Problems in numerical semigroups

HAL Id: tel-01672101

https://hal.archives-ouvertes.fr/tel-01672101v2

Submitted on 17 Apr 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Mariam Dhayni

To cite this version:

Mariam Dhayni. Problems in numerical semigroups. General Mathematics [math.GM]. Université d’Angers, 2017. English. �NNT : 2017ANGE0041�. �tel-01672101v2�

Thèse de Doctorat

Mariam D ^HAYNI

Mémoire présenté en vue de l’obtention du grade de Docteur de l’Université d’Angers sous le sceau de l’Université Bretagne Loire

École doctorale : Sciences et technologies de l’information, et mathématiques Discipline : Mathématiques et leurs interactions, section CNU 25

Unité de recherche : Laboratoire Angevin de Recherche en Mathématiques (LAREMA) Soutenue le 07 Décembre 2017

Problèmes dans la théorie des semigroupes numériques

JURY

Rapporteurs : M. Manuel DELGADO, Professeur, Université de Porto

M. Jorge RAMÍREZ-ALFONSÍN, Professeur, Université de Montpellier Examinateurs : M. Pedro GARCÍA-SÁNCHEZ, Professeur, Université de Grenade

M^meMonique LEJEUNE-JALABERT, Directeur de recherche au CNRS, Université de Versailles-Saint Quentin Directeur de thèse : M. Abdallah ASSI, Maître de Conférences HDR, Université d’Angers

Thèse

pour obtenir le grade de

Docteur ès Mathématiques

présentée à l’Université d’Angers par

Mariam Dhayni

Problèmes dans la théorie des semigroupes numériques

soutenue le 07 Décembre 2017 devant le jury composé de :

M. Manuel Delgado Professeur à l’Université de Porto Rapporteur

M. Jorge Ramírez-Alfonsín Professeur à l’Université de Montpellier Rapporteur M. Pedro García-Sánchez Professeur à l’Université de Grenade Examinateur Mme. Monique Lejeune-Jalabert Directeur de recherche au CNRS Examinateur M. Abdallah Assi Maître de Conférences HDR à l’Université d’Angers Directeur de thèse

préparée au Larema - UMR CNRS 6093

Remerciements

I would like to express my deepest gratitude to my advisor, Abdallah Assi, for all his help and guidance through these years which led me to accomplish this work.

I would like to thank the committee members : Manuel Delgado, Jorge Ramírez-Alfonsín, Pedro García- Sánchez and Monique Lejeune-Jalabert for accepting to be my PhD thesis committee and for their insightful comments and discussions throughout this process.

My special thanks goes out to all my friends especially Elena Dayoub, Zeina Hammoud, Ahmad Moussa and Ali Abbas for being by my side and helping me in whatever way they could throughout this PhD.

I greatly appreciate the support and guidance that I received from my friend Muhammad Ghader throughout my years of study.

To my family, a heartfelt thank you for encouraging me in all of my pursuits and inspiring me to follow my dreams. I am especially grateful to my parents, who provided me with unfailing support and continuous encouragement. I always knew that you believed in me.

Finally, my PhD could not have been accomplished without the financial support from the Ministry of Higher Education Research and Innovation of France, I would like to express my sincere gratitude.

Table des matières

Remerciements iii

Introduction 1

1 Basics and notations 3

2 Wilf ’s conjecture 9

2.1

Equivalent form of Wilf’s conjecture

2.2

Technical results

2.3

Numerical semigroups with w

≥ w

+ w

and (2 +

)ν ≥ m

2.4

Numerical semigroups with w

≥ w

+ w

and (

)ν ≥ m

2.5 Numerical semigroups with 2 + ^bwxmc(y−x−1)+(y−2)+b^wy_mc(x−1) b^wx_mc+b^wy_mc+2 ν ≥mandwm−1−m≥wx+wy 28 2.6

Numerical semigroups with m − ν >

3 Numerical semigroup of the form < m, m+ 1, . . . , m+l, k(m+l) +r > 35 3.1

Apéry set of S

3.2

Frobenius number of S

3.3

Genus of S

3.4

Determination of symmetric and pseudo-symmetric numerical semigroups

3.4.1 Determination of symmetric numerical semigroup . . . 47

3.4.2 Determination of pseudo-symmetric numerical semigroup . . . 52

3.5

Pseudo-Frobenius Numbers

Bibliography 78

Introduction

LetNdenote the set of natural numbers, including0. A semigroupS is an additive submonoid of(N,+), that is0∈S and if a, b∈S, then a+b∈S. A numerical semigroupS is a submonoid ofNof finite complement, i.e.,N\S is a finite set. It can be shown that a submonoid ofNis a numerical semigroups if and only if the group generated by S inZ(namely the set of elements ^Ps_i=1λ_ia_i, λ_i∈Z, a_i∈S) isZ.

There are many invariants associated to a numerical semigroupS. The Apéry setof S with respect to an elementa∈S is defined as

Ap(S, a) ={s∈S; s−a /∈S}.

