Plane Algebroid Curves in Arbitrary
Characteristic
Curvas Algebr´oides Planas em Caracter´ıstica
Arbitr´aria
Ficha catalográfica elaborada pela Biblioteca do Instituto de Matemática e Estatística
A447 Almeida Garcia,Mahalia Violeta
Plane algebroid curves in abitraty characteristic / Mahalia Violeta Almeida Garcia. – Niterói, RJ : [s.n.], 2016.
66 f.
Orientador: Prof. Dr. Abramo Hefez.
Dissertação (Mestrado em Matemática) – Universidade Federal Fluminense, 2016.
1.Geometria algébrica 2. Curva algébrica. 3. Singularidades( Matemática). I. Título.
CDD 516.352
ACKNOWLGEMENTS
I thank God for giving me strength and ability to understand, learn and complete this dissertation.
I would like to thank Professor Abramo Hefez for his guidance, help and patience in carrying out this dissertation.
I would specially like to thank my parents, Tania Eliana Garcia Agudo and Mario Alberto Almeida Salas, for their support and love.
I am really grateful to my friends and professors from UFF, with whom I have learned a lot and had a good time during these three years.
Abstract
The subject of this Dissertation is the study of germs of plane curves de-fined over arbitrary algebraically closed fields. Classically, this was performed over the field of complex numbers, by using as a main tool the Newton-Puiseux parametrization, related to the normalization of the curve. The theory was then adapted to arbitrary algebraically closed field using the so-called Hamburger-Noether expansions that take track of the entire desingularization process of the curve. In this work, we will use, instead, the notion of contact order among irre-ducible curves by means of the logarithmic distance introduced by J. Chadzynski and A. Ploski in [CP]. This attack works in arbitrary characteristic and avoids the use of the Hamburger-Noether expansions, making proofs simpler and more elegant.
The content of this dissertation is as follows:
In Chapter 1, we introduce the notion of algebroid plane curves, their nor-malization and their intersection theory. We used as a reference for this part the book of A. Seidenberg [Sei] and the survey of A. Hefez [He]. In Chapter 2 and 3, we introduce the notion of semigroup of values of an irreducible plane curve and make a detailed study of their properties, introducing at the end the important notion of Key-polynomials, showing that they are nothing else but some special Ap´ery polynomials. This part is based on [He] and personal notes of this author. In Chapter 4, we introduce the contact order among irreducible plane curves and study its properties, applying them to deduce some results about irreducible plane curves that have high contact order. The whole theory is used to deduce Merle’s and Granja’s theorems [Me] and [Gr] over arbitrary algebraically closed fields. To conclude the work we present a result due to E. Garcia Barroso and A. Ploski about the relation among the Milnor number of an irreducible power series and the conductor of its semigroup of values. In this part, we used the works of E. Garcia Barroso and A. Ploski [GB-P1] and [GB-P2].
Keywords: Singularities in positive characteristic, Milnor number in positive char-acteristic, Singularities of algebroid curves
Resumo
O assunto dessa dissertac¸˜ao ´e o estudo dos germes de curvas planas definidas sobre corpos algebicamente fechados arbitr´arios. Classicamente tal estudo era realizado sobre o corpo dos n´umeros complexos, utilizando-se como principal ferramenta para isso as parametrizac¸˜oes de Newton-Puiseux, relacionadas com a normalizac¸˜ao da curva. Em seguida, a teoria foi adaptada para corpos algebrica-mente fechados arbitr´arios, utilizando-se as chamadas expans˜oes de Hamburger-Noether que levam em conta todo o processo de resoluc¸˜ao da singularidade da curva. Neste trabalho, usaremos ao inv´es a noc¸˜ao de contato entre curvas irre-dut´ıveis por meio da distˆancia logar´ıtmica introduzida por J. Chadzynski e A. Ploski em [CP]. Essa abordagem funciona em caracter´ıstica arbitr´aria e evita o uso das expans˜oes de Hamburger-Noether, tornando as demonstrac¸˜oes mais sim-ples e elegantes.
O conte´udo dessa dissertac¸˜ao ´e o seguinte:
No Cap´ıtulo 1, introduzimos a noc¸˜ao de curvas algebr´oides planas, suas nor-malizac¸˜oes e a sua teoria de intersec¸˜ao. Usamos nessa parte como referˆencia o livro de A. Seidenberg [Sei] e o ”survey de A. Hefez [He]. Nos Cap´ıtulos 2 e 3, introduzimos a noc¸˜ao de semigrupo de valores de uma curva plana irredut´ıvel e empreendemos um estudo detalhado de suas propriedades, introduzindo no fi-nal a importante noc¸˜ao de polinˆomios-chave, mostrando que n˜ao s˜ao nada al´em de polinˆomios de Ap´ery particulares. Nessa parte, baseamos-nos em [He] e em notas pessoais desse autor. No Cap´ıtulo 4, introduzimos a ordem de contato en-tre curvas irredut´ıveis planas e estudamos as suas propriedades, utilizando-as para deduzir alguns resultados sobre curvas irredut´ıveis que possuem ordem de contato alta. Toda essa teoria ´e utilizada para deduzir os teoremas de Merle e de Granja, contidos em [Me] e [Gr], sobre corpos algebricamente fechados arbitr´arios. Para concluir o trabalho, apresentamos um resultado recente devido a E. Garcia Bar-roso e A. Ploski sobre a relac¸˜ao entre o n´umero de Milnor de uma s´erie irredut´vel e o condutor de seu semigrupo de valores. Nessa parte, utilizamos os trabalhos de E. Garcia Barroso e A. Ploski [GB-P1] e [GB-P2].
Contents
1 Algebroid Plane Curves 3
1.1 Power series . . . 3
1.2 Algebroid plane curves . . . 5
1.3 Intersection of curves . . . 6
2 Arithmetical Semigroups 9 2.1 Semigroups . . . 9
2.2 Semigroups with conductor . . . 10
2.3 The Ap´ery sequence of a semigroup . . . 16
2.4 Strongly increasing semigroups . . . 20
3 Semigroup of Values of a Plane Branch 27 3.1 The Ap´ery polynomials . . . 27
3.2 Key-polynomials . . . 31
4 The Contact Among Branches 32 4.1 Log-distance . . . 32
4.2 Branches with high contact . . . 36
4.3 The Theorems of Merle and Granja . . . 39
Chapter 1
Algebroid Plane Curves
In this chapter we will introduce the objects that will be studied in this disserta-tion, namely, the algebroid plane curves defined over arbitrary algebraically closed fields and their intersection theory.
1.1
Power series
The theory of algebroid plane curves relies on the notions of power series in one or two variables, as defined below.
Let K be an algebraically closed field of arbitrary characteristic p ≥ 0 and let t, X and Y be indeterminates over K. We will denote by K[[t]] (respectively, by K[[X, Y]]) the ring of formal power series in one indeterminate t (respectively, in
two indeterminates X and Y), with coefficients in K. We will briefly recall some
of their properties and refer to [He] for the proofs.
The ring K[[t]] has elements of the form P∞
i=0aiti, while the elements of
K[[X, Y]] are all formal sums of the type f = P∞i=0Pi, where each Pi is a
ho-mogeneous polynomial of degree i in the indeterminates X and Y with coefficients
in K. The zero polynomial will be considered to be a homogeneous polynomial of any degree. The ring operations are the usual addition and multiplication of power series.
If f = P∞i=0aiti ∈ K[[t]], then mult( f ) is the least i such that ai , 0. On the
other hand, if f = P∞i=0Pi ∈ K[[X, Y]] \ {0}, then n = mult( f ) is the least i such
that Pi , 0. The homogeneous polynomial Pn is called the initial form of f and
the integer n is called the multiplicity of f . If f = 0, we put mult( f ) = ∞. The
notion of multiplicity for power series plays a role similar to that of the degree for polynomials.
The multiplicity of power series has the following properties:
2. mult( f + g) ≥ min{mult( f ), mult(g)}, with equality sign holding whenever mult( f ) , mult(g).
3. mult( f )= 0 if and only if f is a unit.
The rings K[[t]] and K[[X, Y]] are unitary commutative rings with a unique
maximal ideal hti and M = hX, Yi, respectively. The ring K[[t]] is a principal
ideal domain.
The automorphisms of K[[t]] are of the form t 7→ a1t+ a2t2+ · · · , with a1, 0,
while those of K[[X, Y]] are of the form
X 7→ aX+ bY + · · ·
Y 7→ cX+ dY + · · · ,
where ad − bc , 0.
Two elements f and g in a ring R are said associated if there exists a unit u ∈ R
such that f = ug.
We will say that f ∈ K[[X, Y]] is regular in the indeterminate Y of order n, if
f(0, Y)= Ynu(Y), where u(Y) is a unit in K[[Y]].
Let us recall the following result:
Theorem 1.1 (The Division Theorem). Let f ∈ M be regular in Y of order n.
Given any g ∈ K[[X, Y]] there exist q ∈ K[[X, Y]] and r ∈ K[[X]][Y] with r = 0 or
degYr < n, uniquely determined by f and g, such that
g= f q + r
This result implies the following one:
Theorem 1.2 (The Weierstrass Preparation Theorem). Let f ∈ K[[X, Y]] \ K be
of multiplicity n. Then there exist a K-automorphism φ of K[[X, Y]], a unit u ∈
K[[X, Y]] and a1, . . . , an∈ K[[X]], with mult(ai) ≥ i for i= 1, . . . , n, such that
φ( f )u = Yn+ a
1Yn−1+ · · · + an.
A polynomial of the form
Yn+ a1Yn−1+ · · · + an ∈ K[[X]][Y],
where ai(0) = 0, for i = 1, . . . , n, will be called a distinguished polynomial or
simply a d-polynomial. If in addition, we have mult(ai) ≥ i, for all i, it will
be called a Weierstrass polynomial or simply a polynomial. Notice that a w-polynomial is regular in Y of order equal to its multiplicity.
