a , and set x β = x , x β = x x , x β = x . If one considers L = in (cid:31) ( I S + (cid:104) x , x x , x (cid:105) ) , where (cid:31) is the weighted degreereverse lexicographic order with weights (2 , , , , L = (cid:104) x x , x x , x , x x , x x , x x , x , x x , x (cid:105) . Hence, the monomials which are not in L form the following monomial K -basis of K [ x , . . . , x ] / ( I S + (cid:104) x , x x , x (cid:105) ): D = { x a x c , x c x a | a ∈ N , c ∈ { , , , }} ∪{ x a x x c , x c x x a | a ∈ N , c ∈ { , }} ∪{ x x , x x , x x x , x x , x } . a b b b a a a Figure 1: Ap´ery set Ap S ( B ) in Example 2.3. The dots correspond to theelements in S , the circles to the elements in B and the squares to the ele-ments in Ap S ( B ).Thus, by Theorem 2.2, the Ap´ery set with respect to B is the infinite setAp S ( B ) = { ( i, i + 2 λ ) , ( i + 4 λ, i + 2 λ ) | λ ∈ N , i ∈ { , , , }} ∪{ x + λ (0 , | λ ∈ N , x ∈ { (1 , , (2 , }} ∪{ x + λ (4 , | λ ∈ N , x ∈ { (3 , , (4 , }} ∪{ (3 , , (5 , , (4 , , (6 , } . See Figure 1 for a graphical representation of Ap S ( B ).As a direct consequence of Theorem 2.2 one has that the number ofelements of the Ap´ery set Ap S ( B ) coincides with the dimension of the K -vector space K [ x ] /J . Thus, Ap S ( B ) is finite if and only if K [ x ] /J is 0-dimensional or, equivalently, J ∩ K [ x i ] (cid:54) = (0) for all i ∈ { , . . . , n } . The restof this section is devoted to characterize when this happens. Definition 2.4
Let A = { a , . . . , a n } ⊆ Z n , we define the rational polyhe-dral cone C A as C A = Cone( A ) def = (cid:40) n (cid:88) i =1 α i a i | α i ∈ R ≥ (cid:41) . We say that
F ⊆ C A is a face of C A if there exists w ∈ R m such that x · w ≥ x ∈ C A (where · represents the usual inner product) and8 = { x ∈ C A | x · w = 0 } . An extremal ray of the cone C A is a half-line faceof C A , i.e., a face of the form { α b | α ≥ } with b (cid:54) = . Remark 2.5
We refer the reader to [11] for a fuller treatment of rationalpolyhedral cones. We just mention two properties that are necessary for ourpurposes and that follow from [11, Proposition 1.2.12 and Lemma 1.2.15].1. { } is a face of C A if and only if S ∩ ( −S ) = { } , where S = (cid:104)A(cid:105) .2. Given a set B = { b , . . . , b s } ⊆ C A − { } , then C A = C B if and only iffor each extremal ray r of C A , there exists an i ∈ { , . . . , s } such that b i ∈ r .Before proceeding with the characterization of the finiteness of the Ap´eryset Ap S ( B ), we need a Lemma in which the condition of the semigroup ofbeing pointed plays an important role. Lemma 2.6
Let S = (cid:104) a , . . . , a n (cid:105) ⊆ Z m be a pointed affine semigroup,let B = { b , . . . , b s } ⊆ S − { } . Then, x ∈ S if and only if there exist λ , . . . , λ s ∈ N such that x − λ b − . . . − λ s b s ∈ Ap S ( B ) . Proof.
Since Ap S ( B ) ⊆ S and B ⊆ S , the claim is evident in one direction.So assume that x ∈ S , we will prove that ∃ λ , . . . , λ s ∈ N such that x − s (cid:88) i =1 λ i b i ∈ Ap S ( B ) . (5)By Remark 2.5(1), since S is pointed, then { } is a face of C A . Therefore,there exists w ∈ Z n such that w · x ≥ x ∈ S and if w · x = 0, then x = . Now we prove the lemma by induction on w · x ∈ N . If w · x = 0, then x = ∈ Ap S ( B ) and the result is true for λ = . . . = λ s = 0. Assuming (5)to hold for ˜ x ∈ S such that w · ˜ x < λ (with λ ∈ N and λ > x ∈ S with w · x = λ . We distinguish two cases: if x ∈ Ap S ( B ), thenit suffices to take λ = . . . = λ s = 0. Otherwhise, by definition of the Ap´eryset there exists i ∈ { , . . . , s } such that x − b i ∈ S . Let ˜ x = x − b i . Then w · x = w · b i + w · ˜ x with w · b i > . Thus, w · x > w · ˜ x . We conclude, by the principle of induction, that thereexist β , . . . , β s ∈ N such that ˜ x = (cid:80) sj =1 β j b i ∈ Ap S ( B ), hence x − b i − s (cid:88) j =1 β j b i ∈ Ap S ( B ) . Let I be a nonempty subset of an affine semigroup S , we say that I is an ideal if for every x ∈ I then we have x + S ⊆ I . An ideal I ⊂ S is finitely generated if there exists a finite set B = { b , . . . , b s } such that I = ∪ si =1 b i + S . Clearly, in this setting we have that S − I = r (cid:92) i =1 Ap S ( { b i } ) = Ap S ( B ) . Thus, the complement of Ap S ( B ) in S is just the ideal spanned by B .Now we can proceed with the desired characterization. Interestingly,this result also provides a criterion to determine when I ∪ { } inherits thepointed affine semigroup structure of S , being I a finitely generated idealof S . Theorem 2.7
Let S = (cid:104)A(cid:105) = (cid:104) a , . . . , a n (cid:105) ⊆ Z m be a pointed affine semi-group, let B = { b , . . . , b s } ⊆ S − { } and let I ⊆ S the ideal of S given by I = ∪ si =1 ( b i + S ) . Then, the following statements are equivalent: (1) The Ap´ery set Ap S ( B ) is finite. (2) C A = C B . (3) I ∪ { } is a pointed affine semigroup. Proof. (1) = ⇒ (3) Since I ⊆ S ⊆ Z m , we have that I ∪ { } is pointed, so we justhave to prove that it is a finitely generated monoid. Assuming that Ap S ( B )is finite, that is Ap S ( B ) = { h = , h , . . . , h l } we will prove that I ∪ { } = (cid:104){ h i + b j | ≤ i ≤ l and 1 ≤ j ≤ s }(cid:105) . Let x ∈ I , using Lemma 2.6 we have that there exists λ , . . . , λ s ∈ N such that x − (cid:80) sj =1 λ j b j ∈ Ap S ( B ). That is, there exists i ∈ { , . . . , s } suchthat h i = x − (cid:80) sj =1 λ j b j where not all λ j ’s are zero, since x / ∈ Ap S ( B ).Thus, without loss of generality, one can assume that λ (cid:54) = 0 and we canwrite x = h i + s (cid:88) j =1 λ j b j = h i + b + ( λ − b + s (cid:88) j =2 λ j b j . Thus, x belongs to (cid:104){ h i + b j | ≤ i ≤ l and 1 ≤ j ≤ s }(cid:105) . The otherinclusion is evident. 103) = ⇒ (2) For this part of the proof we are going to use the followingproperty of pointed semigroups: if J is a pointed semigroup and we denote J (cid:63) = J − { } , then the unique minimal system of generators of J is givenby its irreducible elements, that is, J (cid:63) − ( J (cid:63) + J (cid:63) ) . (6)Suppose, contrary to our claim and using Remark 2.5(2), that C A (cid:54) = C B .That is, there exists an extremal ray r of the cone C A such that b i / ∈ r forall i ∈ { , . . . , s } . By Definition 2.4, there exists w ∈ R n such that w · x ≥ x ∈ C A , and if x ∈ C A , then w · x = 0 ⇐⇒ x ∈ r. We define δ = min { w · b i | ≤ i ≤ s } . Note that δ >