The elements ofN\S are called thegaps ofS. The largest gap is denoted by f =f(S) =max(N\S)

and is called theFrobenius numberofS. The numberf(S) + 1is known as the conductor ofS and denoted byc or c(S).The number of gaps

g=g(S) =|N\S|

is known as genusof S. The smallest non zero element m=m(S)of S is called themultiplicity of S and the set{s∈S;s < f(S)}is denoted by n(S). Every numerical semigroupS is finitely generated, i.e., S is of the form

S =< g1, . . . , gν >=Ng1+. . .+Ngν

for suitable unique coprime integersg₁, . . . , g_ν.The number of minimal set of generators ofS is denoted by ν=ν(S)

and is called theembedding dimensionofS. An integerx∈N\Sis called apseudo-Frobenius number if x+S\0 ⊆S. The type of the semigroup, denoted by t(S) is the cardinality of set of pseudo-frobenius numbers. We have formulas linking these invariants.

Frobenius in his lectures proposed the problem of giving a formula for the largest integer that is not repre- sentable as a linear combination with nonnegative integer coefficients of a given set of positive integers whose greater common divisor is one. He also threw the question of determining how many positive integers do not have such a representation. This problem is known as Diophantine Frobenius Problem. Using the terminology of numerical semigroups, the problem is to give a formula, in terms of the elements in a minimal system of generators of a numerical semigroupS, for the greatest integer not inS. This problem, introduced and solved by Sylvester for the case ν = 2 (see [21]), has been widely studied. For ν = 3, in 1962 Brauer and Shockly (see [6]) found a formula for the Frobenius number but their solution was not a polynomial in the generators and it involved magnitudes which could not be expressed by the generators. Later on, more solutions to this case were found by using different methods (for example see [20]). However, all of these methods do not give explicit formula of the Frobenius number in terms of the generators. Generally, it has been proved in [15] thatf(S) is not algebraic in the set of generators ofS.

In [22] 1978 H. S. Wilf proposed a conjecture suggesting a regularity in the setN\S. It says the following : f(S) + 1≤ν(S)n(S).

Although the problem has been considered by several authors (cf. [2], [4], [9], [10], [11], [14], [19], [23] ), only special cases have been solved and it remains wide open. In [9], D. Dobbs and G. Matthews proved Wilf’s

conjecture forν ≤3. In [14], N. Kaplan proved it forc≤2m and in [10] S. Eliahou extended Kaplan’s work for c≤3m.

In Chapter1, we recall some basics about numerical semigroups that will be used through the thesis.

In Chapter 2, we generalize the case covered by A. Sammartano in [19], who showed that Wilf’s conjecture holds for2ν≥m, andm≤8, based on the idea of counting the elements ofS in some intervals of lengthm.

We use different intervals in order to get an equivalent form of Wilf’s conjecture and then we prove it in some relevant cases. In particular our calculations cover the case where 2ν ≥m, proved by Sammartano in [19].

Here are few more details on the contents of this Chapter. Section2.1is devoted to give some notations that will enable us in the same Section to give an equivalent form of Wilf’s conjecture. In Section 2.2, we give some technical results needed in the Chapter. Let Ap(S, m) = {0 = w₀ < w₁ < · · · < wm−1}. In Section 2.3, first, we show that Wilf’s conjecture holds for numerical semigroups that satisfy wm−1 ≥w₁+w_α and (2 +^α−3_q )ν≥mfor some 1< α < m−1wherec=qm−ρ for someq ∈N,0≤ρ≤m−1. Then, we prove Wilf’s conjecture for numerical semigroups withm−ν ≤4in order to cover the case where2ν ≥m. We also show that a numerical semigroup withm−ν = 5verify Wilf’s conjecture in order to prove the conjecture for m= 9. Finally, we show in this Section, using the previous cases, that Wilf’s conjecture holds for numerical semigroups with (2 + ¹_q)ν ≥ m. In Section 2.4, we prove Wilf’s conjecture for numerical semigroups with wm−1 ≥wα−1+wα and (^α+3₃ )ν ≥m for some 1 < α < m−1. In Section 2.5, we show Wilf’s conjecture holds for numerical semigroups withwm−1−m≥w_x+w_y and 2 +^b

wx

mc(y−x−1)+(y−2)+b^wy

mc(x−1) b^wx_mc+b^wy_mc+2

ν≥m. The last Section 2.6aims to verify the conjecture in the case m−ν > ^{(n−2)(n−3)}₂ and also in the casen≤5.

Exact determination of Ap(S, m), f(S),g(S) andP F(S) is a difficult problem. WhenS is generated by an arithmetic sequence < m, m+ 1, . . . , m+l >, Brauer [5] gave a formula for f(S). Roberts [17] extended this result to generators in arithmetic progression (see also [3], [24]). Selmer [20] and Grant [13] generalized this to the case S =< m, hm+d, hm+ 2d, . . . , hm+ld >. In [16], it has been considered the case of semigroups generated by{m, m+d, . . . , m+ld, c} (called almost arithmetic semigroups) where it has been given a method to determine Ap(S, m) and also symmetric almost arithmetic semigroups. In [12], pseudo symmetric almost arithmetic semigroups have been characterized. In Chapter3, we focus our attention on numerical semigroup consisting of all non-negative integer linear combinations of relatively prime positive integersm, m+ 1, . . . , m+l, k(m+l) +r wherek, m, l, rare positive integers and r≤(k+ 1)l+ 1. We give formulas for Ap(S, m), f(S), g(S) and P F(S). We also determine the symmetric and the pseudo symmetric numerical semigroups of this form. Note that our semigroups< m, m+ 1, . . . , m+l, k(m+l) +r >are almost arithmetic semigroups. The advantage is that we are able for this class of semigroups to determine all the invariants with simple formulas.

