Jean-Christophe San Saturnino

DEFECT OF AN EXTENSION, KEY POLYNOMIALS AND LOCAL UNIFORMIZATION by

Keywords: 2000 Mathematics Subject Classification. -13A18, 12J10, 12J20, 14B05, 14E15 . -defect, valued fields, local uniformization, key polynomials

For all simple and finite extension of a valued field, we prove that its defect is the product of the effective degrees of the complete set of key polynomials associated. As a consequence, we obtain a local uniformization theorem for valuations of rank 1 centered on an equicharacteristic quasi-excellent local domain satisfying some inductive assumptions of lack of defect.

Introduction

In order to obtain a local uniformization theorem, F.-V. Kuhlmann systematized the study of the defect of an extension of valued fields. We know that this defect is a power of p, the characteristic of the residual field of the valuation. Another way to approaches the local uniformization problem for a valuation of rank 1 is studying the set of key polynomials associated to the valuation. The key polynomials have been developped in [START_REF] Vaquié | Extension d'une valuation[END_REF] and [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF], a link of the two approach is given in [START_REF] Mahboub | Key Polynomials[END_REF]. When the set of key polynomials do not have limit key polynomials, we proved in [9] that we can obtain a local uniformization theorem for valuation of rank 1. In particular, we also proved that, if the residual field of the valuation is of characteristic zero, we have local unifomization of the valuation, result known since Zariski.

In [START_REF] Vaquié | Famille admissible de valuations et défaut d'une extension[END_REF], M. Vaquié found a link between the defect and the key polynomials: the jump of the valuation. In the language of [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF], this is the degree of a limit key polynomial over the degree of a previous key polynomial whose degree is the same of an infinite family of key polynomials being between this polynomial and the limit key polynomial.

In this paper, we want to study more precisely the link between the defect and the key polynomials. We will express the defect as a product of effective degrees and generalize the result of M. Vaquié. The effective degree is closely related with the Newton polygon associated to a key polynomial. We will use this resust to prove a local uniformization theorem for valuations of rank 1 centered on an equicharacteristic quasi-excellent local domain satisfying some inductive assumptions of defectless of the quotient field.

In the first section, we recall some definitions about the center of a valuation and the graded algebras associated to a valuation.

In the second section we define the defect of a valuation, roughly speaking, for a valuation ν of a field K having a unique extension µ of a field L, the defect is the number who gives the equality in this inequality:

[L : K] ≤ [Γ µ : Γ ν ][k µ : k ν ],
where Γ ν (resp. Γ µ ) is the value group of ν (resp. µ) and k ν (resp. k µ ) is the residual field of ν (resp. µ). We know that is a power of p, the characteristic of k ν .

In the third section we first recall the definitions of key polynomials given in [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF] or [9]. With this definitions, we give some characterizations of a complete set of key polynomials, this is very useful to prove if a set of polynomials is a complete or a 1-complete set of key polynomials. For a key polynomial Q, we also define the effective degree of a polynomial P corresponding to this key polynomial. When we write the standard expansion of P in terms of Q, the greatest power of Q where the valuation of the monomials in Q is maximal is the effective degree of P . We prove after that some graded algebras associated to a valuation are euclidean for the effective degree. In the end of this section, we recall that the sequence of effective degrees is decreasing and so, there exist a constant value. If this value is 0 or 1, the set of key polynomial considered is 1-complete.

In the fourth section, we first prove that the defect of an extension of a valuation is the product of the effective degree of the limit key polynomials provided by the valuation. The proof uses the notion of key polynomials of W. Mahboub (see [START_REF] Mahboub | Une construction explicite de polynômes-clés pour des valuations de rang fini[END_REF]), so it is true for valuation of any rank. The prinicpal idea is that the Newton plygon has only one face if the field is henselian. We compute after some defects with four examples of M. Vaquié, W. Mahboub and S.D. Cutkosky.

In the next section we prove that there is no limit key polynomials for valuation of rank one over a defectless field.

In the final section we generalize the results of [9] and prove a local uniformization theorem for a valuation of rank 1 with some inductive assumptions of local uniformization in lower dimension and no defect for a well defined extension. With this we can prove again the local uniformization for a valuation of rank 1 in characteristic zero. I want to thank M. Vaquié who suggested me to study the precise link between the defect and the limit key polynomial and B. Teissier for his advice on clarification and structuration of the notion of key polynomials. I also thank W. Mahboub for his ability to produce examples of key polynomial wich verify or not my ideas. Finally, I want to thank M. Spivakovsky for our discussions and his careful reading.

Notation. Let ν be a valuation of a field K. We write

R ν = {f ∈ K | ν(f ) 0}, this is a local ring whose maximal ideal is m ν = {f ∈ K | ν(f ) > 0}.
We then denote by k ν = R ν /m ν the residue field of R ν and Γ ν = ν(K * ). For a field K, we will denote by K an algebraic closure of K, by K sep a separable closure of K and by Aut(K|K) the group of automorphisms of K|K. If R is a ring and I an ideal of R, we will denote by R I the I-adic completion of R. When (R, m) is a local ring, we will say the completion of R instead of the m-adic completion of R and we will denote it by R. For all P ∈ Spec(R), we note κ(P ) = R P /P R P the residue field of R P . For α ∈ Z n and u = (u 1 , ..., u n ) a n-uplet of elements of R, we write:

u α = u α1 1 ...u αn n . For P, Q ∈ R [X] with P = n i=0 a i Q i and a i ∈ R[X]
such that the degree of a i is strictly less than Q, we write:

d • Q (P ) = n. If Q = X, we will note simply d • (P ) instead of d • X (P ).
Finally, if R is a domain, we denote by F rac(R) its quotient field.

Center of a valuation, graded algebra associated and saturation

In this section, Γ will be a totally ordered commutative group. Definition 1.1. -Let R be a ring and P a prime ideal. A valuation ν : R → Γ ∪ {∞} centered on P is the data of a minimal prime ideal P ∞ of R included in P and a valuation of the quotient field of R/P ∞ centered on P/P ∞ . The ideal P ∞ is the support of the valuation, that is

P ∞ = ν -1 (∞).
If R is a local ring with maximal ideal m, we will say that ν is centered on R to say that ν is centered on m. Definition 1.2. -Let R be a ring and ν : R → Γ ∪ {∞} a valuation centered on a prime ideal of R. For all α ∈ ν(R \ {0}), we define the ideals:

P α = {f ∈ R | ν(f ) α}; P α,+ = {f ∈ R | ν(f ) > α}.
Then we define the graded algebra of R associated to ν by:

gr ν (R) = α∈ν(R\{0}) P α /P α,+ . The algebra gr ν (R) is a domain. For f ∈ R \ {0}, we write in ν (f ) its image in gr ν (R).
If R is a local domain, we define another graded algebra by: Definition 1.3. -Let R be a local domain, K = F rac(R) and ν : K * ։ Γ ∪ {∞} a valuation of K centered on R. For all α ∈ Γ, we define the R ν -submodules of K following:

P α = {f ∈ K | ν(f ) α} ∪ {0}; P α,+ = {f ∈ K | ν(f ) > α} ∪ {0}.
We define then the graded algebra associated at ν by:

G ν = α∈Γ P α /P α,+ .
For f ∈ K * , we write in ν (f ) his image in G ν .

Remark 1.4. -We have the natural injective function:

gr ν (R) ֒→ G ν .
Definition 1.5. -Let G be a graded algebra without zero divisors. We called saturation of G the graded agebra G * define by:

G * = f g f, g ∈ G, g homogène, g = 0 .
We say that G is saturated if G = G * .

Remark 1.6. -For all graded algebra G, we have:

G * = (G * ) * .
In particular, G * is always saturated.

Example 1.7. -Let ν be a valuation centered on a local ring R. Then:

G ν = (gr ν (R)) * .
In particular, G ν is saturated.

Defectless fields

Definition 2.1. -( [START_REF] Vaquié | Famille admissible de valuations et défaut d'une extension[END_REF], Définition 1.2) A valued field (K, ν) is said to be henselian if for all algebraic extension L|K, there exsits a unique valuation µ of L which extends ν.

We call henselization of (K, ν) all extension (K h , µ) of (K, ν) such that (K h , µ) is henselian and for all henselian valued field (L, ν ′ ) and all immersion σ : (K, ν) ֒→ (L, ν ′ ), there exists a unique immersion σ ′ : (K h , µ) ֒→ (L, ν ′ ) which extends σ.

(K h , µ) σ ′ / / (L, ν ′ ) (K, ν) σ : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ O O Remark 2.2.
-All the henselizations of a given valued field are isomorphic, it is its smaller henselian extensions. Moreover, the henselization (K h , µ) of (K, ν) is an immediate extension, that is Γ ν = Γ µ and k ν = k µ (for a proof, we can see Theorem 7.42 of [START_REF] Kuhlmann | Valuation Theory[END_REF]).