0, since b i / ∈ r for all i ∈ { , . . . , s } . We can deduce the following statements:(a) If b ∈ I and w · b = δ , then we claim that b / ∈ I + I and we canconclude by (6) that b belongs to the minimal system of generators of I ∪ { } . Indeed, if b ∈ I + I then, we can write b = b i + s + b j + s with s , s ∈ I and i, j ∈ { , . . . , s } . Hence w · b = w · b i + w · s + w · b j + w · s ≥ δ > δ, a contradiction.(b) If we take b i such that w · b i = δ and a j ∈ r , then w · ( b i + λ a j ) = δ for all λ ∈ N .Using (a) and (b) we have actually showed that the minimal system ofgenerators of I ∪ { } is infinite, which contradicts our assumption.(2) = ⇒ (1). By Theorem 2.2, to prove that Ap S ( B ) is finite it suffices toshow that K [ x ] / ( I S + (cid:104) x β , . . . , x β s (cid:105) ) is a finite dimensional K -vector space.Equivalently, we will show that there exists g i ∈ K [ x i ] such that g i ( x ) ∈ I S + (cid:104) x β , . . . , x β s (cid:105) for all i ∈ { , . . . , n } . In fact, we will see that thereexists γ i ∈ Z + such that x γ i i ∈ I S + (cid:104) x β , . . . , x β s (cid:105) for all i ∈ { , . . . , n } .Since C A = C B and a i ∈ C A , then ∃ ν , . . . , ν s ∈ R ≥ : a i = s (cid:88) j =1 ν j b j . Moreover, (as a consequence of Caratheodory’s theorem) we can assumethat ν , . . . , ν s ∈ Q ≥ . Thus, multiplying by an adequate positive integer ν we deduce that ν a i = s (cid:88) j =1 δ j b j where the δ j ∈ N are not all zero . x νi − (cid:81) sj =1 ( x β j ) δ j ∈ I S and we conclude that x νi ∈ I S + (cid:104) x β , . . . , x β s (cid:105) . (cid:5) From now on, S = (cid:104) a , . . . , a n (cid:105) ⊆ Z m is a pointed semigroup given by itsminimal set of generators, we consider the following subsets of S : L S = { x ∈ S | x has two different factorizations of the same length } . T S = { x ∈ S | x has two different factorizations } . The following lemma, whose proof is easy, shows that both T S and L S are either empty or ideals of S . Lemma 3.1
Let S be a pointed affine semigroup. Then,1. either T S = ∅ or T S is an ideal of S , and2. either L S = ∅ or L S is an ideal of S . Proof.
We just do the proof for L S , being the proof of T S analogue. Let x ∈ L S , then there exists λ = ( λ , . . . , λ n ) , β = ( β , . . . , β n ) ∈ N n such that: x = λ a + . . . λ n a n = β a + . . . β n a n with (cid:96) ( λ ) = (cid:96) ( β ) . Moreover, for any y ∈ S there exist ν = ( ν , . . . , ν n ) ∈ N n such that y = ν a + . . . ν n a n . Thus, x + y = ( λ + ν ) a + . . . + ( λ n + ν n ) a n = ( β + ν ) a + . . . + ( β n + ν n ) a n , with (cid:96) ( λ + ν ) = (cid:96) ( β + ν ) . So x + y ∈ L S . (cid:5) By (4), we have that b ∈ T S if and only if there exists a binomial f ∈ I S with deg S ( f ) = b . The next proposition shows how to obtain the set T S froma set of S -homogeneous generators of I S . Since I S is a binomial ideal onemay consider binomial generating sets of I S ; indeed, all its reduced Gr¨obnerbases consist of binomials. Proposition 3.2
Let { g , . . . , g r } be a binomial generating set of I S . Then, T S = (deg S ( g ) + S ) ∪ . . . ∪ (deg S ( g r ) + S ) . roof. Since g i is a binomial in I S , then it is S -homogeneous and deg S ( g i ) ∈T S . Since T S is an ideal of S , one inclusion holds. To prove the converse,consider b ∈ T S , then there exists f = x λ − x ν ∈ I S with deg S ( λ ) = b .Now, since I S = (cid:104) g , . . . , g r (cid:105) with g i = x α i − x β i and f ∈ I S then, one ofthe binomials of g i divides x λ . That is, x λ = x α i x γ for some γ ∈ N m andsome i ∈ { , . . . , r } . Or equivalently, λ = α i + γ for some γ ∈ N m and some i ∈ { , . . . , r } . Thus, b = deg S ( λ ) = deg S ( α i + γ ) = deg S ( g i ) + γ a + . . . + γ n a n (cid:124) (cid:123)(cid:122) (cid:125) s ∈S . (cid:5) One clearly has that L S ⊂ T S . In the following result we are going tosee how one can obtain L S by means of T ˜ S for a certain affine semigroup ˜ S .More precisely, consider the following pointed affine subsemigroup of Z m +1 associated to S : ˜ S = (cid:104) ( a , , ( a , , . . . , ( a n , (cid:105) ⊆ Z m +1 . (7)Note that { ( a , , ( a , , . . . , ( a n , } is the minimal set of generators of ˜ S .The idea under considering the semigroup ˜ S comes from the fact thatthe toric ideal I ˜ S is generated by the homogeneous binomials in I S (see,e.g., Remark 2.1). Moreover, we will exploit the fact that factorizations ofthe same length of an element in S correspond to homogeneous binomialsin I S and, thus, to binomials in I ˜ S . These ideas in the particular context ofnumerical semigroups have been extensively used in the study of the shiftedfamily of a numerical semigroup (see, e.g., [28, 10]). Lemma 3.3
Let S = (cid:104) a , . . . , a n (cid:105) ⊆ Z m be a pointed semigroup and ˜ S thesemigroup defined as above. Then, L S = (cid:8) x ∈ Z m | ( x , x ) ∈ T ˜ S for some x ∈ N (cid:9) . Proof.