If f ∈ K[[X]][Y] is a d-polynomial, then f is reducible in K[[X, Y]] if and only if f is reducible in K[[X]][Y]. This, together with the Weierstrass Preparation Theorem, imply that K[[X, Y]] is a unique factorization domain.
1.2
Algebroid plane curves
An algebroid plane curve is the equivalence class ( f ) of associated power series
to a given power series f ∈ M= hX, Yi ⊂ K[[X, Y]].
Since the multiplicity of a formal power series remains invariant when we multiply it by unit, we may define the multiplicity of an algebroid plane curve ( f ) as being the multiplicity of f . An algebroid curve of multiplicity one will be called smooth. When the multiplicity is greater than one, we will say that the curve is singular.
Let ( f ) be an algebroid plane curve. We say that the curve ( f ) is irreducible if the formal power series f is irreducible in K[[X, Y]]. Notice that this notion is independent of the representative f of ( f ). An irreducible algebroid plane curve will also be called a plane branch or shortly a branch.
Let ( f ) be an algebroid plane curve and consider the decomposition of f into irreducible factors in K[[X, Y]]
f = f1· · · fr.
The algebroid plane curves ( fj), for j = 1, . . . , r, above defined, are called the
branches of the curve ( f ). The curve ( f ) will be called reduced if ( fi) , ( fj) for
i , j, that is, when fiand fj are not associated if i , j.
Most properties of an algebroid plane curve are preserved after we change coordinates in K[[X, Y]] through a K-automorphism. This motivates the next fun-damental definition.
Two algebroid plane curves ( f ) and (g) will be said equivalent, writing in such case ( f ) ∼ (g), if there exists a K-automorphism φ of K[[X, Y]] such that
(φ( f ))= (g).
Since any branch is equivalent to a curve defined by a w-polynomial, when convenient, we may suppose that its equation is a w-polynomial.
Given an algebroid plane curve ( f ) of multiplicity n, that is, f = Pn+Pn+1+· · · ,
where each Pi is a homogeneous polynomial in K[X, Y] ⊂ K[[X, Y]] of degree i
and Pn , 0, then the curve (Pn) is uniquely determined by the curve ( f ) and will
be called the tangent cone of the curve ( f ).
Since any homogeneous polynomial in two indeterminates with coefficients in
an algebraically closed field decomposes into linear factors, we may write
Pn =
s
Y
i=1
wherePs
i=1ri = n, ai, bi ∈ K, for i, j= 1, . . . , s, and aibj − ajbi , 0 if i , j. So,
the tangent cone of ( f ) consists of the lines (aiX+ biY), i= 1, . . . , s, each counted
with multiplicity ri, called the tangent lines of ( f ).
It is known that a branch ( f ) has a unique tangent line, counted with multi-plicity equal to mult( f ).
Let f be an element in the maximal ideal M= hX, Yi of K[[X, Y]], and let h f i
be the ideal generated by f . We define the coordinates ring of the curve ( f ) as being the K-algebra
Of =
K[[X, Y]]
h f i .
We will denote the residual class of Y by y and the residual class of X by x.
The ring Of is a local ring with maximal ideal Mf = M. When f is
irre-ducible, the ideal h f i is prime and Of is an integral domain. In this case, the field
of fractions of Of will be denoted by Kf. The next result will tell us that the ring
Of is an important invariant of the equivalence classes of algebroid plane curves.
Let ( f ) and (g) be two algebroid plane curves. We have that ( f ) ∼ (g) if and
only if Of and Ogare isomorphic as K-algebras.
Another important structure of Of is the following:
Suppose that f ∈ K[[X, Y]] is regular in Y of order n, then Of is a free
K[[X]]-module of rank n generated by the residual classes yi of the Yi, i= 0, . . . , n − 1, in
Of. In other words,
Of = K[[X]] ⊕ K[[X]]y ⊕ · · · ⊕ K[[X]]yn−1.
1.3
Intersection of curves
Let f , g ∈ M. The following conditions are equivalent: i) f and g are relatively prime;
ii) The dimension of K[[X,Y]]h f,gi as a K-vector space is finite.
The intersection index of f and g is the integer (including ∞) I( f , g)= dimK
K[[X, Y]] h f, gi
Notice that if f or g is a unit in K[[X, Y]], then h f , gi= K[[X, Y]] and therefore
We will say that two algebroid curves ( f ) and (g) are transversal if ( f ) and (g) are smooth and their tangent lines are distinct.
Let f , g, h, u, v ∈ K[[X, Y]], with u and v units. and φ an automorphism of K[[X, Y]]. The intersection index has the following properties:
i) I( f , g) < ∞ if and only if f and g are relatively prime in K[[X, Y]]; ii) I( f , g) = I(g, f );
iii) I(φ( f ), φ(g))= I(u f, vg) = I( f, g);
iv) I( f , hg)= I( f, g) + I( f, h);
v) I( f , g) = 1 if and only if ( f ) and (g) are smooth with distinct tangents;
vi) I( f , g − h f )= I( f, g).
Let ( f ) be an irreducible algebroid curve. A parametrization of ( f ) is a
pair (φ(t), ψ(t)) ∈ K[[t]]2 such that φ(t) , 0 or ψ(t) , 0, φ(0) = ψ(0) = 0
and f (φ(t), ψ(t)) = 0. We say that the parametrization (φ(t), ψ(t)) is a good
parametrization if the field of fractions of the ring K[[φ(t), ψ(t)]] is equal to the
field of fraction K((t)) of the ring K[[t]]. The following results are fundamental.
Theorem 1.3 (Normalization Theorem). If f = f (X, Y) ∈ K[[X, Y]] is an
irre-ducible power series, then there exists a good parametrization(φ(t), ψ(t)) of ( f ).
Moreover, if (α(s), β(s)) ∈ K[[s]] is a parametrization of ( f ), then there exists a
power seriesσ(s) ∈ K[[s]] such that σ(0) = 0, α(s) = φ(σ(s)) and β(s) = ψ(σ(s)).
From this it follows that if (φ(t), ψ(t)) and (α(s), β(s)) are both good parametriza-tions of ( f ), then the map t 7→ σ(s) is an isomorphism from K[[t]] onto K[[s]].
Theorem 1.4. Let f = Yd + a
1(X)Yd−1+ · · · + a0(X) ∈ K[[X]][Y]. Then for some
φ(t) ∈ K[[t]], f (φ(t), Y) splits completely into linear factors in K[[t]][Y].
We describe below another way to compute the intersection index among curves.
Let f (X, Y) ∈ K[[X, Y]] be an irreducible power series and (φ(t), ψ(t)) a good
parametrization of ( f ). Then for any power series g = g(X, Y) in K[[X, Y]] we
have
I( f , g)= mult(g(φ(t), ψ(t))).
Taking g = X (respectively, g = Y) we get from the above formula that
If f is irreducible and I( f , g)= I( f, h) < ∞, then there exists a constant c ∈ K such that
I( f , g − ch) > I( f , g).
Theorem 1.5. Let f (X, Y) ∈ K[[X, Y]] be an irreducible power series such that
f(0, Y) , 0 and let (α(t), β(t)) with α(t) , 0 be a parametrization of ( f ), then for
every power series g(X, Y) ∈ K[[X, Y]] we have
mult(g(α(t), β(t)))= I( f , g)
I( f , X)mult(α(t)).
Given a good parametrization (φ(t), ψ(t)) of an irreducible algebroid curve ( f ), we define a numerical function as follows:
νf: Of \ {0} → N ∪ {∞},
g 7→ mult(g(φ(t), ψ(t)))
where N denotes the set of non negative integers. One has νf(g) = ∞ if and only
if g = 0.
This numerical function is independent from the choice of the good
parametriza-tion since, in this case, as we menparametriza-tioned above, mult(g(φ(t), ψ(t)))= I( f, g).
From the fact that νf(g)= I( f, g), the function νf has the following properties:
For all g1, g2 ∈ Of, one has
1. νf(g1g2)= νf(g1)+ νf(g2);
2. νf(u)= 0 if, and only if, u is a unit in Of;
3. νf(g1+ g2) ≥ min{νf(g1), νf(g2)} with equality verified whenever νf(g1) ,
νf(g2).
Theorem 1.6. Let f, g ∈ M. We have that
I( f , g) ≥ mult( f ) mult(g)
Chapter 2
Arithmetical Semigroups
Semigroups play an important role in the theory of algebroid curves, as we will see in the course of this dissertation. Much of the presentation here in this chapter was influenced by the article of G. Angermuller [An].
2.1
Semigroups
Let {0} ( G ⊂ N. We say that G is a numerical semigroup if it is closed under addition.
Our main concern will be with semigroup associated to an irreducible curve ( f ) as defined below:
Gf = {I( f, h); h ∈ K[[X, Y]] \ h f i}.
That Gf is a semigroup follows immediately from the additivity of the
inter-section index.
In this chapter we will initiate the study of semigroups under an arithmetical point of view.
The element min(G \ {0}) is called the multiplicity of G, and will be denoted by mult(G).
If v0, . . . , vg ∈ N, then
G= hv0, . . . , vgi= {λ0v0+ · · · + λgvg; λ0, . . . , λg ∈ N}
is clearly a numerical semigroup, called the semigroup generated by v0, . . . , vg,
which in turn are called the generators of G.
Proposition 2.1. Given any numerical semigroup G, there exist a finite set of ele-ments v0, . . . , vgin G such that
i) v0< · · · < vg, and vi . vj mod v0, for i , j;
ii) G = hv0, . . . , vgi;
iii) {v0, . . . , vg} is contained in any set of generators of G.
