Cyclotomic exponent sequences of numerical semigroups

arXiv:2101.08826v1 [math.AC] 21 Jan 2021 CYCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS ALEXANDRU CIOLAN, PEDRO A. GARC´ IA-S ´ ANCHEZ, ANDR´ ES HERRERA-POYATOS, AND PIETER MOREE Abstract. We study the cyclotomic exponent sequence of a numerical semigroup S, and we compute its values at the gaps of S, the elements of Swith unique representations in terms of minimal generators, and the Betti elements b∈Sfor which the set {a∈Betti(S) : a≤Sb}is totally ordered with respect to ≤S(we write a≤Sbwhenever a−b∈S, with a, b ∈S). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, Garc´ıa-S´anchez and Moree (2016) stating that S is cyclotomic if and only if it is a complete intersection. 1. Introduction Anumerical semigroup Sis a submonoid of N(the set of non-negative integers) under addition, with finite complement in N. The non-negative integers that are not in Sare its gaps, and the set of gaps is denoted by G(S). The largest gap is the Frobenius number of S, denoted by F(S). The number of gaps of S, also known as the genus of S, is denoted by g(S). A numerical semigroup admits a unique minimal generating system; its elements are called minimal generators, and its cardinality the embedding dimension, denoted by e(S). The smallest positive integer in Sis the multiplicity of Sand is denoted by m(S). For an introduction to the theory of numerical semigroups the reader is referred, e.g., to [17]. To a numerical semigroup Swe can associate its Hilbert series, defined as the formal power series HS(x) = Ps∈Sxs∈Z[[x]],and its semigroup polynomial, given by PS(x) = (1−x)Ps∈Sxs.(Indeed, since all elements larger than F(S) are in Sand F(S) is not, PS(x) is a monic polynomial of degree F(S) + 1.) In the sequel we say that a formal identity of the form A(x) = B(x) is true if it holds in Z[[x]]. For notational convenience we will often denote the infinite sum 1 + xd+x2d+··· by (1 −xd)−1,where d∈N. It is not difficult to conclude that the coefficients of PSare in {−1,0,1}and that consecutive nonzero coefficients alternate in sign. On noting the formal identity HS(x) = (1 −x)−1−Ps∈G(S)xs, we have (1) PS(x) = 1 + (x−1) X s∈G(S) xs, 2010 Mathematics Subject Classification. 20M14, 11C08, 11B68. Key words and phrases. Numerical semigroups, cyclotomic polynomials, Betti elements, complete intersections. The second author is supported by the project MTM2017–84890–P, which is funded by Ministerio de Econom´ıa y Competitividad and Fondo Europeo de Desarrollo Regional FEDER, and by the Junta de Andaluc´ıa Grant Number FQM–343. The third author was supported by an Initiation to Research Fellowship from the University of Granada in the academic year 2017-2018. He is currently supported by an Oxford-DeepMind Graduate Scholarship and an EPSRC Doctoral Training Partnership. 1 2 CIOLAN, GARC´ IA-S´ ANCHEZ, HERRERA-POYATOS, AND MOREE and so PS(1) = 1. In addition, PS(0) = 1 and, by [7, Lemma 11] (see also Lemma 3.1), there exist unique integers ejsuch that the formal identity (2) PS(x) = ∞ Y j=1 (1 −xj)ej holds. We call the sequence e={ej}j≥1the cyclotomic exponent sequence of Sand we will use this notation throughout the paper. Cyclotomic exponent sequences were introduced in [7] as a tool for studying cyclotomic numerical semigroups; we will come back to this later in this section. The purpose of this paper is to initiate the study of any numerical semigroup by means of its cyclotomic exponent sequence. Our ultimate goal is to characterize special families of numerical semigroups in terms of properties of their cyclotomic exponent sequences. Our first main result determines the exponent sequence of Sat gaps and minimal generators. Theorem 1.1. If S6=Nis a numerical semigroup and eis its cyclotomic exponent sequence, then a) e1= 1; b) ej= 0 for every j≥2not in S; c) ej=−1for every minimal generator jof S; d) ej= 0 for every j∈Sthat has only one factorization and is not a minimal generator. Our second main result determines eat certain Betti elements (see Section 2.3 for a definition of the latter and the related notion of R-classes). In order to introduce our findings, we need the following definitions. Let (X, ≤) be a partially ordered set. We define the set U(X) as U(X) = {x∈X:↓xis totally ordered}, where ↓x={y∈X:y≤x}. We note that (3) Minimals≤X= Minimals≤U(X). We write a≤Sbif b−a∈S. Since Sis a cancellative monoid free of units, the relation ≤S defines an order relation on Z.Moreover, for any s∈S, the set ↓s(considered in (S, ≤S)) is finite. If Sis a numerical semigroup, we define the set E(S) = {d∈N:d≥2, ed6= 0, d is not a minimal generator}, notation which we will use throughout. Our next result relates the partially ordered sets (Betti(S),≤S) and (E(S),≤S). Theorem 1.2. Let Sbe a numerical semigroup with cyclotomic exponent sequence e. Then U(Betti(S)) = U(E(S)). Moreover, for every b∈U(Betti(S)), the exponent ebis equal to the number of R-classes of bminus 1. A direct consequence of (3) and Theorem 1.2 is that Minimals≤SBetti(S) = Minimals≤SE(S). In order to prove Theorem 1.2 we need to understand the graph of factorizations ∇bof the elements bin U(Betti(S)) (see Section 2.3 for a definition of ∇b). Our main technical result on this matter is Theorem 5.9, which shows that when b∈U(Betti(S)) \Minimals≤SBetti(S), the graph ∇bhas exactly one connected component that is not a singleton. As a consequence of Theorem 1.2 we are able to characterize some families of numerical semigroups solely in terms of their cyclotomic exponent sequences. Before stating our next result, let us define these families. In what follows, Sis a numerical semigroup. We say that Sis Betti-sorted if Betti(S) is totally ordered with respect to ≤S,and that Sis Betti-divisible if Betti(S) is totally ordered with respect to the divisibility order in N. These two families of numerical semigroups were introduced in [10], where the authors showed that they are complete intersections (see Section 2.4 for a definition). CYCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 3 The third family we consider is that of numerical semigroups with a unique Betti element, which is obviously a subset of each of the two previous families. This family was studied in [10] and [11]. Theorem 1.3. For a numerical semgiroup Sthe following assertions hold: a) The semigroup Sis Betti-sorted if and only if E(S)is totally ordered by ≤S. b) The semigroup Sis Betti-divisible if and only if E(S)is totally ordered by the divisibility order. c) The semigroup Shas a unique Betti element if and only if E(S)is a singleton. Our work also has some consequences for cyclotomic numerical semigroups. Definition 1.1. A cyclotomic numerical semigroup is a numerical semigroup whose cyclotomic exponent sequence ehas finite