Let x ∈ L S . Then, there exist λ , β ∈ N n such that x = λ a + . . . + λ n a n = β a + . . . + β n a n with (cid:96) ( λ ) = (cid:96) ( β ) = s ∈ N . Thus, ( x, s ) ∈ Z m +1 and( x, s ) = λ ( a ,
1) + . . . + λ n ( a n , β ( a ,
1) + . . . + β n ( a n , ,
13r equivalently, ( x, s ) ∈ T ˜ S . The other inclusion may be handled in the sameway. (cid:5) Now, we can state the following proposition, which allows us to obtain L S from the degrees of a set of generators of the ideal I ˜ S . Proposition 3.4
Let ˜ S ⊆ Z m +1 be the semigroup associated to S ⊆ Z m defined as above. Then L S = ∅ if and only if I ˜ S = (0) . Moreover, if { g , . . . g s } is a binomial generating set of I ˜ S . Then, L S = (deg S ( g ) + S ) ∪ . . . ∪ (deg S ( g s ) + S ) . Proof.
Assume that I ˜ S (cid:54) = (0) and let h = x α − x β ∈ I ˜ S . Then deg ˜ S ( x α ) =deg ˜ S ( x β ) and we observe thatdeg ˜ S ( h ) = deg ˜ S ( x α ) = ( a , α + . . . + ( a n , α n == ( α a + . . . + α n a n (cid:124) (cid:123)(cid:122) (cid:125) deg S ( h ) ⊆ N m , α + . . . + α n (cid:124) (cid:123)(cid:122) (cid:125) (cid:96) ( α ) ) . Thus, deg S ( h ) ∈ L S . Conversely, if one takes b ∈ L S one can easily cons-truct a binomial in I ˜ S .We have that L S = (cid:8) x ∈ Z m | ( x , x ) ∈ T ˜ S for some x ∈ N (cid:9) by Lemma3.3. Thus applying Proposition 3.2 with ˜ S we conclude the proof. (cid:5) As a consequence of Proposition 3.4, we get the main result of this sec-tion. This result describes the set
S − L S as a particular Ap´ery set of S . Theorem 3.5
Let
S ⊆ Z m be a pointed affine semigroup and { g , . . . , g s } a binomial generating set of I ˜ S . Consider B = { b , . . . , b s } with b i :=deg S ( g i ) for all i ∈ { , . . . , s } . Then, S − L S = Ap S ( B ) . Proof.
By Proposition 3.4 we have that L S = (cid:83) si =1 ( b i + S ). Therefore S − L S = s (cid:92) i =1 Ap S ( { b i } ) = Ap S ( B ) . (cid:5) This result together with Theorem 2.2 gives an algorithm to determineall the elements of
S − L S . More precisely, consider { g , . . . , g s } a bino-mial generating set of I ˜ S and denote g i = x α i − x β i for all i ∈ { , . . . , s } S ( B ) being B = { b , . . . , b s } with b i = deg S ( g i ). In order to use Theorem 2.2 as it is stated, one needsa factorization of b , . . . , b s . Nevertheless, this does not involve any extracomputations. Indeed, I ˜ S is an S -homogeneous ideal and, hence, α i and β i are two factorization of b i for all i ∈ { , . . . , s } .Let us illustrate this with an example. Example 3.6
Consider, as in Example 2.3, the affine semigroup S = (cid:104) a , . . . , a (cid:105) ⊆ Z with a = (0 , , a = (1 , , a = (1 , , a = (3 , , a = (4 ,
2) and let uscompute L S and S − L S . For this purpose, we first consider I ˜ S with ˜ S = (cid:104) (0 , , , (1 , , , (1 , , , (3 , , , (4 , , (cid:105) . It turns out that I ˜ S is minimallygenerated by { g , g , g , g } , where: g = x − x x , g = x x − x x , g = x − x x , g = x x − x x . If one sets B = { b , b , b , b } with b i := deg S ( g i ) then one gets that b = 3 a = (3 , , b = a + a = (4 , , b = 3 a = (9 ,
6) and b = a + 2 a = (6 , L S = ∪ i =1 ( b i + S ) = ((3 ,
6) + S ) ∪ ((4 ,
4) + S ) ∪ ((9 ,
6) + S ) ∪ ((6 ,
6) + S ) . Moreover, since b = (4 ,
4) + (2 , ∈ b + S , we get L S = ∪ i =1 ( b i + S ) = ((3 ,
6) + S ) ∪ ((4 ,
4) + S ) ∪ ((9 ,
6) + S ) . Thus, setting B (cid:48) = { b , b , b } we have that S − L S = Ap S ( B (cid:48) ) and this setis exactly the one we computed in Example 2.3. So the squared grid pointsin Figure 1 correspond to the elements of S − L S .As a direct consequenece of Theorems 2.2 and 3.5, we have that. Corollary 3.7
Let
S ⊆ Z m , be a pointed affine semigroup. Then, (cid:93) ( S − L S ) = dim (cid:0) K [ x ] / ( I S + in (cid:31) ( I ˜ S )) (cid:1) , where in (cid:31) ( I ˜ S ) represents the initial ideal of I ˜ S with respect to any monomialorder. S notbeloging to L S . It is also worth mentioning that this happens if and only if L S ∪ { } inherits the affine monoid structure of S . Corollary 3.8
Let S = (cid:104)A(cid:105) ⊆ Z m be a pointed affine semigroup. Then, thefollowing are equivalent: (1) S − L S is a finite set. (2) for every extremal ray r of C A there are (at least) three elements of A in r . (3) L S is a pointed affine semigroup. Proof.