Proof. We are going to define v0, . . . , vgas follows:
v0 = mult(G) and vi = min(G \ hv0, . . . , vi−1i), i = 1, . . . , g.
i) Suppose that i , j. We may assume that i < j. Then it is clear that vi .
vj mod v0 for i < j because, otherwise, vj would be in hv0, . . . , vii, which is a
contradiction. Notice that vi, vi+1 ∈ G \ hv0, . . . , vi−1i so, vi < vi+1, because vi is
the least element with this property. This shows that v0 < · · · < vg.
ii) Since vi . vj mod v0 for i , j, then for some g < v0 this process must stop.
Therefore G= hv0, . . . , vgi.
iii) Let {u0, . . . , ur} be a set of generators of G, so
vi = λi,0u0+ · · · + λi,rur, with λi,0, . . . , λi,r∈ N, i ∈ {0, . . . , g}.
On the other hand,
uj = α0, jv0+ · · · + αg, jvg, with α0, j, . . . , αg, j ∈ N, j ∈ {0, . . . , r}.
Then,
vi = λi,0u0+ · · · + λi,rur
= λi,0(α0,0v0+ · · · + αg,0vg)+ · · · + λi,r(α0,rv0+ · · · + αg,rvg)
= (λi,0α0,0+ · · · + λi,rα0,r)v0+ · · · + (λi,0αi,0+ · · · + λi,rαi,r)vi +
+ · · · + (λi,0αg,0+ · · · + λi,rαg,r)vg.
Since vi < hv0, . . . , vi−1i then some of the coefficients of vi, vi+1, . . . , vg must be
nonzero. Since vi < vi+1 < · · · < vg, it must be that of vi and the others are zero.
This implies that for some l ∈ {0, . . . , r}, λi,l = 1 and the others λi, j are zero. This
shows that vi = ul.
The set {v0, . . . , vg} of Proposition 2.1 will be called the minimal system of
generatorsof G and the integer g will be called the genus of the semigroup. Notice
that from Proposition 2.1 (i) one has that g ≤ mult(G) − 1.
2.2
Semigroups with conductor
Given a numerical semigroup G, then the elements of N \ G are called the gaps of G. A semigroup may have finitely or infinitely many gaps.
When the number of gaps is finite then there exists an element c ∈ G, called the conductor of G, such that
a) c − 1 < G;
b) if z ∈ N and z ≥ c then z ∈ G.
Proposition 2.2. Let G be a numerical semigroup. The following assertions are equivalent:
i) G has a conductor;
ii) The elements of G have gcd equal to one; iii) There exist two consecutive integers in G.
Proof. i) ⇒ ii) If G has a conductor then gcd(G)= 1, since G ⊂ hgcd(G)i.
ii) ⇒ iii) Let v0, . . . , vgbe the minimal system of generators of G. Then gcd(G)=
1 implies gcd(v0, . . . , vg)= 1. So there exist integers λ0, . . . , λgsuch that
λ0v0+ · · · + λgvg= 1,
transferring to the right hand side of the equality the negative terms, the result follows immediately.
iii) ⇒ i) Let a and a+ 1 be two elements in G then the set
{0, a+ 1, 2(a + 1), . . . , (a − 1)(a + 1)}
is a complete residue system modulo a. So, any integer n ≥ (a − 1)(a+ 1) may
be written as n = λ(a + 1) + µa with 0 ≤ λ < a. So, µ ≥ 0 and consequently
n ∈ G.
Remark 2.3. Notice that in the proof of the Proposition 2.2 we got the estimate
c ≤ (a − 1)(a+ 1) for the conductor of G, when a and a + 1 are elements of G.
Example 2.4. Important examples of numerical semigroups with conductors are
the semigroups Gf associated to a plane branch( f ).
Indeed, let (φ(t), ψ(t)) be a good parametrization of the branch ( f ), then we
have K((t))= K((φ(t), ψ(t))), so there exist P = P(X, Y), Q = Q(X, Y) ∈ K[[X, Y]],
with Q . 0 mod f , such that
t = P(φ(t), ψ(t))
Q(φ(t), ψ(t)).
It then follows that
I( f , P)= mult(P(φ(t), ψ(t))) = mult(Q(φ(t), ψ(t))) + 1 = I( f, Q) + 1,
implying that in Gf there are two consecutive integers, which in view of
Let G = hv0, . . . , vgi be a numerical semigroup with conductor. We define
below two sequences of numbers associated to the set of generators of G. Put e0 = v0, n0 = 1, and for i = 1, . . . , g,
ei = gcd(v0, . . . , vi) and ni =
ei−1
ei
.
Remark 2.5. From the definition of the e0
is it is clear that ei|ei−1, so ni ∈ N for all
i = 1, . . . , g. Also, eg = gcd(v0, . . . , vg) = 1. We also have n0· · · niei = v0. In
particular, n0· · · ng = v0and ni+1· · · ng= ei.
Given a numerical semigroup G = hv0, . . . , vgi, then any element in G may be
represented in several ways in the form
λ0v0+ · · · + λgvg, λ0, . . . , λg∈ N.
But, when G is a semigroup with conductor we will show in Proposition 2.7 below that the elements of G may be represented uniquely as a combination of special type of the elements v0, . . . , vg.
Lemma 2.6. Let v0, . . . , vg ∈ N with gcd(v0, . . . , vg) = 1 and let ei and ni, i =
0, . . . , g, be their associated integers. For every m ∈ N there is a unique solution for the congruence
m ≡
g
X
i=1
sivi mod v0, with 0 ≤ si < ni, i = 1, . . . , g.
Proof. By induction on g. If g= 1 we have e1= gcd(v0, v1)= 1 and n1= v0, then
there exist integers λ and µ such that λv0+ µv1 = 1. So,
mλv0+ mµv1 = m.
Dividing mµ by v0we get
mµ = qv0+ s1 with q ∈ Z and 0 ≤ s1< v0.
Thus,
m= mλv0+ mµv1 = mλv0+ (qv0+ s1)v1= s1v1+ (mλ + q)v0 ≡ s1v1mod v0.
Let us now suppose the result true for g ≥ 1 and let v0, . . . , vg+1be positive integers
satisfying the hypotheses of the lemma. Consider the sequence v00 = v0
eg, . . . , v
0 g = vg
eg and notice that gcd(v
0 0, . . . , v
0
g) = 1. So, by the inductive hypothesis, for every
integer m0, there exist integers si and λ, such that
m0 = g X i=1 siv 0 i + λv 0 0, with 0 ≤ si < n 0 i, i = 1, . . . , g,
where n0i = e0i−1 e0 i = gcd(v 0 0, . . . , v 0 i−1) gcd(v0 0, . . . , v 0 i) = gcd(v0, . . . , vi−1)/eg gcd(v0 0, . . . , v 0 i)/eg = ni.
Since v0, . . . , vg+1 are relatively prime, there exist integers λ0, . . . , λg+1 such
that 1 = λ0v0+ · · · + λg+1vg+1, then for every integer m, we have
m= mλ0v0+ · · · + mλg+1vg+1.
Dividing mλg+1by ng+1= eg, we get
mλg+1= qng+1+ sg+1with 0 ≤ sg+1 < ng+1and q ∈ Z.
Since egdivides v0, . . . , vgand ng+1 = eg, there exists an integer m0 such that
m0eg= mλ0v0+ · · · + mλgvg+ qng+1. Then m= mλ0v0+ · · · + mλgvg+ mλg+1vg+1 = mλ0v0+ · · · + mλgvg+ (qng+1+ sg+1)vg+1 = m0 eg+ sg+1vg+1 = g X i=1 si vi eg + λv0 eg eg+ sg+1vg+1. So, m ≡ g+1 X i=1 sivi mod v0, with 0 ≤ si < ni, i = 1, . . . , g + 1.
The uniqueness follows from the facts that every integer is congruent modulo
v0toP
g
i=1sivi, for some 0 ≤ si < ni, i = 1, . . . , g, and that
] g X i=1 sivi mod v0; 0 ≤ si < ni, i = 1, . . . , g = n1n2· · · ng = v0.
Proposition 2.7. Let v0, . . . , vg be relatively prime natural numbers and let c be
the conductor of the semigroup G = hv0, . . . , vgi. Then
i) Every natural number m has a unique representation as
m=
g
X
i=0
ii) c ≤Pg
i=1(ni− 1)vi− v0+ 1.
Proof. i) By Lemma 2.6 we have m ≡ Pg
i=1sivi mod v0 with 0 ≤ si < ni, i = 1, . . . , g; so m= g X i=0 sivi, with 0 ≤ si < ni, i = 1, . . . , g and s0∈ Z
and this representation is unique.
ii) Let m be an integer such that m > Pg
i=1(ni− 1)vi − v0+ 1. From (i) we have a
unique representation m = Pgi=0sivi with 0 ≤ si < ni, i = 1, . . . , g and s0 ∈ Z, thus g X i=0 sivi = m > g X i=1 (ni− 1)vi− v0+ 1 ≥ g X i=1 sivi− v0,
which implies that s0v0 > −v0, so s0 ≥ 0. Therefore, m ∈ G.
From Proposition 2.7, if s0 ≥ 0 in the representation of m, then m ∈ G. The
converse is not always true, which means that we may have m ∈ G and s0 < 0, as
we can see in the following example.
Example 2.8. Let G = h8, 10, 11i. We have e0 = 8, e1 = 2 and e2 = 1. So, n1 = 4,
n2 = 2 and 1 − v0+ (n1− 1)v1+ (n2− 1)v2= 1 − 8 + (4 − 1)10 + (2 − 1)11 = 34 >
26 = c (the conductor of G). The element 22 = 2 · 11 is in G and, in the above
representation, it is written as22= 3v1+ 0v2− v0.