support; that is, there exists N∈Nsuch that ej= 0 for every j≥N. These semigroups were introduced and studied in [7] using a different, but equivalent, definition (see Section 2.1). It turns out that every complete intersection numerical semigroup is cyclotomic, the former being a numerical semigroup such that the cardinality of its minimal presentation equals its embedding dimension minus one (see Section 2.3 for a brief recap on the concept of minimal presentations). In [7] the authors made the following conjecture, which they checked to be true for numerical semigroups with Frobenius number not exceeding 70,using the GAP package numericalsgps [8,9]. Conjecture 1.4 ([7, Conjecture 1]).A numerical semigroup is a complete intersection if and only if it is cyclotomic. A version of this conjecture has been established for certain graded algebras, see [6], but the numerical semigroup version remains open. Conjecture 1.4 is equivalent with saying that a numerical semigroup Sis a complete intersection if and only if its cyclotomic exponent sequence ehas finite support. This establishes an equivalence between an algebraic property of a numerical semigroup and one that only involves its cyclotomic exponent sequence. Note that ehas finite support if and only if E(S) is finite. Recall that Theorem 1.3 deals with the case where E(S) is a singleton. As a consequence of our results, we make further progress towards proving Conjecture 1.4 by showing that all members of a certain family of cyclotomic numerical semigroups are complete intersections. More precisely, if the Hasse diagrams of Betti(S) and E(S) with respect to ≤Sare forests, that is, U(Betti(S)) = Betti(S) and U(E(S)) = E(S), then we are able to deduce that Sis a complete intersection (Corollary 7.6). Computations suggest that such forests arise very frequently; for instance, there are 197 complete intersection numerical semigroups with Frobenius number 101 (equivalently, with genus equal to 52), and for 170 of them the Hasse diagram of their set of Betti elements with respect to ≤Sis a forest. Here we should mention that, for any complete intersection numerical semigroup S, we have Betti(S) = E(S), as explained in Section 2.4. The paper is organized as follows. In Section 2we gather some preliminary material used in the rest of the paper. In Section 3we introduce cyclotomic exponent sequences and establish some elementary properties. In Section 4we prove Theorem 1.1. In Section 5we give the proof of Theorem 1.2, which comes in two parts, and we discuss a few tools needed for this purpose, such as minimal Betti elements and restricted factorizations (as this section is the longest, we kindly ask in advance for the reader’s patience). In Section 6we give the proof of Theorem 1.3, while Section 7 is dedicated to applications to cyclotomic numerical semigroups, open questions, and concluding remarks. 2. Preliminaries Here we recall a few properties and notions that are needed throughout the paper. References in the subsection headers give suggestions for further reading. Section 2.1 is exceptional in that it is not needed for the rest of the paper. Its purpose is to show that the original definition of a cyclotomic numerical semigroup S, given in [7] through saying that 4 CIOLAN, GARC´ IA-S´ ANCHEZ, HERRERA-POYATOS, AND MOREE PSadmits a factorization into cyclotomic polynomials as in (5), is equivalent with the definition used here. 2.1. Cyclotomic numerical semigroups and cyclotomic polynomials [7,20].The semigroup polynomial and the Frobenius number of a numerical semigroup of embedding dimension two can be easily determined (see, for instance, [15]). Lemma 2.1 ([15, Theorem 1]).If 2≤a < b are coprime integers, then Pha,bi(x) = (1 −x)(1 −xab) (1 −xa)(1 −xb). Corollary 2.2 (Sylvester, 1884).If 2≤a < b are coprime integers, then F(ha, bi) = ab −a−b. Lemma 2.1 shows that S=ha, biis a cyclotomic numerical semigroup, since its exponent sequence has finite support. The factorization of Pha,biinto irreducibles is easily found by using the well-known factorization (4) xn−1 = Y d|n Φd(x) of xn−1 into cyclotomic polynomials (all of them irreducible over Q). In the special case where a and bare prime numbers, we find that Pha,bi(x) = Φab(x), which then gives a very natural proof of the classical fact that the coefficients of Φab(x) are all in {−1,0,1}and that consecutive non-zero coefficients alternate in sign. The following two results describe some basic properties of the cyclotomic exponent sequence attached to a cyclotomic numerical semigroup. Proposition 2.3. Let Sbe a numerical semigroup and let ebe its cyclotomic exponent sequence. If Sis cyclotomic, then Pj≥1ej= 0. Proof. Let Nbe the largest index jsuch that ej6= 0.Then we have PS(x) = (1 −x)Pj≤NejGS(x), for some rational function GS(x) satisfying GS(1) 6∈ {0,∞} (in fact GS(1) = Qj≤Njej). Since PS(1) = 1, it follows that Pj≥1ej= 0.  Proposition 2.4. Let Sbe a numerical semigroup. Then Sis cyclotomic if and only if PS(x) factorizes in the form (5) PS(x) = Y d∈D Φhd d, where Dis a finite set and hdare positive integers. Proof. Let ebe the exponent sequence of Sand let Nbe the largest index jsuch that ej6= 0.By Proposition 2.3 we have P1≤j≤Nej= 0 and so PS(x) = N Y j=1 (1 −xj)ej= N Y j=1 (xj−1)ej. By (4) it then follows that PS(x) can be written as in (5), where a priori some of the integers hd may be negative. As the complex zeros of the cyclotomic polynomials are all different, this would lead to PShaving a pole, contradicting the fact that PSis a polynomial. CYCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 5 For the other direction, we use the M¨obius function µ(n), which is equal to zero for non-square free integers nand to (−1)rotherwise, where ris the number of prime factors in the prime decomposition of n. By applying M¨obius inversion to (4) one obtains Φn(x) = Y d|n (xd−1)µ(n/d). Using the fact that Pd|nµ(d) = 0 for n≥2, this can be rewritten for n≥2 as Φn(x) = Y d|n (1 −xd)µ(n/d). Since PS(1) 6= 0, Φ1(1) = 0, and Φd(1) 6= 0 for d≥2, we have 1 6∈ D, and so, using the latter identity, it follows that there are integers e1, e2,... such that PS(x) = ∞ Y d=1 (1 −xd)ed, where ed= 0 for d > max D, which means that ehas finite support.  