Being (1) and (3) equivalent by Theorem 2.7, we are going to provethe equivalence between (1) and (2). Let I ˜ S = ( g , . . . , g s ) with g i binomialsand B = { b , . . . , b s } , where b i := deg S ( g i ). By Theorem 3.5 we have that S − L S = Ap S ( B ). Thus, by Proposition 2.7 and Remark 2.5, we havethat S − L S if finite if and only if for every extremal ray of C A there is atleast one element of B . So it just remains to prove that this happens if andonly if there are (at least) three elements of A in r . Take r an extremal ray.First assume that there are (at least) three elements of A in r , say a , a , a .Then taking R = { ( a , , ( a , , ( a , } we have that I R is a height onetoric ideal (see Remark 2.1). Then, there is a binomial f ∈ I R ⊆ I ˜ S and,thus, deg S ( f ) ∈ (cid:104) a , a , a (cid:105) ⊆ r . As a consequence, one of the monomialsappearing in g , . . . , g s can only involve the variables x , x , x and, thusthe S -degree of the corresponding g i belongs to (cid:104) a , a , a (cid:105) ⊆ r . Conversely,if b i in r , then we have that g i = x α i − x β i ∈ I ˜ S and we may assume that x α i and x β i are relatively prime. Since g i is homogeneous and a , . . . , a n are all different, we have that there are at least three variables involved in g i . Moreover, since b i = (cid:80) ni =1 α ij a j = (cid:80) ni =1 β ij a j and r is an extremal ray,we have that a j ∈ r whenever α ij (cid:54) = 0 or β ij (cid:54) = 0. We conclude, thus, thatthere are at least three a i in r , finishing the proof. (cid:5) In this section we apply the results of the previous sections to the setting ofnumerical semigroups. We take S = (cid:104) a , . . . , a n (cid:105) ⊂ N a numerical semigroupgiven by its minimal generating set and we denote by F ( S ) the Frobeniusnumber of S , which is the largest integer not in S , i.e., F ( S ) = max( Z \ S ).16 emma 4.1 Let w ∈ L S , then for all the integers z such that z > w + F ( S ) ,we have that z ∈ L S . Proof.
It suffices to use Lemma 3.1 and the definition of F ( S ) to showthat the assertion follows. (cid:5) Lemma 4.2
Let S = (cid:104) a , . . . , a n (cid:105) ⊆ N be a numerical semigroup then, L S ∪ { } is a numerical semigroup. Proof.
First, by Lemma 3.1 we have that L S ∪ { } is a submonoid of N . Moreover using Lemma 4.1 it is immediate that L S ∪ { } has finitecomplement in N . (cid:5) The results of the article [9] can be deduced aplying Proposition 3.4 inthe setting of numerical semigroups.
Corollary 4.3
Let S = (cid:104) a , . . . , a n (cid:105) ⊆ N be a numerical semigroup givenby its minimal set of generators. • L S = ∅ if and only if n ≤ . • If n = 3 , then L S = ( a ( a − a ) / gcd( a − a , a − a )) + S . Proof.
The height of the ideal I ˜ S is max(0 , n − I ˜ S = (0) if and only if n ≤
2. If n = 3, then I ˜ S is the principal ideal I ˜ S = (cid:16) x ( a − a ) /d − x ( a − a ) /d x ( a − a ) /d (cid:17) , with d := gcd( a − a , a − a ). Thus, by Proposition 3.4, we conclude that L S = deg S (cid:16) x ( a − a ) /d (cid:17) + S = ( a ( a − a ) /d ) + S (cid:5)
Example 4.4
Let S = (cid:104) , , (cid:105) ⊆ N be a numerical semigroup. It is easyto check that I ˜ S = (cid:104) g (cid:105) with g = x − x x . If we set B = { b } with b = deg S ( g ) = 10 then by Proposition 3.4 we have that L S = b + S = 10 + S .This result coincides with the statement of Corollary 4.3. Moreover, byTheorem 3.5 we have that S − L S = Ap S ( B ) with B = { } , that isAp S ( B ) = { , , , , , , , , , } . S ( B ) is by applying Theorem 2.2. That is,Ap S ( B ) is in bijection with D , a monomial K -basis of K [ x , x , x ] /J with J = I S + (cid:104) x β (cid:105) , where β is a factorization of b . In our particular case, I S = (cid:104) x − x , x − x x (cid:105) and, since b = 2 ·
5, then we set β = (0 , ,
0) and x β = x . We consider the degree reverse lexicographic order ≺ and define D as the set of monomials not belonging toin ≺ ( J ) = in ≺ ( I S + (cid:104) x (cid:105) ) = { x , x , x x , x } , that is, D = { , x , x , x , x x , x x , x , x x , x , x x } . We conclude that S − L S is the set of S -degrees of the monomials is D . L S when S is generated by an almostarithmetic sequence In this section we will focus our attention on computing L S in the particularcase of numerical semigroups generated by an almost arithmetic sequence.As a warm-up we begin with the case of arithmetic sequences.Let S be a numerical semigroup generated by an arithmetic sequenceof relative primes, i.e., S = (cid:104) m , . . . , m n (cid:105) ⊆ N where m < . . . < m n isan arithmetic sequence and we assume that gcd( m , . . . , m n ) = 1. In otherwords, m i = m + ( i − e with gcd( m , e ) = 1 for all i ∈ { , . . . , n } . (8)An almost arithmetic sequence is a sequence in which all but one of theelements form an arithmetic sequence. Thus, an arithmetic sequence iscertainly an almost arithmetic sequence. Remark 5.1