Now, we are going to define the notion of nice sequence. Let v0, . . . , vgbe
rel-atively prime non-negative integers and let ei and ni, i = 1, . . . , g, their associated
integers. We say that the sequence v0, . . . , vgis nice if, for all i= 1, . . . , g, we have
nivi ∈ hv0, . . . , vi−1i.
In a numerical semigroup G generated by a nice sequence, we have, with the
above notation, that if m ∈ G then s0 ≥ 0 and we get a formula for the conductor
of G. This fact will be shown in the following proposition.
Proposition 2.9. Let v0, . . . , vg be a nice sequence of integers. If G = hv0, . . . , vgi
and e0
is and n
0
is are their associated integers, then
i) An integer m= Pgi=0sivi, with0 ≤ si < ni, i = 1, . . . , g, and s0∈ Z belongs to
G if and only if s0 ≥ 0.
ii) The conductor c of G is given by
c=
g
X
i=1
Proof. i) Let m= λ0v0+ · · · + λgvg ∈ G with λi ∈ N, i = 0, . . . , g. Dividing λgby
ng we have λg = qgng+ s0gwith 0 ≤ s
0
g< ng. Since the sequence is nice, we get
m1 = λ0v0+ · · · + λg−1vg−1+ qgngvg ∈ hv0, . . . , vg−1i.
This implies that
m1 = λ00v0+ · · · + λ0g−1vg−1for some λ00, . . . , λ 0
g−1 ∈ N.
Now we repeat this procedure with λ0g−1and so on. This shows that one may write
m = Pgi=0s0ivi, with 0 ≤ s0i < ni, i = 1, . . . , g, and s00 ∈ N, so, by the uniqueness in
Proposition 2.7 one has that si = s0i, for all i, hence s0= s00∈ N
ii) From Proposition 2.7 we know that c ≤ Pg
i=1(ni − 1)vi − v0 + 1. On the other
hand, from (i), we have thatPg
i=1(ni− 1)vi− v0< G, so the equality holds.
Another remarkable property that some numerical semigroups have is symme-try, in the following sense:
A numerical semigroup G with conductor c will be called symmetric if For all z ∈ N, z ∈ G ⇔ c − 1 − z < G.
Notice that the implication z ∈ G ⇒ c − 1 − z < G is true in any numerical semigroup with conductor. In fact if z and c − 1 − z are in G, then c − 1 ∈ G, which is a contradiction.
Proposition 2.10. Let G be a numerical semigroup with conductor c. The follow-ing assertions are equivalent:
i) G is symmetric;
ii) 2 ] (G ∩ [0, c))= c;
iii) 2 ] (N \ G) = c;
iv) ] (G ∩ [0, c))= ] (N \ G).
Proof. Notice that (G ∩ [0, c)) ∩ (N \ G) = ∅ and (G ∩ [0, c)) ∪ (N \ G) = [0, c).
This implies that (ii), (iii) and (iv) are equivalent. Consider now the bijection
φ: [0, c − 1] → [0, c − 1]
z 7→ c −1 − z.
We have that G is symmetric if and only if z ∈ G ⇔ c − 1 − z < G if and only
if φ(G ∩ [0, c)) = N \ G.
It follows that if (i) is satisfied, then ] (G ∩ [0, c))= ] φ(G ∩ [0, c)) = ](N \ G),
and consequently (iv) is satisfied.
Conversely, if (iv) is satisfied, and since φ(G ∩ [0, c)) ⊆ N \ G, it follows that
Remark 2.11. i) If G is symmetric, then c is even. Moreover, the symmetry of G is equivalent to the condition that there are as many gaps as non gaps in G.
ii) Let G and H be symmetric semigroups with the same conductor c and such that
H ⊂ G then H = G, indeed we have
z ∈ G ⇒ c −1 − z < G ⇒ c − 1 − z < H ⇒ z ∈ H.
Proposition 2.12. Every semigroup generated by a nice sequence is symmetric.
Proof. Let v0, . . . , vgbe a nice sequence. We know from Proposition 2.7 that the
conductor of G = hv0, . . . , vgi is c = P
g
i=1(ni − 1)vi − v0+ 1. We also know from
Proposition 2.7 that any z ∈ N may be written uniquely as
z=
g
X
i=0
sivi, 0 ≤ si < ni, i = 1, . . . , g, s0 ∈ Z.
We also know, from Proposition 2.9, that z ∈ G if and only if s0 ≥ 0. Then
c −1 − z= g X i=1 (ni− 1)vi− v0+ 1 − 1 − g X i=0 sivi = g X i=1 (ni− 1 − si)vi− (1+ s0)v0. Hence we have z ∈ G ⇔ s0 ≥ 0 ⇔ −(s0+ 1) ≤ −1 ⇔ c − 1 − z < G.
2.3
The Ap´ery sequence of a semigroup
Let G be a semigroup with conductor c and let m be any element in G \ {0}. We
define the Ap´ery sequence of G with respect to m, inductively, as follows: a0 = 0
and aj = min G \ j−1 [ i=1 (ai + mN) , 1 ≤ j ≤ m − 1. The following proposition holds.
Proposition 2.13. The Ap´ery sequence of a semigroup G, with respect to m ∈ G\{0} satisfies the following properties, where [a] is the residual class of a modulo m in N.
ii) ai . aj mod m for 0 ≤ i < j ≤ m − 1;
iii) ai = min([ai] ∩ G);
iv) G= Sm−1j=0(aj+ mN);
v) c= am−1− (m − 1).
Proof. i) Observe that aj, aj+1 ∈ G \ S
j−1
i=0(ai + mN), so aj < aj+1, therefore
a0 = 0 < a1< · · · < am−1.
ii) ai . aj mod m for 0 ≤ i < j ≤ m − 1, since aj < ai+ mN.
iii) Let z ∈ [ai] ∩ G, then z ∈ G \Si−1j=0(aj + mN), so ai ≤ z.
iv) Since aj . ai mod m with j , i, it follows that {m, a1, . . . , am−1} is a complete
residue system modulo m. Hence G \Sm−1
j=0(aj+ mN) = ∅.
v) Notice that am−1− (m − 1) − 1= am−1− m < G. On the other hand, for all r ≥ 1.
Dividing am−1− m+ r by m, we have am−1− m+ r = λm + aifor some λ ∈ Z and
some i = 0, . . . , m − 1. Then r = (λ + 1)m + (ai− am−1) ≥ 1, so λ ≥ 0. Therefore,
am−1− m+ r = λm + ai ∈ G, for all r ≥ 1. Thus c= am−1− m+ 1.
According to Proposition 2.13 (iii) and (iv) the elements of G are of the form
ai + λm for some i = 0, . . . , m − 1 and λ ≥ 0, while the gaps of G are of the form
ai+ λm for some i = 0, . . . , m − 1 and λ < 0.
The set
A= {a0, . . . , am−1}
will be called the Ap´ery set of G with respect to m. When m = n = min(G \ {0}),
the set A will be called simply the Ap´ery set of G.
We could have defined the Ap´ery set with respect to m as
{α ∈ G; α − m < G}.
Indeed, if α ∈ A then α ∈ G and α − m < G, since, by Proposition 2.13, α = min([α] ∩ G). This shows that A ⊂ {α ∈ G; α − m < G}.
On the other hand, if α ∈ {α ∈ G; α − m < G},
α = λm + ai, for some i = 0, . . . , m − 1, λ ≥ 0
and
α − m = λm + ai− m = (λ − 1)m + ai < G,
which implies that λ − 1 < 0. So, λ = 0. Therefore α = ai, which shows that
{α ∈ G; α − m < G} ⊂ A.
i) The sequence v0, . . . , vgis nice;
ii) Every element m ∈ G is uniquely representable in the form
m=
g
X
i=0
sivi 0 ≤ si < n − i, i = 1, . . . , g, s0 ∈ N;
iii) The Ap´ery set of G is given by
A= g X i=1 sivi; 0 ≤ si < ni, i = 1, . . . , g ;
iv) The conductor c of G is given by
c=
g
X
i=1
(ni− 1)vi− v0+ 1.
Proof. (i) ⇒ (ii) was proved in Proposition 2.9.
(ii) ⇒ (iii): By the uniqueness of the representation of the elements in G, each
numberPg
i=1siviis minimal in its congruence class modulo n= v0, so we have
g X i=1 sivi; 0 ≤ si < nii= 1, . . . , g ⊂ A. Now, since ] g X i=1 sivi; 0 ≤ si < ni, i = 1, . . . , g = n1n2· · · ng = n = ] A,
the result follows.
iii) ⇒ iv): The largest element in A isPg
i=1(ni− 1)vi, hence c= av0−1− v0+ 1 = g X i=1 (ni− 1)vi− v0+ 1.
iv) ⇒ (i): We will prove it by induction on g. For g= 1, we have hv0, v1i, e0 = v0,
e1 = 1, n0 = 1, n1 = v0, so n1v1 = v0v1 ∈ hv0i.
Suppose now that the result is true for g − 1. Let v0, . . . , vg be a sequence
of coprime elements with associated integers ni, ei, i = 0, . . . , g, such that the
conductor c of G= hv0, . . . , vgi is given by c= P
g
Now, consider the semigroup G0 = hv00, . . . , v0g−1i, where v0i = vi
eg−1 with i =
0, . . . , g − 1. Its associated integers are e0i = ei
eg−1 and n
0 i = ni.
From Proposition 2.7, we know that the conductor c0of G0satisfies
c0 ≤
g−1
X
i=1
(n0i − 1)v0i − v00+ 1.