The henselization of a given valued field (K, ν) can be constructed explicitly, we give here the construction proposed by Kuhlmann in [START_REF] Kuhlmann | Valuation Theory[END_REF], Chapter 7. Consider µ to be an extension of ν on K. Write:

G d (K|K, µ) = {σ ∈ Aut(K|K) | ∀ α ∈ K, µ(σ(α)) = µ(α)}, K h(µ) = {α ∈ K sep | ∀ σ ∈ G d (K|K, µ), σ(α) = α}.
(K h(µ) , µ) is an henselization of (K, ν), it is an algebraic separable immediate extension of (K, ν). Definition 2.3. -Let K be a field with a valuation ν. Consider L a finite extension of K and write µ 1 , ..., µ g the valuations of L which extend ν. For 1 i g, choose a valuation µ i of K whose restriction on L is µ i . We write K h(µ i ) the field construct previously, it is an henselization of (K, ν). Moreover, it is a subfield of L h(µ i ) := L.K h(µ i ) which is himself an henselization of (L, µ i ). We define the defect of the extension L|K in µ i by:

d L|K (µ i , ν) = L h(µ i ) : K h(µ i ) e i f i ,
where e i = [Γ µi : Γ ν ] and

f i = [k µi : k ν ].
Remark 2.4. -By a theorem of Ostrowski (Lemma 11.17 of [START_REF] Kuhlmann | Valuation Theory[END_REF]), we can show that d L|K (µ i , ν) = p ai where a i ∈ N and p = char(k ν ).

Proposition 2.5. -( [START_REF] Kuhlmann | Valuation Theory[END_REF], Lemma 7.46) Let K be a field with a valuation ν. Consider L a finite extension of K and write µ 1 , ..., µ g the valuations of L which extend ν. Then:

[L : K] = g i=1 d i e i f i ,
where

d i = d L|K (µ i , ν), e i = [Γ µi : Γ ν ] and f i = [k µi : k ν ].
Definition 2.6. -Let K be a field with a valuation ν. Consider L a finite extension of K and write µ 1 , ..., µ g the valuation of L which extend ν. We call global defect of the extension L|K on ν the quotient:

d L|K (ν) = [L : K] g i=1 e i f i
,

where e i = [Γ µi : Γ ν ] et f i = [k µi : k ν ].
We say that L is a defectless extension of K if:

d L|K (ν) = 1.
Remark 2.7. -If the extension L|K is normal, then the global defect is a power of p, where p = char(k ν ), because d L|K (ν) = d L|K (µ i , ν), for all i ∈ {1, ..., g} (see [START_REF] Kuhlmann | Valuation Theory[END_REF], Lemma 11.3) ; otherwise it is a rational number.

Remark 2.8. -d L|K (ν) = 1 ⇔ ∀ i ∈ {1, ..., g}, d L|K (µ i , ν) = 1.
Definition 2.9. -A field K is said to be defectless if all finite extension of K is defectless.

Proposition 2.10. -( [START_REF] Kuhlmann | Valuation Theory[END_REF], Theorem 11.23) All field with a valuation ν such that char(k ν ) = 0 is a defectless field.

Key polynomials and effective degree

Let K ֒→ K(x) be a simple transcendental extension of fields. Let µ ′ be a valuation of K(x), write µ := µ ′ | K . Write G the group of values of µ ′ and G 1 the smallest nonzero isolated subgroup of G. Suppose that the rank of µ is 1 and µ ′ (x) > 0. Finally, for β ∈ G, recall that:

P ′ β = {f ∈ K(x) | µ ′ (f ) β} ∪ {0}; P ′ β,+ = {f ∈ K(x) | µ ′ (f ) > β} ∪ {0}; G µ ′ = β∈G P ′ β /P ′ β,+ ;
and

in µ ′ (f ) is the image of f ∈ K(x) in G µ ′ .
Definition 3.1. -A complete set of key polynomials for µ ′ is a well ordered collection:

Q = {Q i } i∈Λ ⊂ K[x] such that, for all β ∈ G, the additive group P ′ β ∩ K[x] is generated by the products of the form a s j=1 Q γj ij , a ∈ K, such that s j=1 γ j µ ′ Q ij + µ(a) β.

The set is said to be 1-complete if the precedent condition occurs for all β ∈ G 1 and if for all

i ∈ Λ, µ ′ (Q i ) ∈ G 1 .

Theorem 3.2. -([3], Theorem 7.11) There exists a collection

Q = {Q i } i∈Λ which is a 1-complete set of key polynomials.
By the Theorem 3.2, we know that there exists a 1-complete set of key polynomials Q = {Q i } i∈Λ and that the order type of Λ is at most ω × ω. If K is defectless, we will see that the order type of Λ is at most ω at so, there is no limit key polynomials, this is particulary the case if char(k µ ) = 0. For all i ∈ Λ, write

β i = µ ′ (Q i ).
Let l ∈ Λ, write:

α i = d • Qi-1 (Q i ), ∀ i l; α l+1 = {α i } i l ; Q l+1 = {Q i } i l .
We also use the notation γ l+1 = {γ i } i l where the γ i are all zero except a finite number, Q

γ l+1 l+1 = i l Q γi i . Definition 3.3. -A multiindex γ l+1 is said to be standard with respect to α l+1 if 0 γ i < α i+1 , for i l. An l-standard monomial in Q l+1 is a product of the form c γ l+1 Q γ l+1
l+1 , where c γ l+1 ∈ K and γ l+1 is standard with respect to α l+1 . An l-standard expansion not involving Q l is a finite sum β S β of l-standard monomials not involving Q l , where β ranges over a certain finite subset of G + and

S β = j d β,j is a sum of standard monomials of value β such that j in µ ′ (d β,j ) = 0. Definition 3.4. -Let f ∈ K[x

] and i l, an i-standard expansion of f is an expression of the form:

f = si j=0 c j,i Q j i ,
where c j,i is an i-standard expansion not involving Q i .

Remark 3.5. -Such expansion exists by Euclidean division and is unique in the sense of that the c j,i ∈ K[x] are unique. More precisely, if i ∈ N, we can prove by induction that the i-standard expansion is unique.

Definition 3.6. -Let f ∈ K[x], i l and f = si j=0
c j,i Q j i an i-standard expansion of f . We define the i-troncation of µ ′ , denoted by µ ′ i , as being the pseudo-valuation:

µ ′ i (f ) = min 0 j si {jµ ′ (Q i ) + µ ′ (c j,i )}.
Remark 3.7. -We can prove that µ ′ i is a valuation. Moreover, we have:

∀ f ∈ K[x], i ∈ Λ, µ ′ i (f ) µ ′ (f ). Proposition 3.8. -Let {Q i } i∈Λ ⊂ K [x]. Write H = {f ∈ K[x] | µ ′ (f ) / ∈ G 1 }
and, for a graded algebra G, denote by G * his saturation (see Definition 1.5). Consider the following assertions:

1. The set {Q i } i∈Λ is 1-complete; 2. ∀ f ∈ K[x] \ H, ∃ i ∈ Λ, µ ′ i (f ) = µ ′ (f ). 3. For all i ∈ Λ, Q i /
∈ H and there exists h ∈ H monic or zero such that the set

{Q i } i∈Λ ∪ {h} is complete; 4. There exists h ∈ H monic or zero such that G µ ′ = (G µ [{in µ ′ (Q i )} i∈Λ , in µ ′ (h)]) * . Then 1. ⇔ 2. ⇔ 3. and 1. ⇒ 4. . Proof : Show that {Q i } i∈Λ is 1-completed if and only if for all f ∈ K[x] \ H there exists i ∈ Λ, µ ′ i (f ) = µ ′ (f ). Suppose that {Q i } i∈Λ is 1-completed and consider f ∈ K[x] such that µ ′ (f ) ∈ G 1 . Write β = µ ′ (f ), then P ′ β ∩ K[x]
is generated by the products of the form a

s j=1 Q γj ij , a ∈ K, with s j=1 γ j µ ′ Q ij + µ(a) β = µ ′ (f ), thus µ ′ is (f ) µ ′ (f ). Moreover, by definition, for all i ∈ Λ, µ ′ i (f ) µ ′ (f ). We conclude that µ ′ is (f ) = µ ′ (f ). Reciprocally, if all f ∈ K[x] such that µ ′ (f ) ∈ G 1 is as µ ′ i (f ) = µ ′ (f ), for a i ∈ Λ, then, f writes such a sum of monomials in Q i+1 of value at least µ ′ (f )
β. So f is in the ideal generated by all this monomials. We can show in the same way that {Q i } i∈Λ is completed if and only if for all f ∈ K[x] and for all i ∈ Λ,

µ ′ i (f ) = µ ′ (f ). Show that {Q i } i∈Λ is 1-completed if and only if for all i ∈ Λ, Q i /
∈ H and there exists h ∈ H monic or zero such that the set

{Q i } i∈Λ ∪ {h} is completed. If {Q i } i∈Λ is completed then, by definition, for all i ∈ Λ, µ ′ (Q i ) ∈ G 1 and take h = 0. If {Q i } i∈Λ is not completed, there exists an element h ∈ K[x] such that µ ′ (h) / ∈ G 1 and µ ′ i (h) < µ ′ (h)
, for all i ∈ Λ. Take h of minimal degree and monic satifying the previous inequality. For f ∈ K[x], consider the standard expansion in terms of h:

f = s j=0 c j h j . Write m = min{j ∈ {1, ..., s} | c j = 0}. Denote by µ ′ h the troncated valuation in terms of h, since µ ′ (h) / ∈ G 1 then: µ ′ h (f ) = µ ′ h (c m h m ) = µ ′ (c m h m ) = µ ′ (f ). The reciproque is obvious.
Finally, if {Q i } i∈Λ is 1-completed then there exists h ∈ H monic or zero such that the set