This remark will be frequently used throughout this section.Let S be a numerical semigroup generated by the set A = { a , . . . , a n } ⊆ N with a < . . . < a n . Then, as already mentioned in Section 3 we can definethe pointed affine subsemigroup ˜ S of N associated to S as˜ S = (cid:104) ( a , , . . . , ( a n , (cid:105) ⊆ N . The following operations allow us to define semigroups
T ⊆ N defining thesame toric ideal as I ˜ S ⊆ K [ x , . . . , x n ]. • Adding to each element of A the same scalar λ ≤ a , λ ∈ N . If wedefine a (cid:48) i = a i − λ with λ ∈ N and λ ≤ a for all i ∈ { , . . . , n } and weconsider T = (cid:104) ( a (cid:48) , , . . . , ( a (cid:48) n , (cid:105) ⊆ N . Then, I T = I ˜ S .18 Subtracting to each element of A the same scalar λ ≥ a n , λ ∈ N .If we define a (cid:48) i = λ − a i with λ ∈ N and λ ≥ a n and we consider therenaming of variables x i (cid:55)−→ y n +1 − i . Then, I T ⊆ K [ y , . . . , y n ] is equalto I ˜ S ⊆ K [ x , . . . , x n ] (after a renaming of the variables). • Multiplying and dividing all the elements of A by the same scalar.If we define a (cid:48) i = a i λ with λ ∈ N any divisor of d = gcd( a , . . . , a n ) andwe consider T = (cid:104) ( a (cid:48) , , . . . , ( a (cid:48) n , (cid:105) ⊆ N . Then, I T = I ˜ S .A similar property can be deduced if we multiply each element of A by a constant λ ∈ N . Proposition 5.2
Let S = (cid:104) m , . . . , m n (cid:105) ⊆ N be a numerical semigroupgenerated by an arithmetic sequence of relative primes as in Equation (8) .Then L S = { m + λe | ≤ λ ≤ n − } + S , being e := m − m the difference of the arithmetic sequence. Proof.
We define m (cid:48) i = m i − m = ( i − e with gcd( m , e ) = 1 and m (cid:48)(cid:48) i = m (cid:48) i e . Then, by Remark 5.1, we have that I ˜ S = I T = I T with T = (cid:104) (0 , , ( m (cid:48) , , . . . , ( m (cid:48) n , (cid:105) = (cid:104) (0 , , ( e, , . . . , (( n − e, (cid:105) ⊆ N T = (cid:104) (0 , , ( m (cid:48)(cid:48) , , . . . , ( m (cid:48)(cid:48) n , (cid:105) = (cid:104) (0 , , (1 , , . . . , ( n − , (cid:105) ⊆ N . Moreover, I T is known to be the defining ideal of the rational normal curvein P n − K of degree n −
1. Indeed, I T = (cid:104) x i x j − x i − x j +1 | ≤ i ≤ j ≤ n − (cid:105) .Thus, if we apply Proposition 3.4 we obtain L S from the set of generatorsof the ideal I ˜ S . That is, L S = { m i + m j | ≤ i ≤ j ≤ n − } + S = { m + λe | ≤ λ ≤ n − } + S . (cid:5) Example 5.3
Let S = (cid:104) m , m , m , m , m (cid:105) ⊆ N be a numerical semigroupgenerated by an arithmetic sequence of relative primes with m = 10, m =13, m = 16, m = 19 and m = 22 where m i = m + 3( i −
1) for all i ∈ { , . . . , } and gcd( m ,
3) = 1 . (9)The ideal I ˜ S is minimally generated by { g , . . . , g } where: g = x − x x , g = x x − x x , g = x x − x x ,g = x − x x , g = x x − x x , g = x − x x .
19f one sets B = { b , . . . , b } with b i = deg S ( g i ) and we apply Proposition3.4, we have that L S = { m , m + m , m + m , m , m + m , m } + S = { , , , , } + S . Which coincides with the statement of Proposition 5.2.
Remark 5.4
Let A = (cid:104) m , . . . , m n (cid:105) ⊆ N be an arithmetic sequence ofrelative primes and consider the following pointed affine subsemigroup T = (cid:104) (0 , , ( m , , . . . , ( m n , (cid:105) ⊆ N . A binomial generating set of I T is known (see, e.g., [4, Theorem 2.2]). Thatis, I T = (cid:104) x i x j + x i − x j +1 | ≤ i ≤ j ≤ n − (cid:105) + (cid:104) x α x i − x n − k + i x α − en x en +1 | ≤ i ≤ k (cid:105) (10)where the pair ( α, k ) ∈ N is defined as follows: • k is the only integer such that k ≡ − m n mod ( n −
1) and 1 ≤ k ≤ n −
1, and • α = (cid:106) m n − n − (cid:107) ∈ N (where (cid:98)·(cid:99) denotes the floor function).In the rest of the section we will focus on the case of numerical semigroupsgenerated by an almost arithmetic secuence, i.e. S = (cid:104) m , . . . , m n , b (cid:105) ⊆ N with m i = m + ( i − e with gcd( m , e ) = 1 ∀ i ∈ { , . . . , n } and b ∈ N . (11)We distinguish three cases: b < m (Proposition 5.5), b > m n (Proposition5.7) and m < b < m n (Proposition 5.10). Proposition 5.5