If the strict inequality holds, we would havePg−1
i=1(n 0 i− 1)v 0 i − v 0 0 ∈ G 0, hence g−1 X i=1 (ni− 1)vi− v0 = eg−1 g−1 X i=1 (n0i − 1)v0i− v00 ∈ G, which implies that
c −1= g X i=1 (ni− 1)vi− v0= g−1 X i=1 (ni− 1)vi− v0+ (ng− 1)vg ∈ G, a contradiction. Then c0 = g−1 X i=1 (n0i − 1)v0i − v00+ 1,
and from the inductive assumtion one has that v00, . . . , v0g−1 is nice sequence, so
n0iv0i ∈ hv00, . . . , v0i−1i for all i= 1, . . . , g − 1. This means that nivi = eg−1n0iv
0
i ∈ eg−1hv00, . . . , v 0
i−1i= hv0, . . . , vi−1i.
To finish the proof we only have to show that ngvg ∈ hv0, . . . , vg−1i, which is
equivalent to prove that vg ∈ G0, since ng= eg−1. If vg< G0, since G0is symmetric,
it would follows that c0− 1 − v
g∈ G0, so eg−1(c0− 1 − vg) ∈ G, hence c −1= eg−1 g−1 X i=1 (ni − 1) vi eg−1 − v0 eg−1 + e g−1vg− eg−1vg = eg−1 g−1 X i=1 (n0i − 1)v0i − v00− vg + eg−1 vg = eg−1(c0− 1 − vg)+ eg−1vg ∈ G, which is a contradiction.
Corollary 2.15. Let 0 < v0 < v1 < · · · < vg be a nice sequence of coprime
integers such that vi < hv0, . . . vi−1i, for i = 1, . . . , g, then v0, . . . , vg is a minimal
Proof. Let x0, . . . , xg0 be the minimal system of generators of G, then v0 = x0.
Now we proceed by induction on i. Suppose that vj = xj for j = 0, . . . , i − 1.
From the hypothesis we know that xi ∈ G \ hv0, . . . , vi−1i. To show that xi = vi
it is enough to prove that vi ≤ z for all z ∈ G \ hv0, . . . , vi−1i. Since the sequence
v0, . . . , vg is nice, by the unique representation we may write z = P
g
j=0λjvj with
all λj nonnegative. Since z < hv0, . . . , vi−1i, then one of the λj for j = i, . . . , g is
positive, so z= i−1 X j=0 λjxj+ g X j=i λjvj ≥ vi.
2.4
Strongly increasing semigroups
In this section we will study semigroups with an additional property given in the definition below.
A semigroup G with Ap´ery sequence {a0, . . . , an−1}, where n = min(G \ {0}),
that satisfies the condition:
ai+ aj ≤ ai+ j, for all 0 ≤ i, j, i + j ≤ n − 1, (2.1)
is called a strongly increasing semigroup.
From (2.1) we have ai ≥ a1+ ai−1, hence one has
ai ≥ ia1, for all i = 0, . . . , n − 1 (2.2)
Proposition 2.16. If G is a strongly increasing semigroup of multiplicity n, then its Ap´ery sequence satisfies the following equality
ai+ an−1−i= an−1, ∀i = 0, . . . , n − 1.
Proof. We will prove the equality by induction on i. For i = 0, the equality is
trivially satisfied. Suppose that the proposition is verified for 0 ≤ i ≤ k − 1. Since the Ap´ery sequence is a complete residue system modulo n, we have
an−1− ak ≡ aj mod n, for some 0 ≤ j ≤ n − 1,
so,
an−1 = ak + aj+ λn, for some λ ∈ Z.
If λ > 0 then an−1− n= ak + aj+ (λ − 1)n ∈ G, which is a contradiction since
an−1− n is a gap of G, then
an−1 = ak+ aj+ λn ≤ ak+ j+ λn ≤ ak+ j.
Thus, n − 1 ≤ k+ j ⇒ n − 1 − k ≤ j ≤ n − 1, so there exists 0 ≤ r ≤ k − 1 such
that j= n − 1 − k − r. From the inductive assumption, one has
ar+ an−1−r = an−1≤ ak+n−1−k−r = an−1−r,
then ar = 0, so r = 0. This means that an−1 = ak + an−1−k+ λn ≤ ak + an−1−kwith
λ ≤ 0.
Since G is a strongly increasing semigroup, we get ak + an−1−k ≤ an−1, so
an−1 = ak+ an−1−k.
Proposition 2.17. A Semigroup G is symmetric if and only if its Ap´ery sequence with respect to m, for some m ∈ G \ {0}, satisfies the equality
ai+ am−1−i= am−1, for all i= 0, . . . , m − 1.
Proof. Suppose first that G is symmetric. Fix an index i such that 0 ≤ i ≤ m − 1,
and consider the integer am−1 − m − ai, which is a gap of G, since otherwise,
c −1 = am−1 − m would be an element of G. Then am−1− m − ai = aji + λm for
some ji = 0, . . . , m − 1 and λ < 0, so am−1 = ai+ aji+ (λ + 1)m. We have to prove
that λ= −1 and j = m − 1 − i.
If λ < −1, then
c −1 − (ai− m) = am−1− m − (ai− m)= aji + (λ + 1)m < G.
Since G is symmetric, this would imply that ai− m ∈ G, which is a contradiction.
We have shown that for every i = 0, . . . , m − 1 there exists ji = 0, . . . , m − 1
such that am−1 = ai+ aji. Now, because the Ap´ery sequence is increasing am−1 =
am−1− a0 > am−1− a1 > · · · > am−1− am−1 = a0, so j0 > j1 > · · · > ji > · · · > jm−1,
which implies that ji = m − 1 − i.
Suppose now that the equality is true. We know that the conductor is c =
am−1−m+1 ∈ G. Then the condition c−1−z < G is equivalent to c−1−z = aj+λm,
for some j = 0, . . . , m − 1 and λ a negative integer. This, in turn, is equivalent to
z= am−1− aj− (λ+ 1)m = am−1− j− (λ+ 1)m ∈ G.
Corollary 2.18. Every Strongly Increasing Semigroup is symmetric.
Proof. If G is a strongly increasing semigroup, then from Proposition 2.16 its Ap´ery sequence satisfies
an−1 = ai+ an−1−i, ∀i = 0, . . . , n − 1.
Now, from Proposition 2.17, we may conclude that G is symmetric.
If G is a semigroup with conductor, we define the following sequence of num-bers:
x0 = mult(G) and xl = min(G \ gcd(x0, . . . , xl−1)N), for l ≥ 1. (2.3)
Since gcd(G) = 1, because G has a conductor, this process must stop at some
point. Such finite sequence of numbers is called the sequence in G where the gcd varies.
Lemma 2.19. If G is any semigroup with conductor and x0, . . . , xl, is the sequence
in G where thegcd varies, then
xi ≤ an/ei−1, for i = 1, . . . , l,
where ei = gcd(x0, . . . , xi).
Proof. Suppose that for some i= 1, . . . , l one has
xi > an/ei−1,
so n < a1 < · · · < an/ei−1 < xi. From the definition of the xi, we have that ei−1
divides each of the elements a1, a2, . . . , an/ei−1. By euclidean division, we have that
ak = lkn+ mkei−1for unique lk ∈ N and 0 < mk < en
i−1, for k = 1, . . . ,
n
ei−1. From the
fact that ak . ak0 mod n, if k , k0, it follows that mk , mk0 for k, k0 ∈ {1, 2, . . . , n
ei−1}
with k , k0. So there are n
ei−1 distinct elements mk in the set {1, 2, . . . ,
n ei−1 − 1},
which is a contradiction.
Theorem 2.20. Let G be a strongly increasing semigroup and let x0, . . . , xl be the
sequence of G where thegcd varies and put x0= n. If H = hx0, . . . , xli, then
i) xi = an/ei−1, for i= 1, . . . , l;
ii) x0, . . . , xlis a nice sequence and G = H;
iii) x0, . . . , xlis the minimal set of generators of G;
Proof. i) We know that x0 = n = min(G), x1= a1and n1= en1. Let c and c
0
be the conductors of G and H, respectively. From Proposition 2.7 one has
c0 ≤ l X i=1 (ni− 1)xi− x0+ 1. Since H ⊂ G we have c ≤ c0, so an− n = c − 1 ≤ c0− 1 ≤ l X i=1 (ni− 1)xi− n= n ei − 1 ! x1+ l X i=2 (ni− 1)xi− n. (2.4)
Since the semigroup is strongly increasing, from equality (2.2) and inequality (2.4), one has an−1 < an ≤ n e1 − 1 ! a1+ l X i=2 (ni− 1)xi ≤ an/e1−1+ l X i=2 (ni− 1)xi.
Therefore from Lemma 2.19 and from the fact that G is strongly increasing, we get an−1 < l X i=2 (ni− 1)xi+ an/e1−1 ≤ l X i=2
(ni− 1)an/ei−1 + an/e1−1≤
l
X
i=2
(ni− 1)a(ni−1)n/ei−1 + an/e1−1≤ aPl
i=2(ni−1)n/ei−1 + an/e1−1. (2.5)
From Proposition 2.17 we have that
an−1 = an−n/e1 + an/e1−1,
which implies that the inequalities in (2.5) are all equalities, hence
l X i=2 (ni− 1)xi = l X i=2
(ni − 1)an/ei−1,
which in view of Lemma 2.19, implies that xi = an/ei−1, for i = 2, . . . , l, so we are
ii) From all equalities established above and in view of (2.4) we have that c= c0 = l X i=1 (ni− 1)xi− x0+ 1.
From Proposition 2.14, x0, . . . , xl is a nice sequence. And from Proposition 2.12
H is symmetric. Using Corollary 2.18 and from Remark 2.11, we conclude that
H = G.
iii) From (ii) we know that x0, . . . , xl is a nice sequence of coprime integers such
that xi < hx0, . . . , xi−1i, for i = 1, . . . , l. By Corollary 2.15, x0, . . . , xlis the minimal
system of generators of G.
iv) Since G is strongly increasing, from equality (2.2), one has ai ≥ ia1, ∀ i =
0, . . . , n − 1, hence xj = a n e j−1 ≥ ej−2 ej−1 a n e j−2 = nj−1xj−1, for j= 1, . . . , `.