Good references on numerical semigroups are [18] and [1].

1

Basics and notations

Definition 1.0.1. LetS be a subset of N. We say thatS is a submonoidof (N,+) if the following holds :

• 0∈S.

• Ifa, b∈S, thena+b∈S.

Remark 1.0.2. All semigroups considered in this thesis are submonoids of (N,+), hence commutative, that

is,a+b=b+afor all a, b∈S.

Example 1.0.3. Consider the following examples :

• {0} andN are trivially submonoids ofN.

• Letdbe an element ofN, the set dN={da:a∈N} is a submonoid ofN.

Definition 1.0.4. Let S be a submonoid of N. If N\S is a finite set, then S is said to be a numerical

semigroup.

We have the following characterization of numerical semigroups :

Proposition 1.0.5. (See Lemma 2.1 in [18]) Let S 6= {0}, and S 6=N be a semigroup of N and letG be the subgroup of Z generated by S, i.e., (G = {^Ps_i=1λiai, s ∈ N, λi ∈ Z, ai ∈ S}). Then, S is a numerical

semigroup if and only ifG=Z, i.e., (gcd(S)=1).

Proposition 1.0.6. (See Proposition 2.2 in [18]) Let S be a semigroup of N. Then, S is isomorphic to a

numerical semigroup.

Definition 1.0.7. Let S be a numerical semigroup and let A ⊆S. We say that S is generated byA and we writeS =< A >if for all s∈S, there exista₁, . . . , a_r ∈A and λ₁, . . . , λ_r ∈Nsuch that a=^Pr_i=1λ_ia_i. We say that S is finitely generated ifS =< A > withA⊆S and A is a finite set.

Remark 1.0.8. Through this thesis X^∗ will stand for X\ {0}.

Next, we introduce an important tool associated to a numerical semigroup.

Definition 1.0.9. Let n∈S^∗. We define the Apéry setof S with respect to n, denoted by Ap(S, n), to be the set

Ap(S,n) ={s∈S : s−n∈/ S}.

Remark 1.0.10. Given a non zero integern and two integersa and b, we write a≡b mod (n) to denote thatn dividesa−b. We denote by bmod nthe remainder of the division of b byn.

From Definition1.0.9, we can easily see the following.

Lemma 1.0.11. LetS be a numerical semigroup and letn∈S^∗. For all 1≤i≤n, letw(i) be the smallest element ofS such that w(i)≡imod (n). We have the following :

Ap(S,n) ={0,w(1), . . . ,w(n−1)}.

Proposition 1.0.12. LetS be a numerical semigroup. Letn∈S^∗and let Ap(S, n) ={w₀ < w₁. . . < wn−1} be the Apéry set ofS with respect to n. We have the following :

• w₀ =w(0) = 0.

• |Ap(S, n)|=n.

Proposition 1.0.13. (See Lemma 2.6 in [18]) LetSbe a numerical semigroup and letn∈S^∗. For alls∈S, there exists a unique (k, w)∈N×Ap(S,n) such thats=kn+w.

As a consequence of Proposition1.0.13, we obtain the following property.

Corollary 1.0.14. (Theorem 2.7 in [18]) LetS be a numerical semigroup. Then,S is finitely generated.

Definition 1.0.15. Let S be a numerical semigroup and let A ⊆ S^∗. We say that A is a minimal set of generators of S if S =< A > and for all x ∈ A, x cannot be written as a linear combination with

nonnegative integer coefficients of other elements inA.

Corollary 1.0.16. (See Corollary 2.8 in [18]) LetS be a numerical semigroup. Then,S has a minimal set

of generators. This set is finite and unique.

Definition 1.0.17. LetS be a numerical semigroup. We define the following invariants :

• The embedding dimension of S denoted by ν(S), or ν for simplicity, is the cardinality of the minimal set of generators ofS.

• Themultiplicityof S denoted by m(S), orm for simplicity, is the smallest non zero element of S.

Lemma 1.0.18. (See Proposition 2.10 in [18]) Let S be a numerical semigroup with multiplicity m and

embedding dimensionν. We have ν ≤m.

Let us recall some basic and important invariants of numerical semigroups.

Definition 1.0.19. LetSbe a numerical semigroup. We introduce some invariants associated to a numerical semigroupS :

• We define theFrobenius number ofS, denoted by f orf(S) to be max (Z\S).

• We define theconductor ofS, denoted by corc(S) to bef(S) + 1.

• We define theset of gapsof S, denoted byG(S) to beN\S.

• We define thegenus ofS, denoted by g(S) to be the cardinality ofG(S).

• We denote byn(S), the cardinality of {s∈S :s≤f(S)}.

Remark 1.0.20. Note thatf(S)≥1 for all non trivial numerical semigroups.

Lemma 1.0.21. (See in [6], [20]) Let S be a numerical semigroup and let n∈S. Then,

• f(S) = max(Ap(S,n))−n.

• g(S) = 1 n

X

w∈Ap(S,n)

w−1

2(n−1).

Definition 1.0.22. LetS be a numerical semigroup. We say that x∈Nis a pseudo-Frobenius number ifx /∈S and x+s∈S for alls∈S^∗. We denote by P F(S) the set of all pseudo-Frobenius numbers of S.