Recall that a polynomial f(x) of degree dis self-reciprocal if f(x) = xdf(1/x). The cyclotomic polynomial Φnis self-reciprocal for n≥2. As a consequence of this fact and Proposition 2.4, it follows that if Sis cyclotomic, then PS(x) is self-reciprocal. It is not difficult to show that a numerical semigroup is symmetric (that is, for every n∈Z, either nor F(S)−nis in S) if and only if PSis self-reciprocal [15]. Therefore, every cyclotomic numerical semigroup is symmetric. The converse is generally not true; for instance, it can be shown that for every positive integer e≥4, there exists a numerical semigroup of embedding dimension ethat is symmetric but not cyclotomic [12,18]. Proposition 2.4 raises the question whether one can classify cyclotomic numerical semigroups for which PSdecomposes into a small number of irreducible factors. This and similar questions are addressed in [5], where the authors show, for example, that PS= Φnif and only if n=pq and S=hp, qifor distinct prime numbers pand q. 2.2. Ap´ery sets [17].Let Sbe a numerical semigroup and m∈Z. The set Ap(S;m) = {s∈S:s−m6∈ S} is called the Ap´ery set of min S. Given any arithmetic progression modulo m, the numbers in it that are large enough will be in S, whereas the numbers that are small enough will not be in S. Therefore, among them we will find at least one element from Ap(S;m), and so |Ap(S;m)| ≥ m. In the remainder of this subsection we assume that m∈S, in which case S= Ap(S;m) + mN and |Ap(S;m)|=m. It then follows that every integer zcan be uniquely written as z=km +w with k∈Zand w∈Ap(S;m), and that z∈Sif and only if k≥0. We will use this fact several times. From S= Ap(S;m) + mNwe infer that HS(x) = Pw∈Ap(S;m)xwP∞ k=0 xkm, hence (6) (1 −xm) HS(x) = X w∈Ap(S;m) xw, with the right-hand side being the Ap´ery polynomial of min S, see [16]. 6 CIOLAN, GARC´ IA-S´ ANCHEZ, HERRERA-POYATOS, AND MOREE 2.3. Minimal presentations and Betti elements [1,17].Let Sbe a numerical semigroup minimally generated by {n1,...,ne}. There is a natural epimorphism ϕ:Ne→S, defined as ϕ(a1,...,ae) = Pe i=1 aini. The set ker ϕ={(a, b)∈Ne×Ne:ϕ(a) = ϕ(b)}is a congruence, that is, an equivalence relation compatible with addition; hence, Sis isomorphic, as a monoid, to Ne/ker ϕ. Apresentation for Sis a system of generators of ker ϕas a congruence. A presentation is minimal if none of its proper subsets generates ker ϕ. It can be shown that all minimal presentations of a numerical semigroup have the same (finite) cardinality (see, for instance, [17, Chapter 7]). Given ρ⊆Ne×Ne, denote by cong(ρ) the congruence generated by ρ, that is, the intersection of all congruences containing ρ. Define ρ0=ρ∪ {(y, x) : (x, y)∈ρ}and ρ1={(x+u, y +u) : (x, y)∈ ρ0, u ∈Ne}. It turns out that cong(ρ) is the transitive closure of ρ1. A minimal presentation of Scan be constructed as follows. For s∈S, let Z(s) be the set of factorizations of sin S, that is, the fiber ϕ−1(s) (we use Z(S) to denote the set of all factorizations of elements in S, which equals Ne(S)). Define ∇sto be the graph with vertices Z(s) and with edges xy so that x·y6= 0 (dot product; that is, edges join factorizations having minimal generators in common). The connected components of ∇sare called the R-classes of s. The element s∈Sis a Betti element if ∇sis not connected. We denote by Betti(S) the set of Betti elements of S, and by nc(∇s) the number of connected components of ∇s. Assume that s∈Betti(S) and let C1,...,Crbe the connected components of ∇s(thus r= nc(∇s)). Pick xi∈Cifor all i∈ {1,...,r}, and set ρ(s)={(x1, x2),(x2, x3),...,(xr−1, xr)}. Then ρ=Ss∈Betti(S)ρ(s)is a minimal presentation of S. All minimal presentations can be constructed by using the following idea. Think of (xi, xj) as a link connecting Ciand Cj. Then you need all connected components to be connected with these links. The minimal possible choice is to have a spanning tree connecting them all, once the xihave been chosen. Different choices of xiin Ci and different spanning trees will yield different minimal presentations, but they all have the same cardinality (see, for instance, [1, Chapter 4]). As a consequence, all minimal presentations have cardinality equal to Ps∈Betti(S)(nc(∇s)−1). 2.4. Complete intersection numerical semigroups [17].Let Sbe a numerical semigroup with embedding dimension e. It can be shown that the cardinality of any minimal presentation of Shas e−1 as a lower bound (see, for instance, [17, Chapter 8]), and numerical semigroups attaining this bound are called complete intersections. Let S1and S2be two numerical semigroups, and a1,a2be two coprime integers such that a1∈S2, a2∈S1and neither a1, nor a2is a minimal generator. The set a1S1+a2S2is a numerical semigroup known as the gluing of S1and S2. We will write S=a1S1+a1a2a2S2. A complete intersection numerical semigroup Sis either Nor a gluing a1S1+a2S2with both S1and S2complete intersection numerical semigroups (see [17, Chapter 8]). It turns out that Betti(S) = {a1a2} ∪ {a1b1:b1∈Betti(S1)} ∪ {a2b2:b2∈Betti(S2)}, see [2]. It is well-known (see [2]) that (7) Ha1S1+a1a2a2S2(x) = (1 −xa1a2) HS1(xa1) HS2(xa2), which, in terms of semigroup polynomials, can be written as (8) Pa1S1+a1a2a2S2(x) = (1 −x)(1 −xa1a2) (1 −xa1)(1 −xa2)PS1(xa1) PS2(xa2). Consequently, a formula for PSin terms of the minimal generators and Betti elements of Scan be given. If S=n1N+b1n2N+···+be−1neN(with {n1,...,ne}the minimal generating system of S and with binot necessarily distinct integers), then [2, Theorem 4.8] states that (9) HS(x) = Qe−1 i=1 (1 −xbi) Qe i=1(1 −xni). CYCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 7 If S=a1S1+a1a2a2S2is a gluing of S1and S2, then every minimal presentation of Scomes from the union of a minimal presentation of S1, a minimal presentation of S2and a pair of factorizations of a1a2, one in a1S1and the other in a2S2; see, for instance, [17, Chapter 8]. Thus, by (7) and the fact that every minimal presentation of Shas cardinality Pb∈Betti(S)(nc(∇b)−1), the multiplicity of biin the numerator of (9) is precisely nc(∇bi)−1 and the above formula can be rewritten as (10) HS(x) = Qb∈Betti(S)(1 −xb)nc(∇b)−1 Qe i=1(1 −xni). Indeed, this identity characterizes complete intersection numerical semigroups. Proposition 2.5. Let Sbe a numerical semigroup. Then Sis a complete intersection numerical semigroup if and only if HSsatisfies (10). Proof. We prove that if Sverifies (10), then Sis a complete intersection numerical semigroup; the other implication also holds, as we have just seen. Recall that PS(x) = (1 −x) HS(x) is a polynomial and that by (1) we have PS(1) = 1. Thus the factors 1 −xof the numerator and denominator of PS(x) must cancel each other out and we find Pb∈Betti(S)(nc(∇b)−1) = e(S)−1. Consequently, any minimal presentation of Shas cardinality e(S)−1, which means that Sis a complete intersection.  One of our aims is to prove that the Hilbert series of a cyclotomic numerical semigroup always satisfies (10). In this paper we do so for some particular classes of cyclotomic numerical semigroups. 