Let S = (cid:104) b, m , . . . , m n (cid:105) ⊆ N be a numerical semigroupgenerated by an almost arithmetic sequence with b < m . We define • d = gcd( m − b, e ) , • β = (cid:106) m n − b − dd ( n − (cid:107) If d ( n − divides m n − b , then L S = ( { m + λe | ≤ λ ≤ n − } + S ) ∪ (( β + 1) m + S ) (cid:124) (cid:123)(cid:122) (cid:125) D . Otherwise, L S = D ∪ ((( β + 1) m + e ) + S ) . roof. We define m (cid:48) i = m i − b with i ∈ { , . . . , n } and m (cid:48)(cid:48) i = m (cid:48) i d with d = gcd( m (cid:48) , . . . , m (cid:48) n ). Now, by Remark 5.1 we know that I ˜ S = I T where T = (cid:104) (0 , , ( m (cid:48)(cid:48) , , . . . , ( m (cid:48)(cid:48) n , (cid:105) ⊆ N . Moreover, it is easy to check that m (cid:48)(cid:48) < . . . < m (cid:48)(cid:48) n is an arithmetic sequence ofrelative primes. Thus, we can apply Remark 5.4 to obtain a set of generatorsof the ideal I T = I ˜ S = (cid:104) g , . . . , g s (cid:105) and then, Proposition 3.4 to obtain L S .In fact, if we set l ≡ b − m n + dd mod ( n −
1) with l ∈ { , . . . , n − } , then L S = s (cid:91) i =1 (deg S ( g i ) + S )= ( { m i + m j | ≤ i ≤ j ≤ n − } + S ) (cid:124) (cid:123)(cid:122) (cid:125) D ∪ ( { βm + m i | ≤ i ≤ l } + S )Moreover, observe that for i ≥
3, then m + m i = m + m i − . Thus, βm + m i = ( β − m + m + m i − ∈ D With this observation the above formula for L S can be simplified as follows: • If l = 1 (or, equivalently, d ( n −
1) divides m n − b ), then, L S = D ∪ (( β + 1) m + S ) . • If l (cid:54) = 1, then, L S = D ∪ ( { ( β + 1) m , ( β + 1) m + e } + S ) . (cid:5) Example 5.6
Let S = (cid:104) b, m , m , m , m , m (cid:105) be a numerical semigroupgenerated by an almost arithmetic sequence with b = 7, m = 17, m = 20, m = 23, m = 26 and m = 29. Note that the set { m , . . . , m n } with n = 5 elements is an arithmetic sequence of relative primes being e = 3 thedifference between two consecutive terms. We define d = gcd( m − b, e ) = 1 and β = (cid:106) m n − b − dd ( n − (cid:107) = 5 , and observe that d ( n −
1) does not divide m n − b . Then, by Proposition 5.5we have that L S = ( { , , , , } + S ) ∪ ( { , } + S )= { , , , , , , } + S The same conclusion is obtained if we compute a minimal set of gneeratorsof I ˜ S and then apply Proposition 3.4.21 roposition 5.7 Let S = (cid:104) m , . . . , m n , b (cid:105) ⊆ N be a numerical semigroupgenerated by an almost arithmetic sequence. If b > m n , we define • d = gcd( b − m , e ) , • β = (cid:106) b − m − dd ( n − (cid:107) .If d ( n − divides b − m , then L S = ( { m + λe | ≤ λ ≤ n − } + S ) (cid:124) (cid:123)(cid:122) (cid:125) D ∪ (( β + 1) m n + S ) . Otherwise, L S = D ∪ ( { ( β + 1) m n , ( β + 1) m n − e } + S ) . Proof.
The proof of this proposition is similar to that of Proposition 5.7,but taking into account that the transformations that must be carried outnow are: First we define m (cid:48) i = b − m i with i ∈ { , . . . , n } and then, m (cid:48)(cid:48) i = m (cid:48) i d with d = gcd( m (cid:48) , . . . , m (cid:48) n ). Now, by Remark 5.1 we know that I ˜ S is equalto the ideal I T after a renaming of the variables x i (cid:55)−→ y n +1 − i . where T = (cid:104) ( m (cid:48)(cid:48) , , . . . , ( m (cid:48)(cid:48) n , , (0 , (cid:105) ⊆ N We can now proceed analogously to the proof of Proposition 5.5 to get thefinal result. (cid:5)
Example 5.8
Let S = (cid:104) m , m , m , m , m , b (cid:105) be a numerical semigroupgenerated by an almost arithmetic sequence with b = 31, m = 17, m = 20, m = 23, m = 26 and m = 29. Note that the set { m , . . . , m n } with n = 5 elements is an arithmetic sequence of relative primes being e = 3 thedifference between two consecutive terms. We define d = gcd( b − m , e ) = 1 and β = (cid:106) b − m − dd ( n − (cid:107) = 3 . and observe that d ( n −
1) does not divide b − m . Then, by Proposition 5.7we have that L S = ( { , , , , } + S ) ∪ ( { , } + S )= { , , , , , , } + S The same conclusion is obtained if we compute a minimal set of gneeratorsof I ˜ S and then apply Proposition 3.4.22or proving our next result, we use the following fact, which is a directconsquence of [5, Lemma 2.1 and Proposition 2.2]. Remark 5.9
Consider the semigroups T = (cid:104) (0 , , ( a , , . . . , ( a n , , ( b, (cid:105) ⊆ Z T = (cid:104) (0 , , ( a , , . . . , ( a n , (cid:105) ⊆ Z where a , . . . , a n , b ∈ Z + are relatively prime. We set B = gcd( a , . . . , a n ) , if B · b = n (cid:88) i =2 α i a i for some α i ∈ N such that n (cid:88) i =1 α i ≤ B ; then I T = I T · K [ x , . . . , x n +1 ] + (cid:104) x Bn +1 − x B − (cid:80) ni =1 α i n (cid:89) i =2 x α i i (cid:105) ⊆ K [ x , . . . , x n +1 ] . Proposition 5.10
Let S = (cid:104) m , . . . , m n , b (cid:105) ⊆ N be a numerical semigroupgenerated by an almost arithmetic sequence. If m < b < m n , then L S = ( B · b + S ) ∪ ( { m + λe | ≤ λ ≤ n − } + S ) with B = e gcd( e,b − m ) Proof.