We denote the cardinality ofP F(S) by t(S) and we call it the type of S. It results from the definition of

f(S) thatf(S)∈P F(S), and alsof(S) = max (P F(S)).

Corollary 1.0.23. (See Theorem 20 in [11]) LetS be a numerical semigroup with Frobenius numberf(S), type t(S) andn(S) =|{s∈S :s < f(S)}|. Then, we have

f(S) + 1≤(t(S) + 1)n(S).

Wilf’s conjecture :Let the notations be as before. The problem whether f(S) + 1≤n(S)ν(S) is known as Wilf conjecture [22]. For some families of numerical semigroups this conjecture is known to be true, but the general case remains unsolved.

Remark 1.0.24. By Corollary1.0.23, ift(S)≤ν(S)−1, then S satisfies Wilf’s conjecture.

Definition 1.0.25. Leta, b∈N. We define≤_S as follows : a≤_S bif and only if b−a∈S.

Remark 1.0.26. As S is a numerical semigroup, it easily follows that ≤_S is an order relation over S

(reflexive, transitive and anti symmetric).

Definition 1.0.27. Let S be a numerical semigroup and n∈ S^∗. Let Ap(S, n) = {w₀ = 0 < w₁ < w₂ <

. . . < wn−1} be the Apéry set ofS with respect to n. Then, define the following sets :

min_≤S(Ap(S, n)) ={w∈Ap(S, n)^∗ such thatw is minimal with respect to ≤_S}.

max_≤S(Ap(S, n)) ={w∈Ap(S, n)^∗ such thatw is maximal with respect to ≤_S}.

Lemma 1.0.28. (See Lemma 6 in [11]) LetS be a numerical semigroup,n∈S^∗ and Ap(S, n) be the Apéry set ofS with respect ton. Letw∈ Ap(S, n) and u∈S. If there existv∈S such that w=u+v, thenu∈

Ap(S, n).

Corollary 1.0.29. Let x∈Ap(S, n)^∗. We have the following :

• x∈min_≤S(Ap(S, n)) if and only if x6=w_i+w_j for allw_i, w_j ∈Ap(S, n)^∗.

• x∈max≤_S(Ap(S, n)) if and only if wi 6=x+wj for all wi, wj ∈Ap(S, n)^∗. Proof.Letx∈Ap(S, n)^∗.

• Let x ∈ min≤_S(Ap(S, n)). Suppose by the way of contradiction that x =wi+wj for some wi, wj ∈ Ap(S, n)^∗. Then,x=wi+wj withwi ∈Ap(S, n)andwj ∈S which implies thatx /∈min_≤S(Ap(S, n)) and we get a contradiction.

Conversely, suppose thatx 6=wi+wj for all wi, wj ∈Ap(S, n)^∗. Suppose by the way of contradiction that x /∈ min_≤S(Ap(S, n)), then there exist wi ∈ Ap(S, n) and s ∈ S such that x = wi +s. By Lemma1.0.28, it follows thats∈Ap(S, n). Thus,x=w_i+ssuch that w_i, s∈Ap(S, n) which gives a contradiction.

• Let x ∈max≤_S(Ap(S, n)). Suppose by the way of contradiction that w_i =x+w_j for some w_i, w_j ∈ Ap(S, n)^∗. Then,wi =x+wj withwi∈Ap(S, n)andwj ∈Swhich implies thatx /∈max_≤S(Ap(S, n)) and we get a contradiction.

Conversely, suppose that w_i 6=x+w_j for allw_i, w_j ∈ Ap(S, n)^∗. Suppose by the way of contradiction that x /∈ max_≤S(Ap(S, n)), then there exist wi ∈ Ap(S, m) and s ∈ S such that wi = x+s. By Lemma1.0.28, it follows thats∈Ap(S, n). Thus,w_i =x+ssuch that w_i, s∈Ap(S, n) which gives a contradiction.

Thus, the proof is complete.

Proposition 1.0.30. (See Lemma 3.2 in [7] ) Let S be a numerical semigroup with multiplicity m and embedding dimension ν and let n ∈ S^∗. Let Ap(S, n) be the Apéry set of S with respect to n and let {g₁< g₂ < . . . < g_ν} be the minimal set of generators ofS. We have the following :

• g₁ =m.

• min_≤S(Ap(S, n))={g₂, ..., g_ν}.

• max_≤S(Ap(S, n))={w∈ Ap(S, n) such thatw−nis a pseudo-Frobenius number of S}.

From Proposition1.0.30, it follows Corollary1.0.31.

Corollary 1.0.31. Let S be a numerical semigroup with multiplicity m, embedding dimension ν and {g₁=m, g₂, . . . , g_ν} the minimal system of generators ofS. Letn∈S^∗ and Ap(S, n) be the Apéry set ofS with respect ton. We have the following :

• |min_≤S(Ap(S, n))|=ν−1.

• |Ap(S, n)^∗\min_≤S(Ap(S, n))|=n−ν.

• |max_≤S(Ap(S, n))|=t(S).

We introduce in Definitions1.0.32and 1.0.33special kind of numerical semigroups and give some properties of this kind in Lemma1.0.34.