2.5. Other series and polynomials associated to numerical semigroups [19].This subsection is dedicated to introducing a few other objects that arise naturally in connection to numerical semigroups. However, the only reults needed in the sequel are the upcoming definitions and the accompanying identity (11). The reader may therefore choose to omit the discussion on the polynomial KS, which we make here for sake of completeness, and directly skip to Section 2.6. Let Sbe a numerical semigroup minimally generated by a set A. The denumerant of s∈S, denoted by d(s), is the cardinality of Z(s), the set of factorizations of sin S. We can consider the denumerant series Ps∈Sd(s)xs, which verifies the equality (11) X s∈S d(s)xs=Y n∈A ∞ X j=0 xjn =Y n∈A 1 1−xn. This equality is widely used in our work and its proof is straightforward. We note that every Betti element has denumerant exceeding one. Let A={n1<··· < ne}be the minimal system of generators of S. Sz´ekely and Wormald [19] were the first to study the function (12) KS(x) = (1 −xn1)···(1 −xne) HS(x), which, on writing KS(x) = (1 + x+···+xn1−1)(1 −xn2)···(1 −xne) PS(x), turns out to be a polynomial of degree F(S) + Pe j=1 nj. Let Sbe a complete intersection numerical semigroup. From (10) we derive (13) KS(x) = Y b∈Betti(S) (1 −xb)nc(∇b)−1. Corollary 2.6. Let Sbe a complete intersection numerical semigroup minimally generated by {n1,...,ne}. Then F(S) + e X j=1 nj=X b∈Betti(S) b(nc(∇b)−1). 8 CIOLAN, GARC´ IA-S´ ANCHEZ, HERRERA-POYATOS, AND MOREE Proof. The result follows from taking degrees in (13).  The polynomial KShas been explicitly computed for several families of numerical semigroups. For instance, an expression is given in [4] for numerical semigroups of embedding dimension three, and for those of embedding dimension four that are symmetric or pseudo-symmetric. In that paper, KSis related to the Betti numbers of the semigroup ring associated to S(see [3] for a different approach). 2.6. Isolated factorizations [10].Let Sbe a numerical semigroup and let s∈S. We say that a factorization zof sis isolated if z·x= 0 for every factorization xof sdifferent from z. Thus, z is an isolated factorization if and only if {z}is an R-class of ∇s. This means either that shas a unique factorization, or that sis a Betti element with one of its R-classes being a singleton. We denote by I(s) the set of isolated factorizations of s, and by I(Λ) the set of isolated factorizations of the elements of Λ ⊆S. Thus I(Λ) = Is(Λ) ∪Ib(Λ), where Is(Λ) is the set of isolated factorizations coming from elements with a unique factorization, and Ib(Λ) = I(Λ) ∩Z(Betti(S)) that of the isolated factorizations of the Betti elements in Λ. We also denote the cardinality of I(s) by i(s) and we define IBetti(S) as the set of Betti elements with an isolated factorization. Isolated factorizations can be characterized as in Lemma 2.7. First, we need some notation. Let x, y ∈Ne. We say that x≤yif xj≤yjfor every j. This gives an order relation on Ne, known as the cartesian product order. Recall that x < y when x≤yand x6=y. Lemma 2.7. Let Sbe a numerical semigroup and s∈S. A factorization z∈Z(s)is not isolated if and only if there exists x∈Ib(S)such that x < z. In particular, Ib(S) = Minimals≤Z({s∈S:d(s)≥2}). As a consequence, any factorization z∈Ne(S) can be written as z=w+x1+···+xlwith w∈Is(S) and x1,...,xl∈Ib(S). Proof. The first assertion is merely a rephrasing of [10, Lemma 3.1]. Assume that z∈Ne(S)= Z(S). If ϕ(z) has a unique factorization, then z=w∈Is(S). Otherwise, there exists x1∈Ib(S) such that x1< z. We consider now z−x1and start anew. This process must end either with a 0 or with a factorization that is the unique factorization of an element in the semigroup.  We say that an element s∈Sis Betti-minimal if s∈Minimals≤SBetti(S). As a consequence of Lemma 2.7, one can characterize Betti-minimal elements as in Proposition 2.8. Proposition 2.8 ([10, Proposition 3.6]).Let Sbe a numerical semigroup and s∈S. The following statements are equivalent: a) sis Betti-minimal; b) sis a minimal element of IBetti(S)with respect to ≤S; c) shas at least two factorizations and all of them are isolated, that is, nc(∇s) = i(s)≥2. The following result is a particular case of [10, Corollary 3.8] and characterizes the elements having a unique factorization in terms of Ap´ery sets. Corollary 2.9 ([10, Corollary 3.8]).Let Sbe a numerical semigroup. Then {m∈S:d(m) = 1}=\ b∈Betti(S) Ap(S;b) = \ b∈Minimals≤SBetti(S) Ap(S;b). The next lemma allows us to deal with sequences of Betti elements of the form b1≤S· · · ≤Sbt, and will be useful for the study of U(Betti(S)). Lemma 2.10 ([10, Lemma 3.12]).Let Sbe a numerical semigroup. If b1and b2are two Betti elements of Ssuch that b1<Sb2, then x·y= 0 for every x∈Z(b1)and y∈I(b2). CYCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 9 3. Cyclotomic exponent sequences In this section we show how to compute, both theoretically and practically, the cyclotomic exponent sequence of a numerical semigroup, and we give some examples. Practically, they can be computed with the function CyclotomicExponentSequence or, alternatively, with the function WittCoefficients of the GAP [9] package numericalsgps [8], which implements the method given in the upcoming Lemma 3.5. Let a(x), b(x)∈Z[[x]] and p(x)∈Z[x]. We use the notation a(x)≡b(x) (mod p(x)) to indicate that a(x)−b(x)∈p(x)Z[x]. Note that ≡is an equivalence relation. The next lemma, together with the fact that PS(x)≡1 (mod x), shows that for any numerical semigroup there is an expansion of the form (2), where the exponents ejare uniquely determined integers. Lemma 3.1. Let f(x)∈Z[[x]] and suppose that f(x)≡1 (mod x). Then there exist unique integers e1, e2,... such that, in Z[[x]], (14) f(x) = ∞ Y k=1 (1 −xk)ek. Proof. We show how to successively determine the integers e1, e2,...,emsuch that (14) holds modulo xm+1. By assumption, f(x) = 1 −e1x(mod x2) for some e1∈Z. This then gives f(x)(1 −x)−e1≡ 1 (mod x2). Let m≥2. Suppose that we have found integers e1,...,em−1such that f(x) m−1 Y k=1 (1 −xk)−ek≡1 (mod xm). As the right-hand side is of the form 1 −emxm(mod xm+1) for some integer em, we infer that f(x) m Y k=1 (1 −xk)−ek≡1 (mod xm+1). We now turn our attention to the uniqueness claim. For the sake of contradiction, suppose there exists a different sequence of integers fnsuch that f(x) = Q∞ k=1(1 −xk)fk. Let mbe the smallest integer such that fm6=em. Put h(x) = Qm−1 k=1 (1 −xk)ek. We then have f(x)≡h(x) (1 −xm)em(mod xm+1) on the one hand, and f(x)≡h(x) (1 −xm)fm(mod xm+1) on the other. As the two expressions have different coefficients in front of xm, we have reached a contradiction, concluding the proof.  Example 3.2.Let αbe an integer. We have 1 −αx =Q∞ k=1(1 −xk)M(α,k), with M(α, k) = 1 kPj|kµ(k/j)αj. This is the so-called cyclotomic identity, see, e.g., [13]. In case pis a prime number, the fact that M(α, p) must be an integer implies Fermat’s Little Theorem stating that αp≡α(mod p). Remark 3.3.Expansions of the form (14) arise in quite different areas such as automata, group, graph and Lie algebra theory; see [14] for some references. Remark 3.4.Let f(x)∈Z[x] and suppose that f(x)≡1 (mod x). By (14) we have f(x)≡ Qn k=1(1 −xk)ek(mod xn+1). This identity allows one to determine the first ncoefficients of f. On taking n= deg(f), we can even reconstruct fcompletely. 