By Remark 5.1 we know that I ˜ S = I S where S = (cid:104) (0 , , ( B, , . . . , (( n − B, , ( c, (cid:105) , being c = b − m gcd( e,b − m ) and B = e gcd( e,b − m ) = gcd( B, B, . . . , ( n − B ).Let us find explicit α i ∈ { , . . . , n } such that B · c = n − (cid:88) i =1 α i · i · B with n − (cid:88) i =1 α i ≤ B. We take s ∈ { , . . . , n − } such that m s < b < m s +1 ; then ( s − B < c < sB .Performing euclidean division we get c = µs + r with 1 ≤ µ < B and r ∈ { , . . . , s − } . Then, Bc = µ ( sB ) + ( rB ) and µ + 1 ≤ B .Now applying Remark 5.9 we have that I S = I S · K [ x , . . . , x n +1 ] + (cid:104) x Bn +1 − x B − µ − x r +1 x µs +1 (cid:105) S = (cid:104) (0 , , ( e, , . . . , (( n − e, (cid:105) . Moreover, applying again Remark5.1 we know that I S = I S with S = (cid:104) (0 , , (1 , , . . . , ( n − , (cid:105) ⊆ N . Since the toric ideal of S is I S = (cid:104) x i x j − x i − x j +1 | ≤ i ≤ j ≤ n − (cid:105) , we can finally apply Proposition 3.4 to obtain L S from the set of generatorsof the ideal I ˜ S . Thus, L S = ( B · b + S ) ∪ ( { m + λe | ≤ λ ≤ n − } + S ) . Moreover, we have seen that B · b = ( B − µ − m + m r +1 + µm s +1 ∈ m r +1 + m s +1 + S . If m < b < m n − , then 3 ≤ s + 1 ≤ n − ≤ r + 1 ≤ n −
2, hence m r +1 + m s +1 ∈ { m + λe | ≤ λ ≤ n − } , so in this case L S = { m + λe | ≤ λ ≤ n − } + S . (cid:5) Example 5.11
Let S = (cid:104) m , m , m , m , m , b (cid:105) be a numerical semigroupgenerated by an almost arithmetic sequence with b = 19, m = 17, m = 21, m = 25, m = 29 and m = 33. Note that the set m , . . . , m is anarithmetic sequence of difference e = 4. We define B = e gcd( e, b − m ) = 2Then, by Proposition 5.7 we have that L S = (2 b + S ) ∪ ( { m + λe | ≤ λ ≤ } + S )= { , , , , , } + S The same conclusion is obtained if we compute a minimal set of gneeratorsof I ˜ S and then apply Proposition 3.4.24 When is L S a principal ideal? In [9], the authors proved that when S = (cid:104) a , a , a (cid:105) is a three-generatednumerical semigroup, then L S is a principal ideal. In Corollary 4.3, weprovided another proof of the same fact. The idea in our proof is that I ˜ S is a height one ideal and, thus, it is principal. As a consequence, this proofcan be generalized to pointed affine semigroups S = (cid:104) a , . . . , a n (cid:105) ⊆ Z m asfar as I ˜ S is a height one ideal (see also Proposition 3.4). Moreover, followingRemark 2.1, this happens if and only if the rank of the ( m + 1) × n matrixwith columns ( a i ,
1) is n −
1. However, this is not the only situation inwhich L S is a principal ideal, as a consequence of Proposition 3.4 we havethe following. Proposition 6.1
Let S be a pointed affine semigroup and take { g , . . . g r } a binomial generating set of I ˜ S . Then, L S is a principal ideal if and onlyif there exists an i ∈ { , . . . , r } such that deg S ( g j ) ∈ deg S ( g i ) + S for all j ∈ { , . . . , r } . It would be interesting to characterize the affine semigroups, or at leastthe numerical semigroups, such that L S is a principal ideal. One nice featureof numerical semigroups with this property is the following. If L S = e + S with e ∈ Z + , we have that the maximum element not in L S and F ( S ), theFrobenius number of S , are related by max { b ∈ Z | b / ∈ L S } = e + F ( S ).The goal of this section is to provide a partial solution to this question byproviding a family of numerical semigroups such that L S is a principal ideal.We observe that the condition in Proposition 6.1 can be restated asfollows: if one considers ≤ S the partial order y ≤ S z if and only if z − y ∈ S ;then, the set of S -degrees of the generators of I ˜ S has a minimum element.This condition for ˜ S is slightly more general than the one of being an affinesemigroup with one Betti minimal element , explored in [14]. In this section,we build on some ideas of [14, Section 7] to provide in Corollary 6.3 a familyof numerical semigroups such that L S is a principal ideal. The family weare proposing includes the one of three generated numerical semigroups. Proposition 6.2
Let S = (cid:104) b, b + tm , . . . , b + tm n (cid:105) be a numerical semi-group, where b, t ∈ Z + , n ≥ and m i = f i (cid:81) j ∈{ ,...,n } j (cid:54) = i c j ; being (a) c , . . . , c n ∈ N pairwise relatively prime, (b) gcd( f i , c i ) = 1 for all i ∈ { , . . . , n } , (c) m n > m i for all i ∈ { , . . . , n − } , and f n = 1 .Then L S = (cid:83) n − i =1 ( c i ( b + tm i ) + S ) . Proof.
We will prove this result by means of Proposition 3.4. For thispurpose, we are obtaining a generating set for I ˜ S . By Remark 5.1 we havethat I ˜ S = I T , where T = (cid:104) (0 , , ( m , , . . . , ( m n , (cid:105) . We observe thatgcd( m , . . . , m n ) = 1, and for all i ∈ { , . . . , n − } we have thatgcd( m , . . . , m i − , m i +1 , . . . , m n ) m i = c i m i = f i c n m n , and f i c n < c i (because m i < m n ). Thus, applying Remark 5.9, we have that I T = (cid:104) x c i i +1 − x c i − f i c n x f i c n n | ≤ i ≤ n − (cid:105) . Since deg S ( x c i i +1 ) = c i ( b + tm i )for i ∈ { , . . . , n − } , by Proposition 3.4 we are done. (cid:5) Now, we apply this result to a subfamily of the numerical semigroupsdescribed in Proposition 6.2, where we can conclude that L S is a principalideal. Indeed, the family we are considering corresponds to setting f i = 1for all i ∈ { , . . . , n } . We have, hence, that the semigroup we consider S belongs to the so-called shifted family of S (cid:48) = (cid:104) m , . . . , m n (cid:105) , where S (cid:48) is a numerical semigroup with a unique Betti element ; we refer the reader to [15]for more on semigroups with a unique Betti element. Corollary 6.3
Let S = (cid:104) b, b + tm , . . . , b + tm n (cid:105) be a numerical semigroup,where b, t ∈ Z + and m i = (cid:81) j ∈{ ,...,n } j (cid:54) = i c j ; being c , . . . , c n ∈ N pairwise rela-tively prime. Then, L S = c n − ( b + tm n − ) + S . Proof.