Definition 1.0.32. A numerical semigroup is said toirreducible if and only if S cannot be expressed as the intersection of two numerical semigroupsS1, S2 such that S⊂S1, S ⊂S2. Definition 1.0.33. LetS be a numerical semigroup. We have the following :

• S is said to be symmetricif and only if S is irreducible andf(S) is odd.

• S is said to be pseudo-symmetricif and only if S is irreducible and f(S) is even.

Lemma 1.0.34. (See Corollary 4.5 in [18]) Let S be a numerical semigroup with Frobenius number f(S) and genus g(S). We have the following :

• S is symmetric if and only if g(S) = f(S) + 1

2 .

• S is pseudo-symmetric if and only if g(S) = f(S) + 2

2 .

Remark 1.0.35. Consider the following notation that will be used throug this thesis :

• We denote by floor (x) =bxc the largest integer less than or equal to x.

• We denote by ceil (x)= dxe the smallest integer greater than or equal tox.

2

Wilf’s conjecture

In this chapter, we give an equivalent form of Wilf’s conjecture in terms of the elements of the Apéry set of S, embedding dimension and the multiplicity. We also give an affirmative answer to Wilf’s conjecture in some cases.

2.1 Equivalent form of Wilf’s conjecture

Let the notations be as in the introduction. For the sake of clarity we shall use the notations ν, f, n, c... for ν(S), f(S), n(S), c(S).... In this Section, we will introduce some notations and family of numbers that will enable us to give an equivalent form of Wilf’s conjecture at the end of this Section.

Notation.LetS be a numerical semigroup with multiplicity m and conductorc=f+ 1. Denote by q =dc

me.

Thus,qm≥cand c=qm−ρ with 0≤ρ < m.

Given a non negative integerk, we define the kth interval of length m,

I_k = [km−ρ,(k+ 1)m−ρ[={km−ρ, km−ρ+ 1, . . . ,(k+ 1)m−ρ−1}.

We denote by

nk=|S∩Ik|.

Forj∈ {1, . . . , m−1}, we define ηj to be the number of intervalsIk with nk =j.

ηj ={k∈N; n_k=j}.

Let 0 ≤ k ≤ q −2. If s ∈ S∩I_k then s+m ∈ S ∩I_k+1. This implies that n_k ≤ n_k+1. Let for example S = h4,6,13i. We have c(S) = c = 16 = 4·4, hence Ik = [km,(k+ 1)m[ for all k ≥ 0. Moreover, n0= 1, n1= 2, n2 = 2, n3 = 3, andn_k= 4 for all k≥4. We also have η1= 1, η2 = 2, η3= 1.

Proposition 2.1.1. Under the previous notations, we have : i) 1≤n_k≤m−1 for all 0≤k≤q−1.

ii) n_k =m for all k≥q.

iii)

q−1

X

k=0

n_k =n(S) =n.

iv)

m−1

X

j=1

ηj =q.

v)

m−1

X

j=1

jηj =

q−1

X

k=0

n_k=n.

Proof.

i) We can easily verify that if S contains m consecutive elements a, a+ 1, . . . a+m−1, then for all n ≥ a+m, n ∈ S. Since (q −1)m−ρ < f < qm−ρ, then it follows that n_k ≤ m −1 for all 0≤k≤q−1. Moreover, km∈S∩I_k for all 0≤k≤q−1, thus n_k ≥1.

ii) We havef =qm−ρ−1∈Iq−1. From the definition of the Frobenius number, it follows thatn_k=m for allk≥q.

iii) ^Pq−1_k=0nk is nothing but the cardinality of{s∈S;s < f}which isn(S) by definition.

iv) We have 1≤S∩I_k≤m−1 if and only if 0≤k≤q−1. This implies our assertion.

v) The sum^Pm−1_j=1 jη_j is nothing but the cardinality of | ∪^q−1_k=0S∩I_k|=n. This proves our assertion.

Thus, the proof is complete.

Next, we will expressηj in terms of th Apéry set.

Proposition 2.1.2. Let Ap(S, m) ={w₀= 0< w₁ < w₂ < . . . < wm−1}.Under the previous notations, for all 1≤j≤m−1, we have

ηj =bwj+ρ

m c − bwj−1+ρ m c.

Proof. Fix 0 ≤ k ≤ q −1 and let 1 ≤ j ≤ m−1. We will show that the interval I_k contains exactly j elements ofS if and only ifwj−1 <(k+ 1)m−ρ≤w_j.

Suppose that I_k contains j elements. Suppose, by contradiction, that wj−1 ≥ (k + 1)m −ρ. We have wm−1> . . . > wj−1 ≥(k+ 1)m−ρ, thus

wm−1, . . . , wj−1∈ ∪^q_t=k+1I_t.

Hence,I_kcontains at mostj−1 elements ofS (namelyw₀+km=km, w₁+k₁m, w₂+k₂m, . . . , wj−2+kj−2m for some k1, . . . , kj−2 ∈ {0, . . . , k−1}). This contradicts the fact that Ik contains exactlyj elements of S.

Hence,wj−1 <(k+ 1)m−ρ.

Ifw_j <(k+ 1)m−ρ, thenw₀ < . . . < w_j <(k+ 1)m−ρ, thus w₀, . . . , w_j ∈ ∪^k_t=0I_t.

Then,I_kcontains at leastj+ 1 elements ofS which are :w₀+km=km, w₁+k₁m, w₂+k₂m, . . . , w_j+k_jm for some k1, . . . , kj ∈ {0, . . . , k−1}, which contradicts the fact that Ik contains exactly j elements of S.