16 CIOLAN, GARC´ IA-S´ ANCHEZ, HERRERA-POYATOS, AND MOREE Proof. The result is a consequence of Lemma 5.5. Let s∈S. If d(ωs)≥2, then |B(s;b)|= 0. Otherwise, since the isolated factorizations of bare disjoint, we find that every element of B(s;b) is uniquely determined by qsisolated factorizations of b. The proof is completed by counting the number of combinations with repetitions of size qsfrom a set with i(b) elements.  We will use the following observation several times in the proof of Lemma 5.8, which shows connectivity of B(s; Λ) in ∇sunder some hypotheses. Lemma 5.7. Let Sbe a numerical semigroup and let b1be a Betti-minimal element of S. Let Λ⊆IBetti(S)with b1∈Λ. For each s∈Ssuch that b1is the only Betti-minimal element bof S with b≤Ss, and for each x∈I(Λ) such that ϕ(x)≤Ss, there exists z∈B(s; Λ) with x < z. Proof. Let s∈Sbe such that b1is the only Betti-minimal element of Sbelow swith respect to ≤S, and let x∈I(Λ) such that ϕ(x)≤Ss. Set b=ϕ(x). Write s−b=ω+qb1, where ω∈Ap(S;b1) and q∈N. By Corollary 2.9,ωhas only one factorization. We can choose z=w+x+qy ∈B(s; Λ), where Z(ω) = {w}and y∈I(b1). Here we have used that Betti-minimal elements have isolated factorizations by Proposition 2.8. Recall that U(Betti(S)) is the set of b∈Betti(S) such that ↓b={b′∈Betti(S) : b′≤Sb}is totally ordered. In Lemma 5.8 we establish connectivity of B(s; Λ) under some assumptions. This is the main ingredient of the proof of Theorem 5.9. Lemma 5.8. Let Sbe a numerical semigroup. Let u∈U(Betti(S)) and let Λ =↓u. If s∈S\Λis such that u≤Sbfor all b∈Betti(S)\Λwith b≤Ss, then B(s; Λ) is connected in ∇s. Proof. Assume that Λ = {b1<S···<Sbl}, and so u=bl. Write Λi=↓bi={b1<S··· <Sbi}, for i∈ {1,...,l}. We proceed by induction on l, the size of Λ. Note that, by definition of ↓u,b1is Betti-minimal and the only Betti-minimal element with b1≤Su. Let s∈S\Λ be such that u≤Sb for every b∈Betti(S)\Λ with b≤Ss. If b∈Betti(S) with b≤Ss, then either b∈Λ and b1≤Sb, or b6∈ Λ and b1≤Su≤Sb. In any case, we have shown that b1≤Sbfor every b∈Betti(S) with b≤Ss. Hence, either d(s) = 1 or b1is the only Betti-minimal element with b1≤Ss. We will use this fact in our induction. First, we study the case l= 1. Note that either d(s) = 1 or b1is the only minimal element of Swith b1≤Ss. In the first case, we have B(s;b1) = Z(s). In the second case, Lemma 5.7, with Λ = {b1}, yields that B(s;b1) is non-empty. Its connectivity follows from Lemma 5.5. If l≥2, let us assume that the result holds for l−1. If B(s; Λ) = B(s; Λl−1), then we are done by the induction hypothesis. Let us consider the case B(s; Λ) 6= B(s; Λl−1). In this case we have bl≤Ss, so b1is the only minimal Betti element of Swith b1≤Ss. There are two cases depending on the number of factorizations of s−bl. Case 1: d(s−bl)≥2. Notice that, under this assumption, b1is the only minimal Betti element of Swith b1≤Ss−bl. Let z1∈B(s; Λ) \B(s; Λl−1). There is y∈I(bl) such that y < z1. Let x∈I(b1). Since b1≤Ss−bl, Lemma 5.7 provides us with z∈B(s−bl; Λ) such that x < z. Thus, we have z2=z+y∈B(s; Λl) and z1·z26= 0. Moreover, there is z3∈B(s;b1)⊆B(s; Λl−1) with x < z3(Lemma 5.5) and, in particular, z2·z36= 0. From the arbitrary choice of z1and the fact that B(s; Λl−1) is connected, it follows that B(s; Λ) is also connected. Case 2: d(s−bl) = 1. Set ω2=s−bl, and let w2be the unique factorization of ω2. If z∈B(s; Λ) \B(s; Λl−1), then there is y∈I(bl) with y < z. Note that z−yis a factorization of ω2, whence z=w2+y. That is, we have shown that B(s; Λ) \B(s; Λl−1)⊆ {w2+y:y∈I(bl)}. Let us suppose that B(s; Λ) has at least two R-classes in order to obtain a contradiction. Since B(s; Λl−1) and B(s; Λ)\B(s; Λl−1) are connected, the only option is z·y= 0 for every z∈B(s; Λl−1) and y∈B(s; Λ) \B(s; Λl−1). Write bl=ω1+qb1with ω1∈Ap(S;b1) and q∈N(this implies CYCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 17 Z(ω1) = {w1}by Corollary 2.9). We have s=ω1+ω2+qb1. There are two possible subcases, each of which yields a contradiction. Subcase 2.1: d(ω1+ω2) = 1. The factorization z=w1+w2+qx is in B(s; Λ) for any x∈I(b1), but it is not disjoint with any element of B(s; Λ) \B(s; Λl−1), a contradiction. Subcase 2.2: d(ω1+ω2)≥2. In light of Lemma 2.7, there exist b∈Betti(S) and y∈I(b) such that y≤w1+w2. For all x∈I(Λl−1) we have ϕ(x)≤Sbl≤Ssand there is z∈B(s; Λl−1) with x < z (Lemma 5.7). Under the standing assumption, z·w2= 0. In particular, we have w2·x= 0. It follows that the factorization w2is disjoint with any isolated factorization of the Betti elements b1,...,bl−1. Since y≤w1+w2,y∈I(b) and w1, w2∈Is(S), we have w1·y6= 0 and w2·y6= 0. Hence, it follows that yis not an isolated factorization of any of the elements b1,...,bl−1, so b6∈ Λl−1. Since b≤Sω1+ω2≤Ss, either b∈Λ or bl≤Sbby hypothesis. We conclude that bl≤Sb≤Sω1+ω2. Write ω1+ω2=ω+pb1+blwith ω∈Ap(S;b1) and p≥0. Since s=ω2+bl and s=ω1+ω2+qb1=ω+ (p+q)b1+bl, we obtain ω2=s−bl=ω+ (p+q)b1. Recall that d(ω2) = 1. This forces p= 0 = q, a contradiction because bl=ω1+qb1and d(bl)≥2.  We now have the ingredients necessary to prove Theorem 5.9, which determines the number of isolated factorizations of any b∈U(Betti(S)). Theorem 5.9. Let Sbe a numerical semigroup and let b∈U(Betti(S)). Then either bis minimal and all its factorizations are isolated, or the number of isolated factorizations of bequals its number of R-classes minus 1. Proof. Let ↓b={b1<S···<Sbl} ⊆ Betti(S). If l= 1, then bis Betti-minimal and its factorizations are isolated (see Proposition 2.8). Otherwise, we apply Lemma 5.8 to b1,...,bl, and conclude that B(b; Λ) is connected for Λ = {b1,...,bl−1}. By Lemma 5.4, we find that Z(b) = B(b; Λ) ∪I(b),thus the number of isolated factorizations of bis one less than the number of R-classes.  Corollary 5.10. Let Sa numerical semigroup. Write Betti(S) = {b1< b2<··· < bk}. Let us assume that k≥2. a) If b2−b16∈ S, then b2is Betti-minimal; that is, all its factorizations are isolated. b) If b1≤Sb2, then b2has nc(∇b2)−1isolated factorizations. Proof. This is a direct consequence of Theorem 5.9. Theorem 5.9 gives us some information about the smallest Betti elements. Let b1= min Betti(S). It is clear that b1is Betti-minimal. Let us assume that there exists b2= min(Betti(S)\{b1}). Then either b2is Betti-minimal (b2−b16∈ S), or b1≤Sb2. In the latter case we can apply Theorem 5.9 to conclude that i(b2) = nc(∇b2)−1. Therefore, b2always has isolated factorizations. Example 5.11.Let S=h10,15,16,17,19i. Then the Hasse diagram of (Betti(S),≤S) looks as follows: 48 32 57 30 363534 Hence the minimal elements of Betti(S) are 30,32,34,35,36. The factorizations of these elements are all isolated. Theorem 5.9 allows us to conclude that 48 has isolated factorizations and that nc(∇48) = i(48) + 1. Note that this result does not provide information about the factorizations of 57 (↓57 = {30,32,57}). In fact, one can check that 57 has an isolated factorization with the help of the GAP package numericalsgps. 