To conclude the proof, just observe that equality in the above inequality does
not hold, because of the variance of the gcd.
Let x, y ∈ Nr. We will say that x is smaller than y in the reverse lexicographical
order, writing x y, if the last non-zero coordinate of y − x is positive. This
establishes a total order relation on Nr.
Let x0, . . . , xrbe a sequence of positive relatively prime integers. Consider the
integers n0, . . . , nrassociated to x0, . . . , xrand define the set
E(x0, . . . , xr)= {(s1, . . . , sr) ∈ Nr; 0 ≤ si < ni i= 1, . . . , r}.
We put on this set the reverse lexicographical order , and consider N with its natural order ≤.
Lemma 2.21. Let x0, . . . , xrbe a sequence of positive relatively prime integers. The
map
λ: E(x0, . . . , xr) → {0, 1, . . . , x0− 1}
(s1, . . . , sr) 7→ Pri=1sin0. . . ni−1
is an order preserving bijection.
Proof. The inequalities, for j= 1, . . . , r,
j
X
i=1
sin0. . . ni−1≤ (n1− 1)n0+ (n2− 1)n0n1+ · · · + (nj− 1)n0. . . nj−1 =
show that λ preserves orders and that its image is contained in {0, 1, . . . , x0− 1}.
Since λ is order preserving, it follows that it is injective and because both sets E(x0, . . . , xr) and {0, 1, . . . , x0− 1} have the same cardinality x0, it follows that λ
is a bijection.
Lemma 2.22. Let G be a strongly increasing semigroup of multiplicity n and let
x0, . . . , xrbe the sequence of G where thegcd varies. Then the map
ρ: E(x0, . . . , xr) → A
(s1, . . . , sr) 7→ Pri=1sixi
is an order preseving bijection.
Proof. From Theorem 2.20 we know that x0, . . . , xr is a nice sequence, so from
Proposition 2.14 it follows that
A= r X i=1 sixi; 0 ≤ si < ni i= 1, . . . , r .
Let s = (s1, . . . , sr), t = (t1, . . . , tr) ∈ E(x0, . . . , xr) be such that s t.
Sup-pose by reduction to absurdity that ρ(t) < ρ(s). Thus, Pr
i=1tixi < Pri=1sixi and, consequently, 0 > r X i=1 (ti− si)xi > r X i=1 (ti− si)ni−1xi−1 > r X i=1
(ti− si)ni−1ni−2xi−2>
· · · > r X i=1 (ti− si)ni−1ni−2· · · n0x0 = x0 r X i=1 (ti− si)ni−1ni−2· · · n0 . Therefore r X i=1 tini−1ni−2· · · n0< r X i=1 sini−1ni−2· · · n0,
hence t s, which is a contradiction, because λ in Lemma 2.21 is order preserv-ing.
This shows that ρ preserves order. So, ρ is injective. Since ] E = ] A = x0, it
follows that ρ is also surjective.
Theorem 2.23. Let G be a strongly increasing semigroup of multiplicity n and let
x0, . . . , xrbe the sequence of G where thegcd varies, then
r
X
i=1
sixi = aPr
Proof. This follows from the fact that A= Pri=1sixi : 0 ≤ si < ni and that λ and
ρ are order preserving bijections.
Remark 2.24. Let 0 < v0 < · · · < vg be a minimal system of generators of a
strongly increasing semigroup G. Then we have for any integer k, with1 ≤ k ≤ g,
i) (n1− 1)v1+ · · · + (nk− 1)vk . 0 mod ek−1;
ii) (n1− 1)v1+ · · · + (nk− 1)vk < vk+1;
iii) If (n1− 1)v1+ · · · + (nk− 1)vk = l0v0+ l1v1+ · · · + lkvk, with integers l0, lk ≥ 0
and 0 ≤ li < ni, for i ∈ {1, . . . , k − 1}, then l0 = 0 and li = ni − 1, for
i ∈ {1, . . . , k − 1};
iv) If 1 < j ≤ k ≤ g, then ej−1vj < ek−1vk.
Let us prove these assertions.
(i) Suppose that (n1 − 1)v1 + · · · + (nk − 1)vk ≡ 0 mod ek−1, then (nk − 1)vk ≡
0 mod ek−1, since ek−1 = gcd(v0, . . . , vk−1). Thus, (nk− 1) vk ek ≡ 0 mod ek−1 ek , which is a contradiction, since gcdvk ek, ek−1 ek = 1 and 0 < nk− 1 < nk = ek−1 ek .
(ii) Follows easily by induction using the inequalities vi > ni−1vi−1, i= 1, . . . , g.
(iii) From (ii) we get that lk < nk− 1. Now, result follows from the uniqueness of
the writing of the elements of G.
(iv) from Theorem 2.20 we know that, for all i = 1, . . . , g we have
vi > ni−1vi−1 =
ei−2
ei−1
vi−1,
Chapter 3
Semigroup of Values of a Plane
Branch
Recall that the semigroup of values of a branch ( f ) was defined as
Gf = {I( f, g); g ∈ K[[X, Y]] \ h f i} .
We showed that this semigroup has a conductor. In this chapter we will show that it is strongly increasing, hence symmetric (cf. Corollary 2.18). For doing this, we will use the notion of Ap´ery polynomials.
3.1
The Ap´ery polynomials
The elements of the Ap´ery sequence of a branch ( f ) are elements of the semigroup
Gf of ( f ). In this section we will construct some special elements in K[[X, Y]]
whose values in Gf are precisely the elements of the Ap´ery sequence.
Let f ∈ K[[X, Y]] be an irreducible power series of multiplicity n. After a change of coordinates in K[[X, Y]] we may suppose that f is regular in Y of order
nand that n does not divide m= I( f, Y). Recall that, at the end of Section 1.2, we
observed that
Of = K[[X]] ⊕ K[[X]]y ⊕ · · · ⊕ K[[X]]yn−1.
Let us define M−1= {0} and, for k = 0, 1, . . . , n − 1,
Mk = K[[X]] ⊕ K[[X]]y ⊕ · · · ⊕ K[[X]]yk.
So,
K[[X]]= M0 ⊂ M1 ⊂ · · · ⊂ Mn−1= Of.
Recall that I( f , g) = νf(g), where νf is the valuation introduced at the end of
Chapter 1. In this section we will denote νf by ν. The following result, taken from
Theorem 3.1 (Ap´ery). Let f ∈ K[[X, Y]] of multiplicity n, irreducible, regular
of order n in Y and such that n does not divide m = I( f, Y). Then for every
k = 0, 1 . . . , n−1, there exists an element yk ∈ yk+ Mk−1such thatν(yk) < ν(Mk−1).
Proof. For k = 0, if we put y0 = 1, we have 1 = y0 ∈ y0+ M−1with ν(y0) = 0 <
ν(M−1)= {∞}.
Let k < n be given. If ν(yk
) < ν(Mk−1), take yk = yk and we are done. If
ν(yk
) ∈ ν(Mk−1) then there exists φ1 ∈ Mk−1 such that ν(φ1) = ν(yk) hence there
exist c1 ∈ K such that
ν(yk
− c1φ1) > ν(yk).
If ν(yk− c
1φ1) < ν(Mk−1) we are done since we may take yk = yk − c1φ1. But,
if ν(yk− c1φ1) ∈ ν(Mk−1), there exists c2∈ K and φ2 ∈ Mk−1 such that
ν(yk
− c1φ1− c2φ2) > ν(yk− c1φ1),
and so on. At some point this procedure will necessarily stop, because otherwise we would have ν y k− ∞ X i=1 ciφi = ∞, withP∞
i=1ciφi ∈ Mk−1, hence, from the Division Theorem (Chapter 1), yk−P
∞
i=1ciφi =
0 so yk ∈ Mk−1, which is a contradiction.
Observe that since yk ∈ yk+ Mk−1, it follows that
Of = K[[X]] + K[[X]]y1+ · · · + K[[X]]yn−1.
Proposition 3.2 (Azevedo [Az]). Suppose that for k = 0, . . . , , n − 1 we have
elements yk ∈ yk+ Mk−1withν(yk) < ν(Mk−1), where y0 = 1. Then for all i, j, with
0 ≤ i, j ≤ n − 1 and i+ j ≤ n − 1, we have
ν(yi)+ ν(yj) ≤ ν(yi+ j).
Proof. We can write yi = yi+ a and yj = yj+ b with ai ∈ Mi−1and b ∈ Mj−1, then
we have
yiyj = yiyj+ ayj + byi+ ab = yj+i+ c,
where c = ayj+ byi+ ab ∈ M
i+ j−1.
We know that yi+ j = yi+ j+ d for some d ∈ Mi+ j−1, so yiyj = c − d + yi+ j. If
we put e = c − d ∈ Mi+ j−1, it follows that ν(yiyj)= ν(e + yi+ j) ≥ inf{ν(yj+i), ν(e)}.
Since ν(yj+i) < ν(Mi+ j−1) and ν(e) ∈ ν(Mi+ j−1), then ν(yj+i) , ν(e). So, ν(yiyj) =
inf{ν(yj+i), ν(e)}, implying that
ν(yi)+ ν(yj) ≤ ν(yi+ j).
Remark 3.3. We have that ν(yj) > ν(yi), whenever 0 ≤ i < j ≤ n − 1.
Indeed, since for l ≥ 1, ν(yl) < ν(Ml−1), we have that ν(yl) , 0, hence it follows
that ν(yj) ≥ ν(yj−i)+ ν(yi) > ν(yi).