Hence,wj ≥(k+1)m−ρ. Consequently, ifI_kcontains exactlyjelements ofS, thenwj−1<(k+1)m−ρ≤wj. Conversely,wj−1 <(k+ 1)m−ρ implies thatw0 < . . . < wj−1 <(k+ 1)m−ρ, then

w₀, . . . , wj−1 ∈ ∪^k_t=0I_t.

Hence,I_kcontains at leastjelements of S which arew₀+km=km, w₁+k₁m, w₂+k₂m, . . . , wj−1+kj−1m for some k1, . . . , kj−1 ∈ {0, . . . , k−1}. On the other hand, wj ≥ (k+ 1)m−ρ implies that wm−1 > . . . >

w_j ≥(k+ 1)m−ρ, then

wm−1, . . . , w_j ∈ ∪^q_t=k+1I_t.

Thus,Ik contains at mostjelements ofS which are :w0+km=km, w1+k1m, w2+k2m, . . . , wj−1+kj−1m for some k₁, . . . , kj−1 ∈ {0, . . . , k−1}. Hence, if wj−1 < (k+ 1)m−ρ ≤ wj, then I_k contains exactly j elements ofS and this proves our assertion. Consequently,

η_j = |{k∈Nsuch that|I_k∩S|=j}|

= |{k∈Nsuch thatwj−1 <(k+ 1)m−ρ≤wj}|

= |{k∈Nsuch that ^wj−1_m^+ρ <(k+ 1)≤ ^wj_m^+ρ}|

= |{k∈Nsuch that ^wj−1_m^+ρ−1< k≤ ^wj_m^+ρ−1}|

= |{k∈Nsuch thatb^wj−1_m^+ρc ≤k≤ b^wj_m^+ρc −1}|

= b^wj_m^+ρc − b^wj−1_m^+ρc.

Thus, the proof is complete.

Proposition2.1.3gives an equivalent form of Wilf’s conjecture using Propositions 2.1.1and 2.1.2.

Proposition 2.1.3. Let S be a numerical semigroup with multiplicity m, embedding dimension ν and conductorf + 1 =qm−ρ for someq ∈N and 0≤ρ≤m−1. Let w₀ = 0< w₁ < w₂ < . . . < wm−1 be the elements of Ap(S, m). Then,S satisfies Wilf’s conjecture if and only if

m−1

X

j=1

(bw_j+ρ

m c − bwj−1+ρ

m c)(jν−m) +ρ≥0.

Proof.By Proposition 2.1.1, we have f+ 1≤nν ⇔qm−ρ≤ν

q−1

X

k=0

n_k⇔

q−1

X

k=0

m−ρ≤

q−1

X

k=0

n_kν ⇔

q−1

X

k=0

(n_kν−m) +ρ≥0.

Equivalently, we obtain

m−1

X

j=1

ηj(jν−m) +ρ≥0.

By applying Proposition2.1.2, we get

m−1

X

j=1

η_j(jν−m) +ρ≥0⇔

m−1

X

j=1

(bwj+ρ

m c − bwj−1+ρ

m c)(jν−m) +ρ≥0.

Thus, the proof is complete.

2.2 Technical results

Let S be a numerical semigroup and let the notations be as in Section 2.1. In this Section, we give some technical results will be used through the Chapter.

Remark 2.2.1. Let Ap(S, m) = {w₀ = 0< w1 < . . . < wm−1}. The following technical remarks will be used through the Chapter :

i) bw0+ρ m c= 0.

ii) For all 1≤i≤m−1, we havebw_i+ρ m c ≥1.

iii) For all 1≤i≤m−1, we have eitherbw_i+ρ

m c=bw_i

mc orbw_i+ρ

m c=bw_i mc+ 1.

iv) Ifbw_i+ρ

m c=bw_i

mc+ 1, then bw_i+ρ

m c ≥2 andρ≥1.

v) For all 0≤i < j≤m−1, we havebw_i+ρ

m c ≤ bw_j+ρ m c.

vi) bwm−1+ρ m c=q.

Proof.

i) This is becausew₀= 0 and 0≤ρ < m.

ii) We havem < w_i for all 1≤i≤m−1. This implies the result sinceρ≥0.

iii) For all 1≤i≤m−1, letw_i =q_im+r_isuch thatq_i, r_i ∈Nandr_i < m. We haveb^w_mⁱc=q_i. Therefore, bw_i+ρ

m c=bq_im+r_i+ρ

m c=bq_i+r_i+ρ

m c=q_i+br_i+ρ

m c=bw_i

mc+br_i+ρ m c.

Since 0≤ρ, r_i< m, it follows that 0≤ ^ri_m^+ρ <2. Consequently, 0≤ b^ri_m^+ρc ≤1. Hence, bw_i

mc ≤ bw_i+ρ

m c ≤ bw_i mc+ 1.