18 CIOLAN, GARC´ IA-S´ ANCHEZ, HERRERA-POYATOS, AND MOREE Next, we want to give an expression for B(s; Λ∪{b}) in terms of B(s−jb; Λ),for suitable j, which is meant to be of the same shape as that given in recursion (26). Lemma 5.12. Let Sbe a numerical semigroup. Let Λ⊆IBetti(S)with Λ6=∅and b∈IBetti(S)\Λ. Then for every s∈Swe have B(s; Λ ∪ {b}) = qs [ j=0 B(s−jb; Λ) + j X i=1 I(b)!, where qsis the largest integer such that qsb≤Ss. In particular, |B(s; Λ ∪ {b})| ≤ qs X j=0 |B(s−jb; Λ)|i(b) + j−1 j. Proof. It is clear that Sqs j=0 B(s−jb; Λ) + Pj i=1 I(b)⊆B(s; Λ ∪ {b}). Let z∈B(s; Λ ∪ {b}). If z6∈ B(s; Λ), then there is y∈I(b) such that y≤zand z−y∈B(s−b; Λ ∪ {b}). We can repeat this argument a finite number of times until we find y1,...,yj∈I(b) and x∈B(s−jb; Λ) such that z=x+y1+···+yj. Hence, we obtain z∈B(s−jb; Λ) + Pj i=1 I(b).  Example 5.13.Let us consider the numerical semigroup S=h4,5,6i. We have Betti(S) = {10,12}, Z(10) = {(1,0,1),(0,2,0)}and Z(12) = {(3,0,0),(0,0,2)}. Since 72 = 6 ·12, by Lemmas 5.4 and 5.12 we find that Z(72) = B(72; {10,12}) = 6 [ j=0 B((6 −j)12; 10) + j X i=1 I(12)!. Note that (6,0,8) ∈Z(72) and (6,0,8) = 6(1,0,1) + (0,0,2) = 2(3,0,0) + 4(0,0,2). This union is therefore not disjoint and the inequality given in Lemma 5.12 can be strict. The following lemma shows that, under the hypotheses of Theorem 5.9, the upper bound given in Lemma 5.12 can be attained. Note that this recurrent expression has already arisen in (26). Lemma 5.14. Let Sbe a numerical semigroup. Let u∈U(Betti(S)) and let Λ =↓u. Then, for every s∈Sand z∈B(s; Λ), there are unique w∈Is(S)and x1,...,xt∈I(Λ) such that z=w+x1+···+xt. Moreover, we have |B(s; Λ)|= qs X j=0 |B(s−ju; Λ \ {u})|i(u) + j−1 j, where qsis the largest integer such that qsu≤Ss. Proof. By Theorem 5.9, we have Λ ⊆IBetti(S). Let s∈S. If B(s; Λ) = ∅, then we are done. Let us assume that B(s; Λ) 6=∅and let z∈B(s; Λ). The definition of B(s; Λ) ensures the existence of w∈Is(S) and x1,...,xt∈I(Λ) such that z=w+x1+···+xt. We show that this expression is unique. Let w1, w2∈Is(S) and, for each x∈I(Λ), let pxand qxbe non-negative integers such that z=w1+X x∈I(Λ) pxx=w2+X x∈I(Λ) qxx. In light of Lemma 2.10, the supports of the elements of I(Λ) are disjoint. If there is x∈I(Λ) such that px6=qx, then either x < w1, or x < w2, contradicting the fact that w1, w2∈Is(S), see Lemma 2.7. Therefore we have px=qxfor every x∈Λ and w1=w2. CYCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 19 As a consequence, the union given in Lemma 5.12 is disjoint. Moreover, we have B(s−ju; Λ \ {u}) + Xj i=1 I(u)=|B(s−ju; Λ \ {u})|i(u) + j−1 j and the result follows.  5.4. Completing the proof of Theorem 1.2.We now have all the ingredients necessary to complete the proof of Theorem 1.2. Proof of Theorem 1.2.First, we prove that if η∈U(E(S)), then η∈U(Betti(S)) and eηis as in the statement. The tools developed in this part of the proof will be helpful when proving the other inclusion. As we follow the proof idea explained in Section 5.2, we recommend the reader to have a look at Section 5.2 before reading this proof. Let Λ = {d∈ E(S) : d≤Sη}. Since η∈U(E(S)), we can write Λ = {b1<S···<Sbl}and bl=η. For each i∈ {1,...,l}, define Λi={b1,...,bi}and ri−1(s) as in (23), identity which we recall below for the convenience of the reader: X s∈S ri−1(s)xs= HS(x) i−1 Y j=1 (1 −xbj)−ebj. Note that from our hypothesis it follows that b1is minimal in E(S). By Theorem 5.3,b1is Bettiminimal. We prove by induction on i∈ {1,...,l}the following assertions: (a) if b≤Sbifor some b∈Betti(S), then b∈Λi; (b) bi∈U(Betti(S)) and ebi>0; (c) ri(s) = |B(s; Λi)|for every s∈Ssuch that b1is the only Betti-minimal element with b1≤Ss. First, we study the case i= 1. As a consequence of Theorem 5.3,b1Betti-minimal and i(b1) = d(b1) = eb1+ 1 ≥2. In particular, (a) and (b) hold for i= 1. Note that, by (6), we have HS(x) = Pω∈Ap(S;b1)xωP∞ j=0 xjb1. In conjunction with (21) and i(b1) = eb1+ 1, this yields (27) ∞ X s=0 r1(s)xs=X ω∈Ap(S;b1) xω∞ X j=0 xjb1i(b1) =X ω∈Ap(S;b1) xω ∞ X j=0 i(b1) + j−1 jxjb1 =X s∈Si(b1) + qs−1 qsxs, where qsis the unique non-negative integer such that s−qsb1∈Ap(S;b1). Let s∈Sbe such that b1is the only Betti-minimal element with b1≤Ss. Then from Corollary 2.9 it follows that ωs=s−qsb1has only one factorization. By combining Corollary 5.6 and (27), we conclude that r1(s) = |B(s; Λ1)|,as desired. Now assume that (a), (b) and (c) hold for i−1∈ {1,...,l−1}and let us prove that they also hold for i. We prove each of the induction hypotheses for iseparately. In doing so, we will use the identities (25) several times, which, for the sake of readability, we also recall here: ri−1(s) = d(s) when {d∈ E(S) : d≤Ss} ⊆ Λi−1, ri−1(s) = d(s)−eswhen s∈Minimals≤S(E(S)\Λi−1). (a) We proceed by deriving a contradiction. Let us assume that there is b∈Betti(S)\Λi−1such that b <Sbi. Then there exists an element β∈Minimals≤S{b∈Betti(S)\Λi−1:b <Sbi}. 20 CIOLAN, GARC´ IA-S´ ANCHEZ, HERRERA-POYATOS, AND MOREE Let D={d∈Betti(S) : d <Sβ}. Since β <Sbi, from minimality in the choice of βit follows that D⊆Λi−1. If D=∅, then βis Betti-minimal and, by Theorem 5.3,β∈ E(S). We thus obtain β∈ {d∈ E(S) : d <Sbi}= Λi−1, but β6∈ Λi−1by definition, a contradiction. We conclude that ∅ 6=D⊆Λi−1. Hence, we have b1∈d, so b1≤Sβ <Sbiand b1is the only Betti-minimal element that satisfies b1≤Sβ. From our induction hypothesis, we obtain |B(β; Λi−1)|=ri−1(β). Note that {d∈ E(S) : d≤Sβ} ⊆ {d∈ E(S) : d <Sbi}= Λi−1. Hence, by (25), we find that ri−1(β) = d(β), so |B(β; Λi−1)|=d(β). We can apply Lemma 5.8 with u=bi−1and s=β, finding that B(β; Λi−1) is connected in ∇β. But we have shown that |B(b; Λi−1)|=d(b) or, equivalently, Z(β) = B(β; Λi−1). This contradicts the fact that β∈Betti(S). (b) From Lemma 5.4 and (a) it follows that Z(bi)\Ib(bi), so |B(bi; Λi−1)|=d(bi)−i(bi). Moreover, by our hypothesis we have |B(bi; Λi−1)|=ri−1(bi). Note that bi∈Minimals≤S(E(S)\Λi−1) by definition of Λi−1. Hence, by (25) we obtain ri−1(bi) = d(bi)−ebi. We conclude that d(bi)−i(bi) = d(bi)−ebi, that is, 0 ≤i(bi) = ebi. Since ebi6= 0, we have i(bi)≥1 and bi∈Betti(S) because bihas at least two factorizations (Corollary 2.9). In view of (a), we have {b∈Betti(S) : b≤Sbi}= Λi, which by hypothesis is totally ordered, so b∈U(Betti(S)). (c) Let s∈Ssuch that b1is the only Betti-minimal element with b1≤Ss. Note that thanks to (b) we can apply Lemma 5.14 with u=bi. Recall