Proposition 3.4. Let f ∈ K[[X, Y]] be irreducible and regular in Y of order n =
mult( f ). Put y0 = 1 and let y1, y2, . . . , yn−1 be elements of Of such that yk ∈
yk + M
k−1andν(yk) < ν(Mk−1). Denoting by [r] the residual class of the integer r,
modulo n, for all k = 0, . . . , n − 1, we have that
i) ν(Mk)= S
k
i=0(ν(yi)+ nN);
ii) ν(yi) . ν(yj) mod n for all i, j= 0, . . . , n − 1 with i , j;
iii) ν(yk)= min([ν(yk)] ∩ Gf) for all k = 0, . . . , n − 1.
Proof. If n= 1 we have Gf = N and in this case the assertions (i)-(iii) are trivially
satisfied. We may assume that n > 1.
i) By induction on k. For k = 0 we have y0 = 1, then ν(y0) = 0 and ν(M0) =
ν(K[[X]]) = nN. Now, suppose that for some k such that 1 ≤ k ≤ n − 1 we have
ν(Mk−1)= {ν(yi)+ λn; 0 ≤ i ≤ k − 1, λ ≥ 0}.
We know that Mk = Mk−1+ K[[X]]yk, so for any element β ∈ Mk, we may write
β = α + a(X)yk with α ∈ Mk−1 and a(X) ∈ K[[X]]. It is enough to prove that
ν(α) , ν(a(X)yk). Indeed, if ν(α)= ν(a(X)yk)= ν(a(X))+ν(yk), using the inductive
hypothesis, we would have, for some i ≤ k − 1, ν(yi)+ λn = ν(yk)+ µn.
From Remark 3.3, we have λ > µ, then
ν(yk)= ν(yi)+ (λ − µ)n ∈ ν(Mi−1) ⊂ ν(Mk−1),
which is a contradiction.
ii) Suppose that for some integers i, j ∈ {0, . . . , n − 1} with i < j we have ν(yi) ≡
ν(yj) mod n, so for some positive integer λ (see Remark 3.3), ν(yj) = ν(yi)+ λn,
from (i) we get ν(yj) ∈ ν(Mi) ⊂ ν(Mj−1), a contradiction.
iii) From (ii), we have that each residual class modulo n contains exactly one of
the integers ν(yk), k= 0, . . . , n − 1. On the other hand, we know that
N ∩ [ν(yk)] ∪ Gf = {ν(yk)+ λn : λ ≥ 0}.
Therefore,
ν(yk)= min([ν(yk)] ∩ Gf) for all k= 0, . . . , n − 1.
Conditions (ii) and (iii) in Proposition 3.4 say that ν(y0) < · · · < ν(yn−1) is the
Ap´ery sequence of Gf. Also, Proposition 3.2 gives a very important property of
the Ap´ery sequences: Gf is a strongly increasing semigroup.
Polynomials Yi, i = 0, . . . , n − 1, in K[[X]][Y] of degree less than n whose
residual classes mod f are the elements yi, will be called Ap´ery polynomials.
No-tice that these polynomials are not unique.
Corollary 3.5. Let i and j be two distinct integers such that i, j = 0, . . . , n − 1
and letαi(X), αj(X) in K[[X]] \ {0}. Then
ν(αi(X)yi) . ν(αj(X)yj) mod n.
Proof. We have already observed that ν(αi(X)) = λin and ν(αj(X)) = λjn, for
some natural numbers λi and λj. Assuming j > i and ν(αi(X)yi) = ν(αj(X)yj), we
get ν(yi) − ν(yj)= (λj−λi)n, so ν(yi) ≡ ν(yj) mod n, a contradiction according to
Proposition 3.4.
Corollary 3.6.
Of = K[[X]] ⊕ K[[X]]y1⊕ · · · ⊕ K[[X]]yn−1.
Proof. We know that Of = K[[X]] + K[[X]]y1+ · · · + K[[X]]yn−1. It is sufficient to
prove that y0, . . . , yn−1 are independent over K[[X]]. In fact, suppose that we had
a non-trivial relation
α0(X)y0+ α1(X)y1+ · · · + αn−1(X)yn−1 = 0.
From Corollary 3.5, we have ν(αi(X)yi) , ν(αj(X)yj) for i, j ∈ {0, . . . , n − 1}
and i , j. So, there exists i ∈ {0, . . . , n − 1} such that
∞= ν(0) = ν(α0(X)y0+ α1(X)y1+ · · · + αn−1yn−1)= ν(αi(X)yi),
which is a contradiction.
Corollary 3.7. Let 1 = z0, z1, . . . , zn−1∈ Of be such that
a) zk ∈ yk+ Mk−1 for all k= 0, . . . , n − 1; and
b) ν(z0), . . . , ν(zn−1) are pairwise non-congruent modulo n.
Thenν(zk)= ν(yk) for all k = 0, . . . , n − 1.
Proof. From Proposition 3.4, ν(Mk−1) intersects only k residual classes modulo
n and ν(zi) ∈ ν(yi + Mi−1) ⊂ ν(Mk−1) for all i = 0, . . . , k − 1. It follows that
ν(zk) < ν(Mk−1) and by hypothesis zk ∈ yk + Mk−1. From Proposition 3.4 we have
ν(zk)= min([ν(zk)] ∩ Gf)= min([ν(yk)] ∩ Gf)= ν(yk).
3.2
Key-polynomials
In this section we will study properties of the Ap´ery polynomials Yi attached to
a given branch ( f ) defined by an irreducible series f ∈ K[[X, Y]], regular in Y of
order n = mult( f ).
Since the residual class of Yi is in yi + Mi−1, we may assume that Yi = Yi +
Ai,1Yi−1+ · · · + Ai,i ∈ K[[X]][Y]. We have the following result:
Proposition 3.8. Each polynomial Yi is a Weierstrass polynomial.
Proof. We want to prove that mult(Ai, j) > j for all i, j.
First suppose that for some i and some j we have that mult(Ai, j) < j. This
implies that mult(Yi) < i.
If the tangent cone of Yi has a factor not of the form Yr for some r, then we
take a factor g of Yi with tangent cone not containing Y as a factor. Suppose that
Yi = gh, then
ai = I( f, Yi)= I( f, g) + I( f, h) = mult( f ) mult(g) + I( f, h) = n mult(g) + I( f, h).
This implies that ai ≡ I( f , h) mod n, which is a contradiction, since ai is the
least element of Gf in its residual class modulo n.
Then we have shown that the tangent cone of Yi is Yr for some r < i. Now,
applying the Weierstrass Preparation Theorem, we may multiply Yi by a unit u
in order to get uYi = Yr + B1Yr−1 + · · · + Br ∈ K[[X]][Y], where r < i. Hence
ai = I( f, Yi)= I( f, uYi) ∈ ν(Mr) ⊂ ν(Mi−1), which is a contradiction.
So mult(Ai, j) ≥ j. If mult(Ai, j) = j, then mult(Yi) = i and the tangent cone
of Yi is not of the form Yi. Applying the same argument as above we decompose
Yi = gh such that the tangent cone of g does not contain Y as a factor and get, in
the same way, a contradiction.
Suppose that f is an irreducible power series of multiplicity n, let Gf be its
semigroup of values and v0, . . . , vg be the minimal system of generators. After a
change of coordinates in K[[X, Y]],we may assume that f is regular in Y of order n.
Consider now Weierstrass polynomials f−1= X and fi = Yn
ei, for i= 0, . . . , g −
1, so that
degY fi =
n ei−1
and νf( ¯fi)= I( f, fi−1)= an/ei = vi+1. (3.1)
Since νf( ¯fi) = vi+1, i = 0, . . . , g − 1, cannot be written as the sum of two
nonzero elements in Gf, then fiis an irreducible power series.
Irreducible Weierstrass polynomials f−1 = X, f0, . . . , fg−1 (not uniquely
Chapter 4
The Contact Among Branches
The contact among branches was used by M. Merle to prove in [Me] an important factorization theorem over the complex numbers, where in this case, the contact among branches is defined through Puiseux parametrizations, which we do not have in positive characteristic. Later, A. Granja in [Gr] generalized Merle’s result extending it also for arbitrary characteristic, but using in this case Hamburger-Noether expansions which are the substitute for Puiseux parametrizations in any characteristic. The proof we give here is due to A. Ploski and E. Garc´ıa Barroso in [GB-P1], which uses the log-distance and is shorter and more elegant than the original proofs.
4.1
Log-distance
Let A be an nonempty set. A log-distance between elements in A is a function
d: A × A → R ∪ {∞} such that, for all a, b, c ∈ A,
1. d(a, a)= ∞;
2. d(a, b)= d(b, a);
3. d(a, b) ≥ inf{d(a, c), d(c, b)}.
The condition (3), called the strong triangular inequality (STI), is equivalent to the following one:
(3’) At least two of the numbers d(a, b), d(a, c) and d(b, c) are equal and the third is not smaller than them.
Let us prove this equivalence.
Suppose that (3’) holds. Without loss of generality, we may assume that
d(a, c) = d(b, c), then d(a, b) is not smaller than d(a, c) and d(b, c), therefore
Conversely, let us suppose that (3) holds, then
d(a, b) ≥ inf{d(a, c), d(c, b)}, (4.1)
d(a, c) ≥ inf{d(a, b), d(c, b)}, d(c, b) ≥ inf{d(a, c), d(a, b)}.
Because of the symmetry of the above relations, we may assume that d(a, b) ≤
d(b, c) ≤ d(a, c). To prove (3’), we must prove that d(a, b) = d(b, c). Suppose by
reduction to absurdity that d(a, b) < d(b, c). From (4.1), one has d(a, c) < d(c, b), a contradiction.
Lemma 4.1. Let d be a log-distance on a set A and let a1, . . . , am, b1, . . . , bn, c ∈ A,
then at least one of the following conditions holds:
i) There exists j ∈ {1, . . . , n} such that for any i ∈ {1, . . . , m}, d(ai, c) ≤ d(ai, bj).
ii) There exists i ∈ {1, . . . , m} such that for any j ∈ {1, . . . , n}, d(bj, c) ≤ d(ai, bj).