Equivalently,

bw_i+ρ

m c=bw_i

mc orbw_i+ρ

m c=bw_i mc+ 1.

iv) Suppose thatb^wi_m^+ρc=b^w_mⁱc+ 1. By using part ii), we get b^wi_m^+ρc ≥2. In this caseρ≥1 (asρ≥0).

v) By definition, we have wi < wj for all 0 ≤ i < j ≤ m−1. Thus, ^wi_m^+ρ < ^wj_m^+ρ. Consequently, b^wi_m^+ρc ≤ b^wj_m^+ρc.

vi) By Lemma 1.0.21, we have f = max(Ap(S, m))−m = wm−1−m. Hence, b^wm−1_m^+ρc =b^f+m+ρ_m c = b^{qm−ρ−1+m+ρ}_m c=q.

Thus, the proof is complete.

Let 1 < α < m−1. Using Remark 2.2.1, we get the following inequalities which will be used later in the Chapter :

α

X

j=1

(bwj+ρ

m c − bwj−1+ρ

m c)(jν−m)

=

α

X

j=1

bw_j +ρ

m c(jν−m)−

α

X

j=1

bwj−1+ρ

m c(jν−m)

=

α

X

j=1

bwj +ρ

m c(jν−m)−

α−1

X

j=0

bwj+ρ

m c (j+ 1)ν−m

=

α−1

X

j=1

bw_j+ρ

m c(jν−m) +bw_α+ρ

m c(αν−m)−bw₀+ρ

m c(ν−m)−

α−1

X

j=1

bw_j+ρ

m c (j+ 1)ν−m

=bwα+ρ

m c(αν−m)− bw0+ρ

m c(ν−m)−

α−1

X

j=1

bwj+ρ m cν

=bw_α+ρ

m c(αν−m)− bw₀+ρ

m c(ν−m)− bw₁+ρ m cν−

α−1

X

j=2

bw_j+ρ m cν

=bwα+ρ

m c(αν−m)− bw1+ρ m cν−

α−1

X

j=2

bwj+ρ

m cν (asbw0+ρ

m c= 0).

From Remark2.2.1(v), we have b^wj_m^+ρc ≤ b^wα_m^+ρc ∀ 2≤j≤α−1. Hence,

α

X

j=1

(bw_j+ρ

m c − bwj−1+ρ

m c)(jν−m)

≥ bwα+ρ

m c(αν−m)− bw1+ρ m cν−

α−1

X

j=2

bwα+ρ m cν

=bw_α+ρ

m c(αν−m)− bw₁+ρ

m cν− bw_α+ρ

m c(α−2)ν

=−bw1+ρ

m cν+bwα+ρ

m c(2ν−m).

Consequently, we have

α

X

j=1

(bwj+ρ

m c − bwj−1+ρ

m c)(jν−m)≥ −bw1+ρ

m cν+bwα+ρ

m c(2ν−m). (2.2.1) Therefore,

m−1

X

j=α+1

(bw_j+ρ

m c − bwj−1+ρ

m c)(jν−m)

≥

m−1

X

j=α+1

bw_j+ρ

m c − bwj−1+ρ

m c (α+ 1)ν−m (asj ≥α+ 1 and bw_j+ρ

m c ≥ bwj−1+ρ m c)

= (α+ 1)ν−m

m−1

X

j=α+1

bw_j +ρ

m c − bwj−1+ρ m c

= (α+ 1)ν−m

m−1

X

j=α+1

bw_j+ρ m c −

m−1

X

j=α+1

bwj−1+ρ m c

= (α+ 1)ν−m

m−1

X

j=α+1

bw_j+ρ m c −

m−2

X

j=α

bw_j+ρ m c

= (α+ 1)ν−m

m−2

X

j=α+1

bw_j+ρ

m c+bwm−1+ρ

m c−bw_α+ρ m c −

m−2

X

j=α+1

bw_j+ρ m c

= bwm−1+ρ

m c − bwα+ρ

m c (α+ 1)ν−m. Hence, we obtain

m−1

X

j=α+1

(bw_j+ρ

m c − bwj−1+ρ

m c)(jν−m)≥ bwm−1+ρ

m c − bw_α+ρ

m c (α+ 1)ν−m. (2.2.2) The following technical Lemma will be used through the Chapter :

Lemma 2.2.2. Let Ap(S, m) ={w₀ = 0< w1 < . . . < wm−1} and suppose that wi ≥ wj +wk. We have the following :

i) bwi+ρ

m c ≥ bwj+ρ

m c+bw_k+ρ m c −1.

ii) Ifbw_i+ρ

m c − bw_j+ρ

m c − bw_k+ρ

m c=−1, then bw_j+ρ

m c=bw_j

mc+ 1, bw_k+ρ

m c=bw_k

mc+ 1 and ρ≥1.

In particular,

bw_j+ρ

m c ≥2, bw_k+ρ

m c ≥2 andρ≥1.

Proof.

i) Assume thatwi ≥wj+wk. Then,wi+ρ≥wj+wk+ρ. Consequently, w_i+ρ

m ≥ w_j+w_k+ρ

m ⇒ bw_i+ρ

m c ≥ bw_j +w_k+ρ

m c.

Therefore, we have

bw_i+ρ

m c ≥ bw_j +ρ

m c+bw_k mc.

By Remark2.2.1(iii), b^w_m^kc ≥ b^wk_m^+ρc −1. Hence, bwi+ρ

m c ≥ bwj+ρ

m c+bw_k+ρ m c −1.

ii) Suppose that w_i ≥ w_j +w_k and that b^wi_m^+ρc − b^wj_m^+ρc − b^wk_m^+ρc = −1. Suppose by the way of contradiction thatb^wj_m^+ρc 6=b^w_m^jc+ 1 orb^wk_m^+ρc 6=b^w_m^kc+ 1 or ρ <1.By Remark2.2.1(iii) and that ρ≥0, it follows that

bw_j+ρ

m c=bw_j

mc orbw_k+ρ

m c=bw_k

mc orρ= 0.