that in (26) we showed that ri(s) = qs X j=0 ri−1(s−jbi)i(bi) + j−1 j, where qsis the largest integer such that s−qsbi∈S. This equation in combination with Lemma 5.14 yields ri(s) = qs X j=0 |B(s−jbi; Λi−1)|i(bi) + j−1 j=|B(s; Λi)|, which finishes the proof by induction. In the induction we have also shown that ebi= i(bi), see the proof of our hypothesis (b). Since bi∈U(Betti(S)), by Theorem 5.9 we have nc(∇bi) = i(bi) + 1 = ebi+ 1. The fact that eb1= nc(∇b1)−1 has been established in Theorem 5.3. Finally we show that U(Betti(S)) ⊆U(E(S)). Let u∈U(Betti(S)). Let us write ↓u={b∈ Betti(S) : b≤Su}={b1<S··· <Sbl=u}. We prove by induction on ithat bi∈U(E(S)). Note that b1is Betti-minimal and, thus, b1∈U(E(S)) by Theorem 5.3. Let us assume that b1,...,bi−1∈U(E(S)) and let us prove that bi∈U(E(S)). Let Λi−1={b1,...,bi−1}. We consider ri−1(s) as in (23). In light of the induction hypothesis (c), we have ri−1(s) = |B(s, Λi−1)|for every s∈Ssuch that b1is the only Betti-minimal element with b1≤Ss. In particular, we have ri−1(bi) = |B(bi,Λi−1)|and, thus, ri−1(bi) = d(bi)−i(bi), where we used Lemma 5.4. From Theorem 5.9, we find that i(bi)>0. Therefore, ri−1(bi)<d(bi) and, by (24), there exists d∈ E(S)\Λi−1with d≤Sbi. Hence, there is α∈Minimals≤S{d∈ E(S)\Λi−1}with α≤Sbi. By (25) we have ri−1(α) = d(α)−eα. From the induction hypothesis (c), we obtain ri−1(α) = |B(α, Λi−1)|. Since 0 < r1(s)≤ri−1(s) by (27) and (26), we have B(α, Λi−1)6=∅. In view of Lemma 5.4, B(α; Λi−1) = Z(α)\Ib(α), so 1≤ri−1(α) = d(α)−i(α). We find that i(α) = eα6= 0. We have 0 6=eα= i(α). We conclude that αis a Betti element with α≤Sbi. Since α6∈ Λi−1by definition, we must have bi=α∈ E(S) and {d∈ E(S) : d≤Sbi}={b1,...,bi}. We obtain bi∈U(E(S)),as wanted.  Example 5.15.Here we can see Theorem 1.2 in action for a couple of numerical semigroups. a) We consider again the semigroup from Example 5.11. Let S=h10,15,16,17,19i. Recall that the Hasse diagram of (Betti(S),≤S) looks as follows: CYCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 21 48 32 57 30 363534 Hence the minimal elements of Betti(S) are 30,32,34,35 and 36. The set U(Betti(S)) consists of these Betti-minimal elements and the Betti element 48. Therefore, by Theorem 1.2, we conclude that U(E(S)) = {30,32,34,35,36,48}and we can determine the exponents esof these elements from their number of isolated factorizations. b) An interesting application is finding Betti elements from E(S). Let us consider the semigroup S=h3,5,7iof Example 3.7b. We gave the first entries of the cyclotomic exponent sequence of S. The smallest elements of E(S) are 10,12,14,17,19, . . . . Note that U(E(S)) = {10,12,14} since any other element in E(S) can be written as α+ 3jfor some α∈ {10,12,14}and j≥0. Therefore, we have U(Betti(S)) = {10,12,14}= Minimals≤SBetti(S), and each one of these elements has only two factorizations (their cyclotomic exponents are 1). 6. Betti-sorted and Betti-divisible numerical semigroups In this section we prove Theorem 1.3, which characterizes Betti-sorted and Betti-divisible numerical semigroups in terms of their cyclotomic exponent sequences. Recall that Sis Betti-sorted if Betti(S) is totally ordered by ≤S,and that Sis Betti-divisible if Betti(S) is totally ordered by the divisibility order in N. Our characterizations are consequences of Theorem 1.2. We will use the following result on ordered sets. Lemma 6.1. Let (X, ≤)be an ordered set. Then Xis totally ordered if and only if U(X)is totally ordered. Proof. First, if Xis totally ordered, then any subset of X, and in particular U(X), is totally ordered. Now let us assume that U(X) is totally ordered. Suppose U(X)6=Xin order to obtain a contradiction. Then we can choose α∈Minimals≤(X\U(X)). We have {a∈X:a < α} ⊆ U(X) by minimality of α. Thus ↓αis of the form {a1<··· < ak< α}for some k≥0 and a1,...,ak∈U(X). We conclude that α∈U(X), a contradiction. Therefore, X= U(X) and Xis totally ordered.  Lemma 6.2. Let Sbe a numerical semigroup. Then Sis Betti-sorted if and only if E(S)is totally ordered by ≤S. Moreover, if this is the case, then Betti(S) = E(S). Proof. In view of Theorem 1.2 and Lemma 6.1,Sis Betti-sorted if and only if U(Betti(S)) = U(E(S)) is totally ordered by ≤Sor, equivalently, E(S) is totally ordered by ≤S. This gives the following alternative proof of the fact that Betti-sorted numerical semigroups are complete intersections. For the original proof we refer to [10], where in fact the authors show the stronger result that Betti-sorted numerical semigroups are free. Corollary 6.3. If Sis a Betti-sorted numerical semigroup, then Sis a complete intersection. Proof. In view of Lemma 6.2, we have E(S) = U(E(S)). By applying Theorem 1.2 we find that eb= nc(∇b)−1 for every b∈Betti(S) = E(S). Let Abe the minimal system of generators of S. With the help of Theorem 1.1, we conclude that HS(x) = Qb∈Betti(S)(1 −xb)nc(∇b)−1 Qn∈A(1 −xn). Therefore, Sis a complete intersection by Proposition 2.5. 22 CIOLAN, GARC´ IA-S´ ANCHEZ, HERRERA-POYATOS, AND MOREE Lemma 6.4. Let Sbe a numerical semigroup. Then Sis Betti-divisible if and only if E(S)is totally ordered by the divisibility order. Proof. Let Sbe a numerical semigroup such that either Betti(S) or E(S) is totally ordered by the divisibility order. Then, by Lemma 6.2,Sis Betti-sorted and Betti(S) = E(S). It follows that Betti(S) and E(S) are totally ordered by the divisibility order.  Betti-divisible numerical semigroups are rare, but they have a very rich structure. In fact, it can be shown that these are the numerical semigroups that are free for any arrangement of their minimal generators, see [10, Theorem 7.10]. Lemma 6.5. Let Sbe a numerical semigroup minimally generated by A. Then Shas a unique Betti element if and only if E(S)is a singleton. Proof. Let Sbe a numerical semigroup such that Betti(S) or E(S) is a singleton. Then Sis Bettisorted by Lemma 6.2 and Betti(S) = E(S), so both Betti(S) and E(S) are singletons.  Theorem 1.3 now follows by combining Lemmas 6.2,6.4 and 6.5. 