{Q i } i∈Λ ∪ {h} is completed, that is P ′ β ∩ K[x] is generated by ele- ments of the form a s j=1 g γj j where g j ∈ {Q i } i∈Λ ∪ {h}. So we deduce that G µ ′ = (G µ [{in µ ′ (Q i )} i∈Λ , in µ ′ (h)]) * .
Remark 3.9. -Note that 4. ⇒ 1., consider the example 2.2 of [START_REF] Herrera Govantes | Extending a valuation centered in a local domain to the formal completion[END_REF]: Let k a field and ι :

k(u, v) ֒→ k [[t]] a monomorphism defined by ι(v) = t and ι(u) = i 1 c i t i where c i ∈ k * and Q ∞ = u - i 1 c i v i is transcendantal over k(u, v).
Denote by µ ′ the valuation of k(u, v) induced by the t-adic valuation of k [[t]] through ι and µ by the restriction of µ ′ on K = k(v). The set of key polynomials associated to the extension

K ֒→ K(u) is {Q i } i 1 with Q 1 = u and Q i = u - i-1 j=1 c i v i while G µ ′ = k µ [in µ ′ (u)].
For the next definition, we use the terminology of [START_REF] Vaquié | Extensions de valuation et polygone de Newton[END_REF] following the notations of [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF].

Definition 3.10. -Let h ∈ K[x], consider its i-standard expansion h = si j=0 c j,i Q j i .
We call by the i-th effective degree of h the natural number:

δ i (h) = max{j ∈ {0, ..., s i } | jβ i + µ ′ (c j,i ) = µ ′ i (h)}.
By convention, δ i (0) = -∞.

Remark 3.11. -If we denote by:

in i (h) = j∈Si(h,βi) in µ ′ (c j,i )X j ∈ G µ [in µ ′ (Q i ), X]
where

S i (h, β i ) = {j ∈ {0, ..., s i } | jβ i + µ ′ (c j,i ) = µ ′ i (h)}, then δ i (h) = d • X (in i (h)). Moreover, if in µ ′ i (h) = in µ ′ i (g) then δ i (h) = δ i (g).
The definition of the effective degree extends naturally to gr µ ′ i (K[x]).

Remark 3.12. -Remind that, by Proposition 5.2 of [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF], for l ∈ Λ an ordinal number, the sequence (δ l+i (h)) i∈N * decreases. Thus there exists i 0 ∈ N * such that δ l+i0 (h) = δ l+i0+i (h), for all i 1 and we denote this common value by δ l+ω (h) or δ l+ω if no confusion is possible. Note that δ l+ω may to be 0.

The next three lemmas generalize the lemmas 2.12, 2.13 and 2.14 of [2], the proofs are quasi the same.

Lemma 3.13. -Let l ∈ Λ and f, g ∈ K[x]: δ l (f g) = δ l (f ) + δ l (g).
Proof : It is an immediate consequence of the Remark 3.11.

Lemma 3.14. -For all

h ∈ K[x], δ i (h) = 0 if and only if in µ ′ i (h) is a unit of gr µ ′ i (K[x]). Proof : If δ i (h) = 0 then µ ′ i (h) = µ ′ (h) and in µ ′ i (h) = in µ ′ i (c 0,i ) in gr µ ′ i (K[x]). As the polynomial Q i is irreducible in K[x] and d • (c 0,i ) < d • (Q i ), the polynomials Q i and c 0,i are coprime in K[x]. Then, there exists U, V ∈ K[x] such that U Q i + V c 0,i = 1. We deduce that µ ′ i (V c 0,i ) = µ ′ i (1) < µ ′ i (U Q i ) and h is a unit of gr µ ′ i (K[x]
). Reciprocally, it is sufficient to apply the Lemma 3.13 and the Remark 3.11.

Lemma 3.15. -For all i ∈ Λ, the ring gr µ ′ i (K[x]) is euclidean for δ i , ie: ∀g, h ∈ K[x], h = 0, ∃ Q, R ∈ K[x], g = hQ + R in gr µ ′ i (K[x]) and 0 δ i (R) < δ i (h).
Proof : Write h = si j=0 c j,i Q j i , we can suppose without loss of generalities then c j,i = 0

for j > δ i (h) because µ ′ (c j,i Q j i ) > µ ′ i (c j,i Q j i ) and so in µ ′ i si j=δi (h)+1 c j,i Q j i = 0.
We can also suppose that c δi(h),i = 1 by Lemma 3.14. Note that d

• (h) = δ i (h).d • (Q i ). By euclidean division in K[x], there exists P, Q ∈ K[x] such that g = hQ + P with 0 d • (P ) < d • (h). Let P = ti j=0 p j,i Q j i the i-standard expansion of P . If we write R = δi(P ) j=0 p j,i Q j i , we obtain that h = gQ + R in gr µ ′ i (K[x]
) and:

δ i (P ).d • (Q i ) d • (p δi(P ),i )+δ i (P ).d • (Q i ) = d • (R) d • (P ) < d • (h) = δ i (h).d • (Q i ).
Thus,

δ i (R) = δ i (P ) < δ i (h).
The construction of the key polynomials is recursive (see [START_REF] Mahboub | Key Polynomials[END_REF], [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF] and [START_REF] Herrera Govantes | Valuations in algebraic field extensions[END_REF]). For l ∈ N * , we construct a set of key polynomials Q l+1 = {Q i } 1 i l ; there are two cases:

(1) ∃ l 0 ∈ N, β l0 / ∈ G 1 ; (2) ∀ l ∈ N, β l ∈ G 1 .
In case (1), we stop the construction; the set Q l0 = {Q i } 1 i l0-1 is by definition a 1-complete set of key polynomials and Λ = {1, ..., l 0 -1}. Note that the set Q l0+1 is a complete set of key polynomials.

In case (2), if K is defectless, we will prove in Proposition 5.1 that the set Q ω = {Q i } i 1 is infinite and Λ = N * ; the next propositions will ensure us that the set of key polynomials obtained is also 1-complete. The next lemma, very useful, allows us to note that there no exists any increasing bounded sequence with valuations of rank 1. Proof : Let (β i ) i 1 an increasing infinite sequence of ν (R \ P ∞ ) bounded by β. This sequence corresponds to a decreasing infinite sequence of ideals of R/P β . It is sufficient to prove that R/P β have finite length. Write m the maximal ideal of R, ν(m) = min {ν (R \ P ∞ ) \ {0}} and Γ the value group of ν. Note that the group ν (R \ P ∞ ) is archimedean. Indeed, by absurde, if ν (R \ P ∞ ) is not archimedean, there exists α, β ∈ ν (R \ P ∞ ), β = 0 such that, for all n 1, nβ α. Particularly, the set:

{γ ∈ Γ | ∃ n ∈ N \ {0}, -nβ < γ < nβ}
is a non trivial isolated subgroup of Γ. We deduce that ther exists n ∈ N such that:

β nν(m).
Thus, m n ⊂ P β and therefore, there exists a surjective map: R/m n ։ R/P β . Proposition 3.18. -( [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF], Proposition 3.30) Suppose that we have construct an infinite set of key polynomials

Q ω = {Q i } i 1 such that, fo all i ∈ N * , β i ∈ G 1 .
Suppose further that the sequence {β i } i 1 is not bounded in G 1 . Then, the set of key polynomials Q ω is 1-complete.

Proof : It is sufficient to prove that, for all β ∈ G 1 and for all h ∈ K [x] such that µ ′ (h) = β, h belongs to the R µ -submodule of K [x] generated by all the monomials of the form a

s j=1 Q γj ij , a ∈ K, such that µ ′ a s j=1 Q γj ij β. Therefore consider h ∈ K [x] such that µ ′ (h) ∈ G 1 . Write h = d j=0
h j x j , we can suppose, without loss of generalities, that:

∀ j ∈ {0, ..., d}, µ(h j ) 0.
Indeed, otherwise it is sufficient to multiply h by an element of K appropriately chosen. Since the sequence {β i } i 1 is not bounded in G 1 , there exists i 0 ∈ N * such that:

µ ′ (h) < β i0 .
Then denote by h = si 0 j=0 c j,i0 Q j i0 , the i 0 -standard expansion of h. This expansion is obtain by euclidean division, seen the choice made on the coefficients of h and, since the sequence

βi d • (Qi) i 1
is strictly increasing (it is sufficient to take the (i -1)standard expansion of Q i ), we prove easily that:

∀ j ∈ {0, ..., s i0 }, µ (c j,i0 ) 0.
Recall that, by construction of the key polynomials, for j ∈ {0, ..., s i0 }, µ ′ i0 (c j,i0 ) = µ ′ (c j,i0 ). We then deduce that:

∀ j ∈ {1, ..., s i0 }, µ ′ c j,i0 Q j i0 = µ ′ i0 c j,i0 Q j i0 > µ ′ (h). Thus, µ ′ (h) = µ ′ (c 0,i0 ) and so, h is a sum of monomials in Q i0+1 of valuation at least µ ′ (h) (in particular, µ ′ i0 (h) = µ ′ (h)).
We consider now two cases:

(1) ♯{i 1 | α i > 1} = +∞; (2) ♯{i 1 | α i > 1} < +∞.
In case (1), with the Proposition 3.19, we can prove that the infinite set of key polynomials is always 1-complete, independently of the characteristic of k µ . In case (2), if the effective degree is always 1 and if the set of key polynomials Q ω = {Q i } i 1 is not complete, we will prove in the Proposition 3.20 that the sequence {β i } i 1 is never bounded. In this case, with the Proposition 3.18, we deduce that the set of key polynomials

Q ω = {Q i } i 1 is also 1-complete.
Q ω = {Q i } i 1 such that, for all i ∈ N * , β i ∈ G 1 . More, suppose that the set {i 1 | α i > 1} is infinite. Then, Q ω is a 1-complete set of key polynomials. Proof : Let h ∈ K[x],
as in the proof of the Proposition 3.18, it is sufficient to show that µ ′ i (h) = µ ′ (h) for a i 1. But, if we write:

δ i (h) = max S i (h, β i ),
where:

S i (h, β i ) = {j ∈ {0, ..., s i } | jβ i + µ ′ (c j,i ) = µ ′ i (h)}, h = si j=0 c j,i Q j i ,
by (1) of Proposition 5.2 of [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF], we have:

α i+1 δ i+1 (h) δ i (h), ∀ i 1.
We deduce that if δ i (h) > 0 and α i+1 > 1:

δ i+1 (h) < δ i (h), ∀ i 1.
The set {i 1|α i > 1} is infinite and the previous inequality does not occure infinitly, we conclude that there exists i 0 1 such that δ i0 (h) = 0 and so µ ′ i0 (h) = µ ′ (h).

From now, we suppose that we have construct an infinite set of key polynomials Q ω = {Q i } i 1 such that α i = 1,for all i sufficiently large. Thus for this i, we have:

Q i+1 = Q i + z i
, where z i is an i-standard homogeneous expansion, of value β i , not containing Q i . More we suppose that β i ∈ G 1 and there exists h ∈ K[x] such that, for all i 1:

µ ′ i (h) < µ ′ (h).
Write Q ω the monic polynomial of smaller degree possible satisfying the previously inequality. As in the proof of the Proposition 3.19, write:

δ i (Q ω ) = max S i (Q ω , β i ),
where:

S i (Q ω , β i ) = {j ∈ {0, ..., s i } | jβ i + µ ′ (c j,i ) = µ ′ i (Q ω )}, Q ω = si j=0 c j,i Q j i .
By (1) of Proposition 5.2 of [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF], we have:

α i+1 δ i+1 (Q ω ) δ i (Q ω ), ∀ i 1.
Since α i = 1 for i sufficiently large, there exists δ ω ∈ N * such that δ ω = δ i (Q ω ), for i sufficiently large.

δ ω = 1 then the sequence {β i } i 1 is not bounded in G 1 .

Defect and effective degree

In this section, we do not suppose any assumption about the rank of the valuation. We use the construction of key polynomials of W. Mahboub (see [START_REF] Mahboub | Une construction explicite de polynômes-clés pour des valuations de rang fini[END_REF]). Recall that the order type of the set of key polynomials is at most ω × ω. Lemma 4.1. -Let (K, µ) ֒→ (L, µ ′ ) be a finite and simple valued field. Let {Q i } i∈Λ be the well ordered set of key polynomials associated to this extension. Then there exists n 0 ∈ N * such that Λ ωn 0 , ie {Q i } i∈Λ have a finite number of limit key polynomials.

Proof : Let P ∈ K[x] monic and irreducible such that L = K[x]/(P ). Denote also by µ ′ the pseudo-valuation of K[x] for which the set of key polynomials is associated, thus for all Q ∈

(P ), µ ′ (Q) = ∞. Suppose that the set {Q ωi } i 1 is infinite. Such α ωi > 1, for all i 1, the sequence {d • (Q ωi )} i 1 is strictly increasing. Then there exsits n 0 ∈ N * such that d • (P ) d • (Q ωi ), for all i n 0 . Let i n 0 , by euclidean division there exists S i , R i ∈ K[x] such that Q ωi = P S i + R i with R i = 0 or 0 d • (R i ) < d • (P ) d • (Q ωi ). If R i = 0 then µ ′ (Q ωi ) = µ ′ (R i )
which is absurd by minimality of the degree of a limit key polynomial. If R i = 0 then Q ωi ∈ (P ) and the Q ωi are not key polynomials for i > n 0 . Lemma 4.2. -Let (K, µ) be an henselian field and L be a finite and simple extension of K. By definition, µ extends uniquely to L and this extension corresponds to a (pseudo-)valuation in K [x] denoted by µ ′ . Consider {Q i } i∈Λ the well ordered set of key polynomials associated to µ ′ and n 0 ∈ N * the smallest possible such that Λ ωn 0 . Then there exists an index i 0 ∈ Λ having a predecessor, such that:

[L : K] = d • (Q ω(n0-1)+i0 )d ωn0
where:

d ωn0 = δ ωn0 si Λ = ωn 0 et ♯{i 1 | α ω(n0-1)+i = 1} = +∞ 1 si Λ < ωn 0 ou Λ = ωn 0 et ♯{i 1 | α ω(n0-1)+i = 1} < +∞ Proof : By assumption, L = K [x] /(P (x)) with P ∈ K [x]
irreducible and monic. Suppose that Λ is an ordinal having an immediate predecessor, to fix ideas denote it by ω(n 0 -1) + n, n ∈ N * . By definition, such Λ < ωn 0 , d ωn0 = 1. By construction of key polynomials, P = Q ω(n0-1)+n . Thus, i 0 = n and:

[L : K] = d • (P ) = d • (Q ω(n0-1)+i0 )d ωn0 .
Suppose now that Λ is a limit ordinal, denote it by ωn 0 .

If ♯{i

1 | α ω(n0-1)+i = 1} < +∞, then d ωn0 = 1.
As in the proof of Proposition 3.19, there exists i 0 ∈ N * minimal such that µ ′ ω(n0-1)+i0 (P ) = µ ′ (P ). Thus, if we write the ω(n 0 -1) + i 0 -standard expansion of P , we note that Q ω(n0-1)+i0 belongs to the center of the valuation µ ′ which is the ideal (P ). Note that, by definition and construction of the key polynomials and by choice of i 0 , d

• (P ) d • (Q ω(n0-1)+i0
). Since Q ω(n0-1)+i0 ∈ (P ), we conclude that Q ω(n0-1)+i0 = cP , c ∈ K * and so:

[L : K] = d • (P ) = d • (Q ω(n0-1)+i0 )d ωn0 . If ♯{i 1 | α ω(n0-1)+i = 1} = +∞, then d ωn0 = δ ωn0 .
In this case, P = Q ωn0 . Take i 0 ∈ N * sufficiently large such that α ω(n0-1)+i = 1 and δ ω(n0-1)+i (P ) = δ ω(n0-1)+i+1 (P ) for all i i 0 . Recall that this common value is denoted by δ ωn0 . By Proposition 2.12 of [START_REF] Vaquié | Extensions de valuation et polygone de Newton[END_REF], since K is henselian, the Newton polygon ∆ ω(n0-1)+i (P ) have a unique side of slope β ω(n0-1)+i . This is equivalent to:

µ ′ (c 0,ω(n0-1)+i ) = µ ′ (c s ω(n 0 -1)+i ,ω(n0-1)+i ) + s ω(n0-1)+i β ω(n0-1)+i µ ′ (c j,ω(n0-1)+i ) + jβ ω(n0-1)+i , with P = s ω(n 0 -1)+i j=0 c j,ω(n0-1)+i Q j ω(n0-1
)+i and 0 j s ω(n0-1)+i . By Corollary 3.23 of [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF], β ω(n0-1)+i determine always a side of ∆ ω(n0-1)+i (P ) which only have one, so:

µ ′ ω(n0-1)+i (P ) = µ ′ (c 0,ω(n0-1)+i ) = µ ′ (c s ω(n 0 -1)+i ,ω(n0-1)+i ) + s ω(n0-1)+i β ω(n0-1)+i .
We deduce, by definition of δ ω(n0-1)+i (P ), that:

δ ω(n0-1)+i (P ) = s ω(n0-1)+i . But, for i i 0 , d • (Q ω(n0-1)+i ) = d • (Q ω(n0-1)+i0
) and δ ω(n0-1)+i (P ) = δ ωn0 . Thus:

δ ωn0 = s ω(n0-1)+i = d • (P ) d • (Q ω(n0-1)+i0
d ωj = d L|K (µ ′ , µ).
Proof : Recall that we denote by