Sincew_i ≥w_j+w_k, we have

bw_i+ρ

m c ≥ bw_j +w_k+ρ

m c.

Sinceb^wj_m^+ρc=b^w_m^jcorb^wk_m^+ρc=b^w_m^kc orρ= 0, it follows that bw_i+ρ

m c ≥ bw_j+ρ

m c+bw_k+ρ m c, which contradicts the hypothesis. Hence,

bw_j+ρ

m c=bw_j

mc+ 1, bw_k+ρ

m c=bw_k

mc+ 1 and ρ≥1.

Using Remark2.2.1(ii), we get thatb^wj_m^+ρc=b^w_m^jc+ 1≥2,b^wk_m^+ρc=b^w_m^kc+ 1≥2 and ρ≥1.

Thus, the proof is complete.

2.3 Numerical semigroups with w

≥ w

+ w

and (2 +

)ν ≥ m

In this Section, we show that Wilf’s conjecture holds for numerical semigroups in the following cases : 1. wm−1≥w1+wα and (2 +^α−3_q )ν ≥m for some 1< α < m−1.

2. m−ν ≤ 5. (Note that the case m−ν ≤ 3 results from the fact that Wilf’s conjecture holds for 2ν ≥m. This case has been proved in [19]), however we shall give a proof in order to cover it through our techniques).

Then, we deduce the conjecture for m= 9 and for (2 +¹_q)ν ≥m.

Next, we will show that Wilf’s conjecture holds for numerical semigroups with wm−1≥w₁+w_α and (2 +α−3

q )ν ≥m.

Theorem 2.3.1. LetS be a numerical semigroup with multiplicitym, embedding dimensionν and conduc- torf+ 1 =qm−ρ for someq, ρ∈N; 0≤ρ≤m−1. Let w₀ = 0< w₁ < w₂ < . . . < wm−1 be the elements of Ap(S, m). Suppose that wm−1 ≥w₁+w_α for some 1 < α < m−1. If (2 + ^α−3_q )ν ≥m, then S satisfies Wilf’s conjecture.

Proof. We are going to use the equivalent form of Wilf’s conjecture given in Proposition 2.1.3. Since wm−1≥w₁+w_α, by Lemma2.2.2, it follows that

bwm−1+ρ

m c ≥ bw₁+ρ

m c+bw_α+ρ m c −1.

Let x = b^wm−1_m^+ρc − b^w1_m^+ρc − b^wα_m^+ρc.Then, x ≥ −1 and b^w1_m^+ρc+b^wα_m^+ρc = b^wm−1_m^+ρc −x = q−x (by Remark2.2.1 (vi)). Now, using (2.2.1) and (2.2.2), we have

m−1

X

j=1

(bwj +ρ

m c − bwj−1+ρ

m c)(jν−m) +ρ

=

α

X

j=1

(bwj+ρ

m c − bwj−1+ρ

m c)(jν−m) +

m−1

X

j=α+1

(bwj +ρ

m c − bwj−1+ρ

m c)(jν−m) +ρ

≥ −bw1+ρ

m cν+bwα+ρ

m c(2ν−m)+ bwm−1+ρ

m c − bwα+ρ

m c (α+ 1)ν−m+ρ (by (2.2.1) and (2.2.2))

=bw1+ρ m c

−ν+ (α+ 1)ν−m− (α+ 1)ν−m

+bwα+ρ

m c(2ν−m) + bwm−1+ρ

m c − bw_α+ρ

m c (α+ 1)ν−m+ρ

=bw₁+ρ

m c(αν−m) +bw_α+ρ

m c(2ν−m)+ bwm−1+ρ

m c − bw_α+ρ

m c − bw₁+ρ

m c (α+ 1)ν−m+ρ

= (bw₁+ρ

m c+bw_α+ρ

m c)(2ν−m) +bw₁+ρ

m c(α−2)ν + bwm−1+ρ

m c − bwα+ρ

m c − bw1+ρ

m c (α+ 1)ν−m+ρ

= (q−x)(2ν−m) +bw1+ρ

m c(α−2)ν+x (α+ 1)ν−m+ρ.

Consequently,

m−1

X

j=1

(bw_j+ρ

m c − bwj−1+ρ

m c)(jν−m) +ρ≥(q−x)(2ν−m) +bw₁+ρ

m c(α−2)ν+x (α+ 1)ν−m+ρ.

(2.3.1) Since

x=bwm−1+ρ

m c − bw₁+ρ

m c − bw_α+ρ

m c ≥ −1, then we have two cases :

• Ifx=−1, then by Lemma 2.2.2(ii), we haveb^w1_m^+ρc ≥2. From (2.3.1), it follows that

m−1

X

j=1

(bwj+ρ

m c − bwj−1+ρ

m c)(jν−m) +ρ

≥(q+ 1)(2ν−m) + 2(α−2)ν− (α+ 1)ν−m+ρ

=ν(2q+ 2 + 2α−4−α−1)−qm+ρ

=ν(2q+α−3)−qm+ρ