7. Applications to cyclotomic numerical semigroups and open questions We can now use our freshly enriched insight on the connections between cyclotomic exponent sequences and Betti elements to prove that certain cyclotomic numerical semigroups are complete intersections. We do so by showing that these numerical semigroups satisfy the hypotheses of Proposition 2.5 and are, as such, complete intersections. This approach has already been carried out in Corollary 6.3, where we showed that Betti-sorted numerical semigroups are complete intersections. In fact, here we extend Corollary 6.3 to a larger family of numerical semigroups. First, let us consider the following conjectures. Conjecture 7.1. Let Sbe a cyclotomic numerical semigroup and let ebe its cyclotomic exponent sequence. Then n∈Nis a minimal generator of Sif and only if en<0. Conjecture 7.2. Let Sbe a cyclotomic numerical semigroup and let ebe its cyclotomic exponent sequence. Then eb= nc(∇b)−1for all b∈Betti(S). In particular, we have Betti(S)⊆ E(S). These conjectures are motivated by the following result. Proposition 7.3. Conjecture 1.4 holds if and only if Conjectures 7.1 and 7.2 hold. Proof. First, we note that Conjectures 7.1 and 7.2 are directly implied by Conjecture 1.4 and Proposition 2.5. Now let us assume that Sis a cyclotomic numerical semigroup such that Conjectures 7.1 and 7.2 hold for Sand let us prove that Conjecture 1.4 holds for Sor, equivalently, that Sis a complete intersection. Since Conjecture 7.1 holds for S, from Proposition 2.3 and Theorem 1.1 we obtain 0 = X d≥1 ed=−e(S) + X d≥1 ed>0 ed. From these equalities, Conjecture 7.2 and the fact that e1= 1 (by Theorem 1.1), we conclude that e(S) = X d≥1 ed>0 ed≥1 + X b∈Betti(S) (nc(∇b)−1), which shows that the cardinality of any minimal presentation of Sis bounded by e(S)−1,and therefore that Sis a complete intersection (see Section 2.4).  Theorem 1.1 shows one direction of Conjecture 7.1. Here we show that this conjecture holds for a large set of cyclotomic numerical semigroups. CYCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 23 Corollary 7.4. Let Sbe a numerical semigroup minimally generated by A. If U(E(S)) = E(S), then E(S)⊆Betti(S),Sis cyclotomic and Conjecture 7.1 holds for S. Proof. From Theorem 1.2, we find that E(S) = U(Betti(S)) and that eb= nc(∇b)−1>0 for every b∈ E(S). In particular, E(S) is finite and, thus, Sis cyclotomic (Definition 1.1). Let n∈Nwith en<0. We have n6∈ E(S) because eb>0 for every b∈ E(S). Moreover, recall that e0= 0 and e1= 1 by Theorem 1.1, so n≥2. Since E(S) is the set of positive integers jsuch that j≥2, ej6= 0 and jis not a minimal generator, we conclude that nis a minimal generator of S. As already mentioned in the introduction, computations suggest that these numerical semigroups arise very frequently. Corollary 7.5. Let Sbe a cyclotomic numerical semigroup. If Betti(S) = U(Betti(S)) and Conjecture 7.1 holds for S, then Sis a complete intersection. Proof. From Theorem 1.2 and Betti(S) = U(Betti(S)), we obtain Betti(S)⊆ E(S). The result now follows from Proposition 7.3. Corollary 7.6. Let Sbe a numerical semigroup. If Betti(S) = U(Betti(S)) and E(S) = U(E(S)), then Sis a complete intersection. Proof. This follows by combining Corollaries 7.4 and 7.5. Example 7.7.Let S=h8,12,18,25i. Then PS(x) = (1 −x)(1 −x24)(1 −x36)(1 −x50) (1 −x8)(1 −x12)(1 −x18)(1 −x25). Then E(S) = Betti(S) = {24,36,50}. The graph (Betti(S),≤S) is depicted below. 36 50 24 From this graph it follows that Betti(S) = U(Betti(S)) and, thus, Sis a complete intersection. Finally, let us make a few comments on Conjecture 7.1. In [7, Lemma 14] it is shown that this conjecture holds true under several restrictions on S. We notice that the restrictions in part (a) and (b) of Lemma 14 from [7] cannot both hold at the same time, hence the statement of [7, Lemma 14] is void, in the sense that it does not find cyclotomic numerical semigroups satisfying Conjecture 7.1. We conclude this section by improving [7, Lemma 14]. Proposition 7.8. Let Sbe a cyclotomic numerical semigroup with cyclotomic exponent sequence e. Let j∈Nwith ej<0and j < min{d∈N:ed>0}. Then jis a minimal generator of S. As a consequence, if max{d∈N:ed<0}<min{d∈N:ed>0}, then Conjecture 7.1 holds for S. Proof. Let j∈Nwith ej<0 and j < min{d∈N:ed>0}. In view of Theorem 1.1, either jis a minimal generator or d(j)≥2. In the latter case, by Theorem 5.3, there is α∈Minimals≤SE(S) with α≤Sj, and that eα>0. However, this implies that min{d∈N:ed>0} ≤ α < j < min{d∈N:ed>0}, a contradiction. We conclude that jmust be a minimal generator of S. 24 CIOLAN, GARC´ IA-S´ ANCHEZ, HERRERA-POYATOS, AND MOREE 7.1. Open questions. Regarding cyclotomic exponent sequences of arbitrary numerical semigroups, it would be interesting to study the values of these sequences at those Betti elements that are not in U(Betti(S)), where our current techniques fail to yield any result. Coming back to cyclotomic numerical semigroups, by Definition 1.1 a numerical semigroup is cyclotomic if and only if its cyclotomic exponent sequence has finitely many non-zero terms. By Proposition 2.4 this is equivalent with PSbeing a product of cyclotomic polynomials. Question 7.9. Is there a weaker condition than the cyclotomic exponent sequence having finite support that would ensure that PSis a product of cyclotomic polynomials? A possible way to weaken the condition would be, for instance, to require that the exponent sequence has infinitely many zeros. We point out that Conjecture 1.4 remains open, and it seems likely that further tools are needed in order to tackle it. One could start by showing that if Sis a numerical semigroup such that |E(S)| ≤ 2, then Sis a complete intersection. Here we have managed to address the case |E(S)|= 1 in Theorem 1.3, but our techniques are not enough to analyze the case when |E(S)|= 2 and the two elements in E(S) are incomparable with respect to ≤S. As seen in this section, Conjectures 7.1 and 7.2 are equivalent with Conjecture 1.4. It is thus well possible that at least one of the two is considerably easier than Conjecture 1.4, and thus they warrant individual investigation. Acknowledgments Part of the work on this paper was done during an internship in the Fall of 2016 carried out by the third author at the Max Planck Institute for Mathematics in Bonn and during a one-week visit in April 2019. He would like to thank the fourth author for the invitation and the institute staff for their hospitality and support. The project was completed during a stay of the first author at the same institute. Substantial progress on this paper was made in February 2017, when the first and the fourth author were invited by the second and third author for one week to the University of Granada. They are grateful for the hospitality, for the inspiring and cheerful atmosphere and, last but not least, for the excellent tapas and wine! References [1] A. Assi and P. A. 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Thangadurai, On the coefficients of cyclotomic polynomials, in “Cyclotomic fields and related topics” (Pune, 1999), Bhaskaracharya Pratishthana, Pune, 2000, 311–322. Max-Planck-Institut f¨ ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany Email address:[email protected] Departamento de ´ Algebra, Universidad de Granada, E-18071 Granada, Espa˜ na Email address:[email protected] Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK. Email address:andres.herrerapoyato[email protected]x.ac.uk Max-Planck-Institut f¨ ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany Email address:[email protected]