α ωj = d • Q ω(j-1)+i 0 (Q ωj ) with i 0 a ωj-inessentiel index
and j ∈ N * . In [START_REF] Vaquié | Famille admissible de valuations et défaut d'une extension[END_REF], this number corresponds to the jump of order j denote by s (j) . By Proposition 3.4.4 of [START_REF] Mahboub | Une construction explicite de polynômes-clés pour des valuations de rang fini[END_REF], we can always suppose that δ ωj = α ωj , for all j < n 0 . By Proposition 2.9 of [START_REF] Vaquié | Famille admissible de valuations et défaut d'une extension[END_REF]:

d • (Q ω(n0-1)+i0 ) = [Γ µ ′ ω(n 0 -1)+i 0 : Γ µ ][k µ ′ ω(n 0 -1)+i 0 : k µ ] n0-1 j=1 α ωj .
We verified after that if Λ = ω(n 0 -1) + i 0 or if Λ = ωn 0 , we always have:

[Γ µ ′ : Γ µ ] = [Γ µ ′ ω(n 0 -1)+i 0 : Γ µ ], [k µ ′ : k µ ] = [k µ ′ ω(n 0 -1)+i 0 : k µ ].
Thus, applying Lemma 4.2, we obtain:

[L : K] =   [Γ µ ′ : Γ µ ][k µ ′ : k µ ] n0-1 j=1 α ωj   d ωn0 .
Since d ωj = δ ωj = α ωj , we concluded that:

d L|K (µ ′ , µ) =   n0-1 j=1 α ωj   d ωn0 =   n0-1 j=1 d ωj   d ωn0 = n0 j=1 d ωj .
Corollary 4.4. -With the assumptions of Lemma 4.2, we have:

δ ωn0 = α ωn0 .
Proof : By Corollary 2.10 of [START_REF] Vaquié | Famille admissible de valuations et défaut d'une extension[END_REF], the defect is the total jump, applying Corollary 4.3 we have the equality:

n0 j=1 α ωj = d L|K (µ ′ , µ) = n0 j=1 d ωj .
Such δ ωj = α ωj = 0, for all j < n 0 , we deduced that:

δ ωn0 = α ωn0 .
Corollary 4.5. -Let (K, µ) be a valued field and L be a finite and simple extension of K. Write µ (1) , ..., µ (g) the different extensions of µ on L, its corresponds to a (pseudo-)valuation of K [x] denoted by the same way. Consider {Q (i) l } l∈Λ (i) the set of key polynomials associated to µ (i) and n (i) 0 ∈ N * the smallest possible such that Λ (i) ωn (i) 0 , 1 i g. Then:

d L|K (µ (i) , µ) = n (i) 0 j=1 d (i) ωj .
We deduced that:

[L : K] = g i=1 e i f i d (i) ω d (i) ω2 ...d (i) ωn (i) 0
,

where e i = Γ µ (i) : Γ µ and f i = k µ (i) : k µ .
Proof : By Proposition 2.5, we know that:

[L : K] = g i=1 e i f i d i ,
where

d i = d L|K (µ (i) , µ), e i = Γ µ (i) : Γ µ and f i = k µ (i) : k µ . Applying Corollary 4.
3 to the fields L h(µ (i) ) and K h(µ (i) ) we have the equalities.

Example 4.8. -We study the example of W. Mahboub given in [START_REF] Mahboub | Key Polynomials[END_REF]. Let k be a field of characteristic p > 2. Denote by ν the z-adic valuation of k(z) and by µ the valuation of K = k(z, y) define by the set of key polynomials {Q y,i } i 1 given in [START_REF] Mahboub | Key Polynomials[END_REF] extending ν. For e ∈ N * , write:

f = x p e -y 2 -z.
We want to find all the valuations of L = K[x]/(f ) whose extends µ. Consider the valuation µ ′ of L define by the set of key polynomials {Q i } i∈Λ :

Q 1 = x β 1 = 1 - 1 4p
Q i = x p e-1 - i j=0 h j β i = 1 - 1 2 2i p i ; i 2
where h 0 = 0 and

h j = Q 2 y,2j+1
z 2 2j p j -1 , for all j 1. We deduced that, for all i 1:

f = (Q i + i j=0 h j ) p -y 2 -z = Q p i + i j=0 h p j -Q y,2 = Q p i + Q y,2i z 2 2i-2 p i -p . But µ(h p i ) = µ Q y,2i z 2 2i-2 p i -p , so: in i (f ) = X p -in µ (h p i ) = (X -in µ (h i )) p .
Thus, for all i 1, δ i (f ) = p and so δ ω = p. Moreover we are in the situation where Λ = ω and α i = 1, for all i 3. We deduced, with the Corollary 4.5, that:

[k µ ′ : k µ ] = p e-1 ; d L|K (µ ′ , µ) = d ω = δ ω = p.
Since [L : K] = p e , by the Corollary 4.5, we deduced that µ ′ is the unique valuation extending µ on L. Example 4.9. -Let k be a field of characteristic p > 0. We denote by K = k ((y, z)). Consider the valuation ν : K * → Z 2 lex , trivial on k, defined by ν(z) = (1, 0) and ν(y) = (0, 1). Consider the polynomial P ∈ K [w] defined by:

P (w) = w p -z p 2 -p w + y p z p 2 .
Write L = K [w] /(P ), we want to find all the valuations ν (i) of L enxtending ν and all the corresponding sets of key polynomial {Q (i) j } j∈Λ (i) . Find a valuation of L extending ν is equivalent to find a pseudo-valuation of K [w] of kernel P . Let µ a such pseudo-valuation. The only possibles values of w are:

µ(w) =    (p, p) (p, 1) (p, 0)
In this case, the polynomial in 2 (P ) is:

in 2 (P ) = in µ (Q (2) 1 ) X + in µ (y p z p 2 ) in µ (Q (2) 1 )
.

Remark that µ(u) = µ(y p ) = (0, p) then µ(α 0 ) = µ(z p u) = (p, p). Writing h = p-1 m=1 α p-m 0 w m-1 , we remark that h ∈ K[w] represent inµ(y p z p 2 ) inµ(Q (2) 1 )
because µ(h) = µ(w p-2 α 0 ) = (p 2 -p, p). Then denote by:

Q (2) 3 = Q (2) 2 + h. But, (w -α 0 )h = α 0 (w p-1 -α p-1 0 ). So: (w -α 0 )Q (2) 3 = w p -z p 2 -p w + α 0 z p 2 -p -α p 0 . But, α 0 z p 2 -p -α p 0 = uz p 2 -u p z p 2 = (u -u p )z p 2 = y p z p 2 we deduced that: Q (1) ∞ Q (2) 3 = (w -α 0 )Q (2)
3 = P. The sequence of key polynomials stop. The second way to extend ν on L is to consider the valuation ν (2) defined with the key polynomials Q

(2) 1 , Q (2) 2 and Q (2) 3 .
Finally, since Λ (2) = 3 < ω, we deduced that: d L|K (ν (2) , ν) = d (2) ω = 1. Then we obtain:

p = [L : K] = d • (P ) = d • (Q (1)
1 )d (1) ω + d • (Q

3 )d (2) ω = 1 + (p -1). Example 4.10. -We will study an example proposed by S.D. Cutkosky in [1]. Let k a field of characteristic p > 0. Denote by K = k(u, v) and by L = K[y]/(f ) where:

f = y p 2 +1 + y p 2 -yv p + u -v p .
We define a valuation ν over K such that ν(u) = p with:

R ν = i 0 A i , where A i = k [u i , v i ] (ui,vi) and: u 0 = u v 0 = v ;    u pi+j = u pi v j pi+j v pi+j = v pi , 1 j < p;    u p(i+1) = v pi v p(i+1) = u pi v p pi -γ p(i+1) ; γ p(i+1) ∈ k * .
Then we have ν(v pi ) = 1 p i and ν(u pi ) = 1 p i-1 . Remark that, for i 0:

v p(i-1) -γ p(i+1) v p pi = v p pi v p(i+1)
, where we write v -p = u. Then:

ν(v p(i-1) -γ p(i+1) v p pi ) = 1 p i-1 - 1 p i+1 .
We want to extend ν on L. There exist a first valuation denoted by ν * defined by the sequence of key polynomials {Q l } l∈Λ :

Q 1 = y, β 1 = ν * (Q 1 ) = 1 p ; Q l = Q l-1 -γ 1 p p+2l-3 l-2 j=0 v p(2j+1) , β l = ν * (Q l ) = l-1 j=0 1 p 2j+1 . Remark that lim l→+∞ β l = p p 2 -1
, so there exists a first limit key polynomial which is:

Q ω = y p -v, β ω = ν * (Q ω ) = 1.
We continue to define the set of key polynomials, for i 1, by:

Q ω+i = Q ω+i-1 + γ 1 p p+2i-2 i-1 j=0 v p(2j) , β ω+i = ν * (Q ω+i ) = i j=0 1 p 2j .
Remark that lim i→+∞

β ω+i = 1 + 1 p 2 -1
, so there exists a second limit key polynomial which is:

Q ω2 = y p 2 + u -v p = Q p ω + u, β ω2 = ν * (Q ω2 ) = p + 1 p .
We end the set of key polynomials with, for n 1:

Q ω2+n = Q ω2+n-1 + (-1) n uy n , β ω2+n = ν * (Q ω2+n ) = p + n + 1 p .
The last and third limit key polynomial is Q ω3 = f and β ω3 = +∞. Remark that for all n 0:

f = (y + 1)Q ω2+n + (-1) n+1 uy n+1 . So, in K [[y]], f = (y + 1)Q ∞ where Q ∞ = y p 2 + u -v p -uy n 1
(-1) n y n . Remark that for l 1:

y p -v = Q p l -v p(2(l-1)) l-2 j=0 v p p(2j+1) , then: in l (y p -v) = X p -in ν   v p(2(l-1)) l-2 j=0 v p p(2j+1)   =   X -in ν   l-2 j=0 v p(2j+1)     p .
We deduce that for all l 1:

α l = 1, d ω = δ ω = δ l (y p -v) = p = α ω .
In the same way, for i 1:

y p 2 + u -v p = Q p ω + u = Q p ω+i -v p(2i-1) i-2 j=0 v p p(2j) , in ω+i (y p 2 + u -v p ) = X p -in ν   v p(2i-1) i-2 j=0 v p p(2j)   =   X -in ν   i-1 j=0 v p(2j)     p .
We deduce that for all i 1:

α ω+i = 1, d ω2 = δ ω2 = δ ω2+i (y p 2 + u -v p ) = p = α ω2 .
Finally, for all n 0:

in ω2+n (f ) = X + in ν (-1) n+1 uy n+1 .
We deduce that for all n 1:

α ω2+n = 1, d ω3 = δ ω3 = δ ω3+n (f ) = 1.
By the Corollary 4.3:

d L|K (ν * , ν) = d ω d ω2 d ω3 = p × p × 1 = p 2 .
Remark that [L : K] = p 2 + 1, so there exists a second valuation ν (2) of L who extends ν and defectless. By the equality f

= (y + 1)Q ∞ in K [[y]
], the valuation verifies ν (2) (y) = 0, so this is the trivial valuation.

Key polynomials and defectless fields

Consider K ֒→ K(x) a simple transcendental field extension. Let µ ′ be a valuation of K(x), write µ := µ ′ | K . We denote by G the value group of µ ′ and G 1 the smallest isolated non zero subgroup of G. We suppose that µ is of rank 1, µ ′ (x) > 0.

Proposition 5.1. -If K is defectless, there exists a 1-complete set of key polynomials {Q i } i∈Λ such that Λ is a finite set or N * . In particular, there is no limit key polynomial for valuations of rank 1 over defectless fields.

Proof : Apply the process of [START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF]. If there exists i 0 ∈ N, such that β i0 / ∈ G 1 , we write Λ = {1, ..., i 0 -1} and, by definition, {Q i } i∈Λ is 1-completed. Otherwise, for all i ∈ N, β i ∈ G 1 and we write Λ = N * . If ♯{i 1 | α i > 1} = +∞, by Proposition 3.19, the set {Q i } i∈Λ is 1-completed. If ♯{i 1 | α i > 1} < +∞, write Q ω the monic polynomial of smallest degree such that, for all i 1:

µ ′ i (Q ω ) < µ ′ (Q ω ).
Since K is defectless, the extension K ֒→ L = K[x]/(Q ω ) is defectless. Consider µ ′ as the composition of a valuation µ (1) of value group G 1 centered on K[x]/(Q ω ) and a valuation θ of value group G/G 1 centered on K[x] (Qω ) . The set of key polynomials for µ (1) is the same as the set of key polynomials for µ ′ except that µ (1) (Q ω ) = ∞. Thus, d L|K (µ (1) , µ) = 1. By Corollary 4.5, we deduce that: (1) , µ) = 1.

δ ω = d L|K (µ
Applying Proposition 3.20, we conclude that the sequence {β i } i 1 is unbounded in G 1 and so, by Proposition 3.18, the set {Q i } i∈Λ is a 1-completed set of key polynomials.

Corollary 5.2. -If car(k µ ) = 0, there exists a 1-completed set of key polynomials {Q i } i∈Λ such that Λ is a finite or N * . In particular, there is no limit key polynomials for valuations of rank 1 whose residual field is of characteristic zero.

Proof : Apply Proposition 2.10 and Proposition 5.1.

Local uniformization of quasi-excellent local domain without defect

We extend here the results of the section 8 of [9] for a valuation satisfying some inductive assumption about defect. More precisely, in [9], to obtain a theorem of monomialization, the valuation needs to have a complete set of key polynomials without limit key polynomial: this is the case if the valuation is defectless. As a corollary, we find the local uniformization in characteristic zero.

Let (R, m, k) be a local complete regular equicharacteristic ring of dimension n with m = (u 1 , ..., u n ). Let ν be a valuation of K = F rac(R), centered on R, of value group Γ and Γ 1 the smallest non-zero isolated subgroup of Γ. Write:

H = {f ∈ R | ν(f ) / ∈ Γ 1 }.
H is a prime ideal of R (see the proof of Theorem 6.4). Moreover suppose that:

n = e(R, ν) = emb.dim (R/H) , that is to say:

H ⊂ m 2 . Write r = r(R, u, ν) = dim Q n i=1
Qν(u i ) .

The valuation ν is unique if ht(H) = 1; it is the composition of the valuation µ : L * → Γ 1 of rank 1 centered on R/H, where L = F rac(R/H), with the valuation θ : K * → Γ/Γ 1 , centered on R H , such that k θ ≃ κ(H). By abuse of notation, for f ∈ R, we will denote by µ(f ) instead of µ(f mod H). By the Cohen's theorem, we can suppose that R is of the form:

R = k [[u 1 , ..., u n ]] .
For j ∈ {r + 1, ..., n}, write {Q j,i } i∈Λj the set of key polynomials of the extension k ((u 1 , ..., u j-1 )) ֒→ k ((u 1 , ..., u j-1 )) (u j ), Q j,i = {Q j,i ′ |i ′ ∈ Λ j , i ′ < i}, Γ (j) the value group of ν |k((u1,...,uj)) and ν j,i the i-troncation of ν for this extension.

Lemma 3 .

 3 16. -Let ν be a valuation of rank 1 centered on a local noetherian ring R. Denote by P ∞ the support of ν. Then, ν (R \ P ∞ ) does not contain any increasing bounded infinite sequence. Remark 3.17. -Recall the Definition 1.1, the data of a valuation ν centered on a local ring (R, m) is the data of a minimal prime ideal P ∞ of R (the support of the valuation) and of a valuation ν ′ of the quotient filed of R/P ∞ such that R/P ∞ ⊂ R ν ′ and m/P ∞ = R/P ∞ ∩ m ν ′ .

Proposition 3 .

 3 19. -([START_REF] Herrera Govantes | Key polynomials for simple extensions of valued fields[END_REF], Corollary 5.8) Suppose that we have construct an infinite set of key polynomials

Proposition 3 .

 3 20. -([3], Proposition 6.8) Under the previously assumptions, if

For the definition of

  local framed sequences, one may consult Définition 7.1 and the sections 4.1 and 4.2 of [9]. Theorem 6.1. -Suppose that, for R n-1 = k [[u 1 , ..., u n-1 ]] we have:

Corollary 4.6. -Under the assumptions of Corollary 4.5 and denoting by p = car(k µ ), then, for 1 i g and 1 j n (i) 0 , there exists e (i) ωj ∈ N such that:

ωj .

Proof : It is an immediate consequence of Corollary 4.5 and Remark 2.4.

Example 4.7. -We study the example 3.2. of [START_REF] Vaquié | Famille admissible de valuations et défaut d'une extension[END_REF]. Let k be an algebraic closed field of characteristic 0. Consider K = k(y) equipped with its y-adic valuation denote by µ. The polynomial:

. The valuation extends only on two valuations µ (1) and µ (2) over the field L = K[x]/(P ) given by the sequences of key polynomials {Q

i } i∈Λ (1) and {Q (2) i } i∈Λ (2) such that:

where u 0 = 0, u l = l j=1 c j y j+2 , for all l 1, c 1 = c 2 = 1 and c j = 4 j-2 3...(2j -3) 6...(2j) for all j 3. Following the notations of the Corollary 4.5, with Λ (1) = Λ (2) = ω, and α

= 1 for all i 2, we deduced that:

Moreover, since a field of caracteristic zero is defectless, we found the fact that:

d L|K (µ (1) , µ) = 1 = d (1) ω and:

We still find the result of the Corollary 4.5, that is to say:

But if µ(w) = (p, 1) then µ(w p ) = µ(y p z p 2 ) = (p 2 , p) > µ(z p 2 -p w) = (p 2 , 1), the monomial w p is not of minimal value in P . The two only values possible of w are:

We are in the situation of the example 5.3.2 of [START_REF] Mahboub | Une construction explicite de polynômes-clés pour des valuations de rang fini[END_REF]. In this case, we prove that the key polynomials are:

where u j = i i=1 y p i and µ(Q (1) j ) = (p, p j ),for all j 2. Moreover the polynomial P is a limit key polynomial. Finally, the polynomial in j (P ) is irreducible for all j 1, the valuation extends uniquely since µ(w) = (p, p). Thus, a first way to extend ν over L is to consider the valuation ν (1) defined by the key polynomials

The other solutions of the equation are the u -l with l ∈ {1, ..., p -1}. Writing α l = z p (u -l), for l ∈ {0, ..., p -1}, we can prove that:

So, the roots of P are α 0 , ..., α p-1 and if we write Q

(1) ∞ = w -α 0 = w -z p u then:

Finally, since in j (P ) = -z p 2 -p (X -y p i z p ), for all j 1 and Λ (1) = ω, we deduced that:

2) Suppose that µ(w) = (p, 0).