Proof. Suppose that neither (i) nor (ii) holds. Then for any j ∈ {1, . . . , n} there
exists and index p( j) ∈ {1, . . . , m} such that d(ap( j), c) > d(ap( j), bj) and for any i ∈
{1, . . . , m} there exists s(i) ∈ {1, . . . , n} such that d(bs(i), c) > d(ai, bs(i)). Applying
the STI condition to ap( j), bj, c and to ai, bs(i), c, we get
d(ap( j), c) > d(bj, c) = d(ap( j), bj), (4.2)
and
d(bs(i), c) > d(ai, c) = d(ai, bs(i)). (4.3)
Without loss of generality, we may assume that
d(ap(1), b1)= max
j∈{1,...,m}{d(ap( j), bj}.
From (4.2) and (4.3) and again (4.2), we get
d(ap(1), b1) < d(ap(1), c) < d(bs(p(1)), c) = d(ap(s(p(1)), bs(p(1))),
a contradiction.
Example 4.2. The function d : K[[s]] × K[[s]] → R ∪ {∞} given by
d(α(s), β(s))= mult(α(s) − β(s))
Indeed, suppose that α(s) − β(s)= sau, α(s) − γ(s)= sbvand γ(s) − β(s)= scw,
where u, v and w are units. Suppose that a = min(a, b, c). From the last two
equalities above, we get that sau= α(s) − β(s) = sbv+ scw. Since a= min(a, b, c),
it follows that a = b or a = c and if a = b and since b ≥ a and c ≥ a, the result
follows.
Theorem 4.3. Let (l) be a smooth branch. If dl: K[[X, Y]]] × K[[X, Y]] → R ∪ {∞}
is given by
dl( f , g)=
I( f , g)
I( f , l) I(g, l), (4.4)
then dlis a log-distance in the set of all branches different from (l).
Proof. Without loss of generality, we may assume after a change of coordinates
in K[[X, Y]] that l= X.
Because of the symmetry of the intersection index, we have that dX( f , g) =
dX(g, f ). It is also clear that dX( f , f )= +∞.
It then suffices to check the STI condition. Let ( f ), (g), (h) be three branches
different from (X), so they are all regular in Y. After multiplication by units, we
may assume that they are distinguished polynomials in Y of degrees m = I( f, X),
n = I(g, X) and p = I(h, X). Using Theorem 1.4, there exist power series α(s),
αi(s), i= 1, . . . , m, βj(s), j = 1, . . . , n and γk(s), k= 1, . . . , p, such that
f(α(s), Y)= Qmi=1(Y − αi(s)),
g(α(s), Y)= Qnj=1(Y − βj(s)),
h(α(s), Y)= Qkp=1(Y − γk(s)).
Let us consider the log-distance of Example 4.2 and apply Lemma 4.1 to α1(s),. . .,αm(s), β1(s), . . . , βn(s) and γ(s) = γk(s), where k ∈ {1, . . . , p} is fixed.
Then
1) There exists j ∈ {1, . . . , n} such that, for all i ∈ {1, . . . , m},
mult(αi(s) − γ(s)) ≤ mult(αi(s) − βj(s)); or
2) There exists i ∈ {1, . . . , m} such that, for all j ∈ {1, . . . , n},
mult(βj(s) − γ(s)) ≤ mult(αi(s) − βj(s)).
If (1) holds, thenPm
i=0mult(αi(s) − γ(s)) ≤P
m
i=0mult(αi(s) − βj(s)); that is,
Since h(α(s), γ(s)) = 0 and g(α(s), βj(s))= 0, by Theorem 1.5, we get
I(h, f )
I(h, X)mult(α(s))= mult( f (α(s), γ(s))) ≤ mult( f (α(s), βj(s)))=
= I(g, f ) I(g, X)mult(α(s)), therefore, dX( f , h)= I(h, f ) I(h, X) I( f , X) ≤ I(g, f ) I(g, X) I( f , X) = dX( f , g). (4.5)
On the other hand, if (2) holds, in the same way, one shows that dX(g, h) =
I(h, g)
I(g, X) I(g, X) ≤
I(g, f )
I(g, X) I( f , X) = dX( f , g) (4.6)
Using (4.5) and (4.6) we have that, in any case, dX(g, f ) ≥ inf{dX(g, h), dX( f , h)}
Corollary 4.4. The function
d( f , g)= I( f , g)
mult( f ) mult(g) is a log-distance in the set of all plane branches.
Proof. Given ( f ),(g) in the set of all branches, take (l) different from the tangents
of ( f ) and (g), then dl( f , g)= I( f , g) I( f , l) I(g, l) = I( f , g) mult( f ) mult(g) (4.7) is a logarithmic distance.
We define the relative contact index of two branches ( f ) and (g) with respect
to (l) as being the number dl( f , g). The number d( f , g) will be called simply the
contact indexof f and g.
In general, d( f , g) is a rational number greater or equal than 1. One has that
d( f , g) = 1 if and only if f and g have distinct tangents. Notice that this notion
of contact among branches is not the same as the classical one that measures the coincidence of the Puiseux parametrizations of the two branches up to a certain order.
Example 4.5. Let ( f ) be a plane branch and let f0, . . . , fg−1 be a sequence of
key-polynomials for f . Then
d( f , fi−1)=
ei−1vi
n2 , i = 1, . . . , g.
The numbers appearing on the right hand side of the above equality will have an important role in the next sections.
4.2
Branches with high contact
We are going now to study pairs of branches with relative high contact. We will show, in particular, that given two irreducible branches ( f ) and (h) such that d( f , h) is sufficiently high, then the semigroups Gf = hv0, . . . , vgi and Gh = hv00, . . . , v0g0i,
where v0 = n = I( f, X) and v00 = n0 = I(h, X), are closely related up to a certain
order.
After a change of coordinates, if necessary, we may assume without loss of generality that f and g are regular in Y of order equal to their multiplicity. We will denote by ei, ni (respectively e0i, n
0
i) the integers attached to Gf (respectively
to Gh) and by f0, . . . , fg−1key-polynomials of f (respectively h0, . . . , hg0−1of h).
In the sequel we will need the following remark.
Remark 4.6. If the equalities vi
n = v0
i
n0, for all i ∈ {1, . . . , k}, where k ≤ min{g, g
0}, hold, then en i = n0 e0 i for all i ∈ {1, . . . , k}.
The proof of this assertion is by elementary arithmetic.
Lemma 4.7. Let n = I( f, X) > 1, and suppose that dX( f , h) >
ek−1vk
n2 for some integer
k ∈ {1, . . . , g}. Then
i) n I(h, fi−1)= n0vi, for all i ∈ {1, . . . , k};
ii) n0 ≡ 0 mod en
k;
iii) dX( fi−1, hi−1)=
ei−1e0i−1I( fi−1,hi−1)
nn0 , for0 < i ≤ min{k, g
0};
iv) We have n0 > 1 and v1
n = v01 n0;
v) Let 0 < i < k, i < g0
and assume that vj
n = v0
j
n0, for all j ≤ i. Then i < g
0 and vi+1 n = v0i+1 n0 .
Proof. (i) Fix i ∈ {1, . . . , k}. From our hypothesis, by Remark 2.24 (iv) and by
Example 4.5, we have dX( f , h)= I( f , h) nn0 > ek−1vk n2 > ei−1vi n2 = dX( f , fi−1).
From the STI condition applied to f , fi−1and h, we have
I(h, fi−1)ei−1
n0n = dX(h, fi−1)= dX( f , fi−1)=
ei−1vi
n2 ,
ii) From part (i), we have
n0ek = n0gcd(vi; i= 0, . . . , k) = n gcd(I(h, fi−1); i= 0, . . . , k) ≡ 0 mod n.
iii) One has
dX( fi−1, hi−1)=
I( fi−1, hi−1)
I( fi−1, X) I(hi−1, X) =
ei−1e0i−1I( fi−1, hi−1)
nn0 .
iv) That n0> 1 follows from (ii). Now, Applying (i) and Example 4.5, one gets
dX(h, f0)= e0v1 n2 = v1 n < N and dX(h, h0)= e00v01 (n0)2 = v01 n0 < N.
On the other hand,
dX( f0, h0)=
e0e00I( f0, h0)
nn0 = I( f0, h0) ∈ N,
so, by the STI condition applied to h, h0and f0, one gets
v1 n = v01 n0. v) Since vj n = v0j n0, by Remark 4.6, ej n = e0j
n0. By (ii) there exists an integer l > 0 such
that n0 = ln ek. Thus, e0i = n 0ei n = l n ek ei n = l ei ek > 1,
since i < k, then, obviously, i < g0. From what we obtained above, we have
dX(h, fi)= eivi+1 n2 , dX(h, hi)= e0iv0i+1 (n0)2 and dX( fi, hi)= eie0iI( fi, hi) nn0 .
We claim that dX(h, fi) , dX( fi, hi), because if the equality was true, we would
have eivi+1 n2 = eie0iI( fi, hi) nn0 , then vi+1= ne0iI( fi, hi) n0 = n0eiI( fi, hi) n0 = eiI( fi, hi),
which is absurd, since vi+1 . 0 mod ei.
We also claim that dX(h, hi) , dX( fi, hi), because otherwise
e0iv0i+1 (n0)2 = eie0iI( fi,hi) nn0 , thus v0i+1 = n0e iI( fi, hi) n = ne0iI( fi, hi) n = e 0 iI( fi, hi),
which is absurd too, by the same reason.
Therefore, by the STI condition, dX(h, fi) = dX(h, hi), hence
eivi+1 n2 = e0 iv 0 i+1 (n0)2 , so vi+1 n = v0 i+1 n0 .