Clearly we are under the hypotheses of Proposition 6.2 with f i = 1for all i ∈ { , . . . , n } . Set D i := c i ( b + tm i ) for all i ∈ { , . . . , n − } . Weassume without loss of generality that m < · · · < m n or, equivalently, that c > c > · · · > c n . To get the result it suffices to prove that D i ∈ D n − + S or, equivalently, that D i − D n − ∈ S for all i ∈ { , . . . , n − } . Take i ∈{ , . . . , n − } , we have that D i − D n − = ( c i − c n − ) b + t ( c i m i − c n − m n − ) = ( c i − c n − ) b ∈ S (cid:5) Let us illustrate this result with an example.
Example 6.4
Let S = (cid:104) , , , (cid:105) , we have that S satisfies the hypothe-ses of Proposition 6.1 with b = 17 , t = 2 , n = 3 , c = 5 , c = 3 and c = 2.Thus, L S = (3 ·
37) + S = 111 + S . 26ndeed, as we proved in Proposition 6.2 and Corollary 6.3, I ˜ S = (cid:104) g , g (cid:105) with g = x − x x and g = x − x x and we have that deg S ( g ) ∈ deg S ( g ) + S , because deg S ( g ) = 3 ·
37 = 111 , deg S ( g ) = 5 ·
29 = 145 and145 = 111 + 2 · ∈
111 + S .Moreover, since the Frobenius number of S is F ( S ) = 107, we have thatmax { b ∈ Z | b / ∈ L S } = 111 + 107 = 218.One could build further families of numerical semigroups such that L S is a principal ideal by choosing appropriate values of f , . . . , f n − in Propo-sition 6.2.We observe that Corollary 6.3 includes the case of three generated nu-merical semigroups and, hence, generalizes the formula obtained in Corollary4.3. Indeed the numerical semigroup S = (cid:104) a , a , a (cid:105) with a < a < a cor-responds to b = a , n = 2, t = gcd( a − a , a − a ), m = c = ( a − a ) /t and m = c = ( a − a ) /t and, in this context, we have that L S = c ( b + tm ) + S = ( a ( a − a ) / gcd( a − a , a − a )) + S . max { b ∈ Z | b / ∈ L S } for S a numericalsemigroup is N P -hard
Let S = (cid:104) a , . . . , a n (cid:105) be a numerical semigroup, as we saw in Corollary4.3, then L S = ∅ if and only if n ≤
2. Thus, when n ≥
3, by Lemma4.1 if follows that N − L S is a finite set. Hence, for n ≥ F ,(cid:96) = max { b ∈ Z | b / ∈ L S } is well defined.The goal of this short section is to show that the problem of comput-ing the largest element in Z − L S is an N P -hard problem, under Turingreductions.In [24] (see also [25, Theorem 1.3.1]), Ram´ırez Alfons´ın proves that theproblem of determining the Frobenius problem is
N P -hard. His proof con-sists of a Turing reduction from the Integer Knapsack Problem (IKP), whichis well-known to be an
N P -complete problem (see, e.g., [22, page 376]). TheIKP is a decision problem that receives as input ( a , . . . , a n ) ∈ N n , t ∈ N and asks if there exist x , . . . , x n ∈ N such that (cid:80) ni =1 x i a i = t . We definehere a related decision problem, we call this problem IKP ,(cid:96) : • Input: ( a , . . . , a n ) ∈ N n , t ∈ N , and • Question: do there exist distinct ( x , . . . , x n ) , ( y , . . . , y n ) ∈ N n suchthat (cid:80) ni =1 x i a i = (cid:80) ni =1 y i a i = t and (cid:80) ni =1 x i = (cid:80) ni =1 y i ?27he easy fact thatIKP(( a , . . . , a n ) , t ) = True ⇔ IKP ,(cid:96) (( a , . . . , a n , a , . . . , a n ) , t ) = True ,implies that IKP ,(cid:96) is an N P -hard problem.Moreover, a careful inspection of the proof of [25, Theorem 1.3.1] showsthat if we replace IKP by IKP ,(cid:96) , and F ( (cid:104) a , . . . , a n (cid:105) ) by F ,(cid:96) ( (cid:104) a , . . . , a n (cid:105) )the proof also holds. This fact together with the N P -hardness of IKP ,(cid:96) yields the following: Proposition 7.1
Let S = (cid:104) a , . . . , a n (cid:105) be a numerical semigroup with n ≥ . The problem of computing F ,(cid:96) ( S ) = max { b ∈ Z | b / ∈ L S } is N P -hard.
We finally remark that one can define F i ( S ) = max { b ∈ Z | b has not i factorizations } , andF i,(cid:96) ( S ) = max { b ∈ Z | b has not i factorizations of the same lenght } and, following the same argument presented here, one can prove that thecomputational problem of computing F i ( S ) or F i,(cid:96) ( S ) for bounded values of i are all N P -hard.
This paper is devoted to the study of the set L S of elements having (at least)two factorizations of the same length in an affine monoid S . In Proposition3.4, we prove that L S is the ideal in S spanned by the S -degrees of thegenerators of I ˜ S , a homogeneous ideal associated to S . As a consequence,in Theorem 3.5, we get that the set S − L S coincides with an Ap´ery set of S and we provide a Gr¨obner basis approach to compute this set (based on2.2) and characterize when this set is finite (Corollary 3.8).When one has an explicit expression of the generators of the toric ideal I ˜ S , then one can also describe L S . This is the case of S being a numericalsemigroup generated by an almost arithmetic sequence (see Proposition 5.5,Proposition 5.7 and Proposition 5.10) or a shifting of a semigroup with aunique Betti element (Corollary 6.3). In this last family, it turns out that L S is a principal ideal. It would be interesting to characterize when L S is a principal ideal, a first result in this direction is Proposition 6.1. Wefinish by proving in Proposition 7.1 that for a numerical semigroup, thecomputational problem of determining the maximum integer not in L S isan N P -hard problem. 28he main idea in this paper is that the homogeneous ideal I ˜ S capturesthe information of the elements of S having (at least) 2 factorizations of thesame length. This object does not seem well suited to study the elementsof S having (at least) k -factorizations of the same length when k ≥
3. Weleave this as an open problem for further research. The works [2, 20] studygeneralized Frobenius numbers and, thus, they might shed some light onthis problem.
Acknowledgements
We wish to thank M.A. Moreno-Fr´ıas, who presented the results of [13] inthe seminar GASIULL at Universidad de La Laguna and introduced andencouraged us to work on this topic.
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