Denote by

-z p 2 -p . Then:

1 ). Definition 6.2. -For a local noetherian ring (R, m), with m = (u) = (u 1 , ..., u n ) and f 1 , ..., f s ∈ m, we call the monomial property for R and f 1 , ..., f s this three statements:

1. R is regular; 2. For 1 j s, f j is a monomial in u times a unit of R; 3. For 1 j s, f 1 divides f j in R. 1) , u (1) , k (1) ρ1 / / . . .

such that S (i) /H (i) and f 1 , ..., f s have the monomial property, where H (i) is the implicit prime ideal of S (i) and f j are the images of f j mod H (i) , 1 j s.

2.

We say that S and f 1 , ..., f s have the local uniformization property of dimension l if there exists a local framed sequence:

(S, u, k) = S (0) , u (0) , k (0) ρ0 / / S (1) , u (1) , k (1) ρ1 / / . . .

ρi-1

/ / S (i) , u (i) , k (i) , such that S (i) and f 1 , ..., f s have the monomial property.

Then S and f 1 , ..., f s have the quotient local uniformization property of dimension l.

Proof : By Theorem 2.1 of [START_REF] Herrera Govantes | Extending a valuation centered in a local domain to the formal completion[END_REF], µ extends uniquely in a valuation µ centered on S/H. By the Cohen's structure theorem, we know that there exists a complete regular local ring of characteristic zero R and a surjective morphism ϕ: ϕ : R ։ S/H.

Write H = ker ϕ, since H is a prime ideal (Theorem 2.1 of [START_REF] Herrera Govantes | Extending a valuation centered in a local domain to the formal completion[END_REF]), H is a prime ideal of R. Choose R such that dim(R) = l. Write K the quotient field of R, K is of the form k ((x 1 , ..., x l )).Let θ a valuation of K, centered on R H , such that k θ = κ(H). If we look µ as a valuation centered on R/H by the morphism ϕ, we can consider the valuation ν = µ • θ centered on R and of value group Γ. Then, Γ 1 is the non-zero smallest isolated subgroup of Γ and:

Then, T contains x 1 , ..., x l and: Proof : Write u = (u 1 , ..., u n ) a minimal set of generators of m and H the implicit prime ideal of S. Write u = (y, x) such that x = (x 1 , ..., x l ), l = emb.dim S/H and the images of x 1 , ..., x l in S/H induce a minimal set of generators of (m S)/H. By Theorem 2.1 of [START_REF] Herrera Govantes | Extending a valuation centered in a local domain to the formal completion[END_REF], µ etends uniquely in a valuation µ centered on S/H and, since car (k µ ) = 0, then car(k µ ) = 0. We saw that there exists a surjective morphism:

where H = ker ψ. The quotient field of R is of the form k ((x 1 , ..., x l )). By Proposition 2.10, we deduce that the valued fields k ((x 1 , ..., x j-1 )) , µ |k((x1,...,xj-1)) are defectless. To conclude, it is sufficents to apply recursively on j ∈ {1, ..., n} the Theorem 6.5 for k ((x 1 , ..., x j-1 )) , µ |k((x1,...,xj-1)) .


Eduardo Cabrera-Granado

Yxing Chen

Christian Köhler

Ihar Babushkin

Luc Bergé

Stefan Skupin

THz and far-infrared radiation from ionizing multi-color pulses

Research on intense terahertz (THz) electromagnetic sources has received an increasing attention owing to numerous applications, for example, in time-domain spectroscopy, biomedical imaging or security screening [START_REF] Tonouchi | Cutting-edge terahertz technology[END_REF]. Among the various techniques employed to generate THz radiation, focusing intense two-color femtosecond pulses in air or noble gases provides interesting features like absence of material damage, large generated bandwidth (up to ∼ 100 THz) and high amplitudes of the emitted THz pulses (> 100 MV/m) [START_REF] Kim | High-Power Broadband Terahertz Generation via Two-Color Photoionization in Gases[END_REF]. First reported by Cook et al. [START_REF] Cook | Intense terahertz pulses by four-wave rectification in air[END_REF], THz emission from intense two-color pulses was initially attributed to optical rectification via third-oder nonlinearity. However, it was shown later that the plasma built-up by tunneling photoionization is necessary to explain the high amplitudes of the THz field [START_REF] Kress | Terahertz-pulse generation by photoionization of airwith laser pulses composed of both fundamental and second-harmonicwaves[END_REF][START_REF] Roskos | Broadband thz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications[END_REF][START_REF] Kim | Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions[END_REF], and a quasi-dc plasma current generated by the temporally asymmetric two-color field is responsible for THz emission [START_REF] Bergé | 3D numerical simulations of THz generation by two-color laser filaments[END_REF][START_REF] Borodin | Transformation of terahertz spectra emitted from dual-frequency femtosecond pulse interaction in gases[END_REF].

Here, THz emission in gases via ionizing multi-color femtosecond pulses is analyzed by means of semianalytical models and finite-difference-time-domain simulations in 1D and 2D geometries. We find the emission in backward direction having a much smaller spectral bandwidth than in forward direction and explain this by interference effects. Forward THz radiation is generated predominantly at the ionization front and thus almost not affected by the opacity of the plasma, in excellent agreement with results obtained from a unidirectional pulse propagation model [START_REF] Koehler | Directionality of terahertz emission from photoinduced gas plasmas[END_REF]. Moreover, we show that produced THz signals interact with free electron trajectories and thus influence significantly further THz generation upon propagation, i.e., make the process inherently nonlocal. This self-action plays a key role in the observed strong spectral broadening of the generated THz field in forward direction. Diffraction limits the achievable THz bandwidth by efficiently depleting the low frequency amplitudes in the propagating field.


Clement Oury

Clément Oury

Head

Roswitha Poll

Counting the uncountable: statistics for web archives

Keywords: Libraries, web archives, statistics, quality measures, ISO/TR 14873

Formore than a decade, libraries have started to "collect the web". National libraries in particular select, collect and store publications and websites from their national domain, seeing this as a task similar to traditional legal deposit. The collection policies and collecting methods vary, so that it is difficult to compare the quantity and quality of the respective web archives. In order to harmonize the evaluation of web archives, ISO TC 46 SC 8 has produced a Technical Report that standardizes the terminology and statistics and offers tested indicators for assessing the quality of web archiving. The paper describes the aimsand contents of the ISO Report.

General

Standards forassessingthequantityandqualityoflibraryservices

In the last decades, ISO has developed a set of standards for assessing the quantity and quality of library services. Thiswork was and is done within the Technical Committee 46 Information and documentation, Subcommittee 8 Quality -Statistics and performance evaluation.

 ISO 2789International library statistics

Thisis the central standard for all statistical procedures in libraries. The 5 th edition is being published.

 ISO 11620 Library performance indicators

This standard describes quality indicators for library services. The 3 rd edition is planned for the end of 2013.

 ISO/TR 28118 Performance indicators for national libraries

The Technical Report (published 2009) describesquality indicators for the specific tasks of national libraries. As a number of these indicators have been taken over into ISO 11620, no new edition of the TR is planned at the moment.  ISO/DIS 16439Methods and procedures for assessing the impact of libraries This new standard describes the ways to identify and provide proof ofa positive impact of libraries on individuals and society. Publication is planned for the end of 2013.

The standards, especially ISO 2789, are intended to provide definitions and statisticalprocedures for all types of library collections and library services. New activities are continuously integrated into the new versions, e. g. digitisation of analogue material or services for mobile devices.

There is one library activity that has developed recently and has not yet found its place in the existing standards for library evaluation, namely web archiving.Evidently, this huge new task did not seem to fit in with the usual role of libraries. Archiving the web means selecting and capturingInternet resources, storingthem in web archives, preserving them and managing sustainable access to the archives.The collecting processes are managed automatically at regular intervals by harvesting software.

Web archiving startedtowards the end of the nineties, mainly in national libraries. Most national libraries are responsible under legal deposit law for collecting and preserving the printed cultural heritage of their country. This task has now beenlegally or voluntarilyextended to Internet resources. The reason for the collection is the same as for print material: the danger of itemsin this case websitesdisappearing and being lost for future generations.

The libraries started their archiving activities from different approaches, but cooperation soon proved to be crucial in this quickly changing area. Therefore the technical and legal problems were taken up by IIPC,the International Internet Preservation Consortium, founded in 20031 .

A first step towards standardisation in web archives: the WARC format

The first challenges which heritage institutions needed to face were of a technical nature: how to harvest, store, access and preserve the immense set of data available on the web. This is the reason why the first achievements towards standardisation were in the technical domain. When pioneer institutions started web archiving, in the early 2000s, there was at least a de facto standard: the ARC file, a container file designed by Internet Archive in 1996 to concatenate, store and handle the thousands of files harvested on the web2 .

The ARC file was indeed the reference format for all tools developed and used by Internet Archive and its partners in the framework of the IIPC: the Heritrix crawler used to harvest websites, the Wayback Machine used to access web archives… However in the middle of the 2000s emerged a strong need for the adoption of a standard format: -first, the ARC format needed to evolve to better take into account new requirements for collection description and long term preservation: new characteristics, new functionalities and record types were expected; -as the ARC format specification was short, simple and highly adaptable, the creation and usage rules of ARC files were mainly dictated by the way tools developed by Internet Archive were working; other institutions needed a better overview of the evolutions of the format;

-finally, as web archiving was a new activity, being able to build collections and tools based on an internationally standardised format was a way to give strong institutional confidence on the long term maintenance of web archives and web archiving activities.

A draft standard for an evolution of the ARC format came out of the original discussions of the members of the IIPC, called WARC (Web ARChive format) 3 . After a few years of standardisation process, the first version of the "ISO 28500 WARC file format" was published on 15 May 2009. This standard has been adopted by most heritage institutions and is now even used for other kind of collections: storage of e-journals, of digitized material… 4

2. The new Technical Report: ISO/TR 14873

Why introduce standardisation for web archives?

Web archiving institutions are not only facing technical issues, and other standardisation needs emerged when they started gathering a large amount of resources. It became rapidly critical to identify quantitative and qualitative measures for the evaluation of this new library service. Collecting the web is a complex and expensive activity, and funding institutions ask for evidence of the cost-effectiveness of the service and of its value for society. Definitions and collecting methods for statistical data and quality indicators were needed.

At first, standardisation was needed in order to provide a necessary clarification of terms and measures used to define and assess the different parts of a web archiving process. Although the majority of web archiving institutions use similar crawling technologies, and even the same software, they do not necessarily describe their activity the same way. For example, some of them were crawling or harvesting, whereas others were collecting or archivinginstitutions were sometimes using different words for the same kind of activity; or the same words for different kind of activities. It is even more complex when it comes to the result of the crawl: should institutions call the copy of a website performed by a crawler a "capture", a "version" or an "archive" of this website?

However, this was only a matter of definitions for new tools and new procedures. The traditional library paradigms were more severely challenged by the fact that institutions had to deal with totally different kinds of documents. The web offers several levels of granularity at which institutions may identify "documents" that could be assimilated to "classical" library holdings. The website appears at first sight as a possible documentary unit; but some institutions may only choose to harvest parts of websites, or only individual pagesthis is especially the case for very large websites: institutional websites or publication platforms. How to count those parts or those pages along with websites? Finally, each individual component of a webpagei.e. each web filemay also be viewed as a separate document: a PDF, a JPEG or PNG image, etc.

These two issuesthe lack of maturity of the terminology, and the complex granularity of documents on the webmay first be seen as problematic within institutions. How to define a content policy if the borders of the content itself are fuzzy? How to explain to readers what they can find in the collection if librarians are not even able to precisely describe it? They become even more complex when a dialogue or a comparison must be established between institutions.

Standardisation was therefore identified as one of the ways to face this kind of issue. The clarification of definitions, and the proposal of agreed statistics and indicators were intended to facilitate:

-international measures and comparisons between institutions; -therefore, the identification of best evaluation practices within institutions -and a better understanding and advocacy of web archiving initiatives in a wider environment, either national, regional or international. Indeed, having a standard for statistics and quality indicators was in itself a way to obtain international recognition of web archives as heritage and research library material, and as an activity in the durability and reliability of which library managers could have confidence. This is the kind of role that the standardisation of WARC, the storage and preservation format for web archives, played few years ago.

The standardisation process

To this end, ISO TC 46 SC 8 decided to address the topic of web archiving. A working group was launched in December 2009. In order to ensure that the needs of the widest range of libraries were taken into account, members of this working group included both long-time practitioners and beginners in the field of web archiving. However, they were all working in national libraries (France, Germany, Spain, Sweden, United Kingdom; with the specific case of the state library of Bavaria). The draft documents were therefore distributed to the other members of the IIPC, in order to ensure that the chosen terminology, statistics and indicators were consistent with the current practices in university libraries, or in other research and heritage institutions (national archives, not for profit foundations, etc.). The working group notably got detailed and valuable comments from the Library of Congress and the University of North Texas libraries, which had worked on metrics for web archives in the framework of the End of Term Project5 . SC 8 decided that instead of incorporating web archiving directly into the existing library standards, the topic should be first handled in a Technical Report. This type of ISO publication has less formal regulations than a standard and can best serve the purpose of quickly proposing an instrument for the evaluation of web archiving activities. After a phase of testing, the definitions and procedures may be integrated into new editions of ISO 2789 and ISO 11620.

A report addressing all components of web archiving activities

ISO/TR 14873"Statistics and quality issues for web archiving", which will be released in 2013,proposes statistics and indicators not only forall individualsdirectly involved in web archiving, but also for the management of the collecting institutions and for funding authorities. This wide potential audience is taken into consideration in the terminology and descriptions used, and more general information is given about the background, practicesand problems of web archiving than would be needed for experts in this topic. After defining the terminology, the Report describes the following: a. The different collecting strategies: bulk harvesting, selective harvesting, or a combinationof both Bulk harvesting means that the crawl (the process of browsing and copying web resources) is intended to capture entirely a top level domain or a subset (e. g. .fr, .de).

In selective harvesting web resources are collected according to certain criteria, such as relevance to specific subjects or events (e.g. Special software is needed for finding and replaying resources in the web archive. The software must be able to identify and retrieve unique resources. But as the archived resources have been "frozen" at a specified point of time, they cannot show the same interactivity as live versions (e.g.message boards, discussion forums).

c. Preservationmethods

The value of web archives generally reveals itself after a certain period of time, when the original websites are no longer available online. Therefore, maintaining a long term access to these collections is a critical issue. Compared to other kinds of digital documents, the specific difficulties of preserving web archives are related to the huge amount of data involved and the diversity of file formats and media types.

d. The legal basis for web archiving

Web archiving is mainly regulated by national legislation on copyright and/or legal deposit. This legislation can include or exclude certain resources and can restrict web archiving to permission-based collecting which requires permissions from the rights holders prior to the harvesting.

In countries where no specific legislation is available, institutions may choose a permission-based approach where authorization to harvest resources should be obtained from the rights holders. Alternative approaches to permission management include the so called "opt out" or "notice and take down" model, where resources are harvested and made available on the basis of assumed or implied permissions, and canbe taken down when requested by the rights holders.

Measuring quantity in web archiving

As is the case with the statistics for all other library services, the data collected for web archiving should be appropriate for the evaluation of the service, for comparison over time and between libraries, for the support of internal library management, and for promotingthe value of the service. The Technical Report recommends the continuous collection of the following statistics:

Collection: statistics onthe size and growth of the web archive

The first question about web archives is usually: How big is such an archive, and how much is added every day, month or year? Traditional library materials are counted as volumes, audio-visual documents or manuscripts, the electronic resources as eBooks, e-journals or databases. Quantifying the size of a web archive demands other measures. Indeed, as mentioned above, it isnot straightforward to define what the discrete "document" ison the web. Each file may be considered a separate document: a jpeg may be considered an image, a pdf may be considered a monograph… On the other hand, html pages groups together several, sometimes tens of different files; and generally individual files are meaningless if they are not considered in relation to the other files to which they are connected. Libraries do not count the number of pages in the books they hold in their stacks… except perhaps for the calculation of digitization budgets.

It is thus tempting to refer to the notion of website. However a website is merely an intellectual entity: it is a set of interconnected pages produced by a single publisher (a person, a group or an organisation), generally (but not necessarily) hosted on the same domain and/or on the same host. There is no real technical way to count the number of websites within a web archive. Therefore, the following measures are recommended:

-number of targets (i.e. number of intellectual entities selected by librarians; which generally correspond to a website); and number of captures of these targets (the number of captures depends on the crawling frequency); -number of domains or hosts (again used as a substitute for the number of websites, this measure is also applicable to bulk crawls, contrary to the number of targets); -number of URLs, i.e. number of responses to the http request sent by the crawler (even 404 errors should be included); -number of bytes (compressed and uncompressed): this measure is useful whenplanning storage, and is comparable to the linear metrefor the management of stacks in a library.

Collection: statistics onthe contents

The contents of a web archive can also be described and counted accordingto the following criteria: geography, format,language, and chronology.

-geographical distribution, indicated by top level or second level domains Resources may be hosted on generic top level domains (TLDs), such as .com or .org; or on country code TLDs, such as .dk, .de… The country code TLDs indicate the geographical distribution of the resources in a Web archive. This measure is especially useful for national libraries which are entitled to collect the entire intellectual output of their country. For a more thorough characterization, it is also possible to look at the secondlevel domains, which are subdivisions within the top level domains for specific categories of organisations (e.g. .gov.uk for governmental websites).

-differentiation by format type This differentiation can use resource types (e.g. text, image, audio) or file formats(e.g. html, jpeg).It is also especially useful for preservation issues.

-differentiation by language

Identification of languages within a web archive and within one top level domain helps the understanding of the cultural diversity in a country or the proximity to other countries; it is however technically challenging. Characterisation by language has always been used for library collections.

-chronological distribution Chronology here does not mean the date of origin of a resource, but the point of time at which it was archived. This statistic is useful for assessing the uniqueness of resources: If they have been archived years ago, they are the more likely to have disappeared on the live web. The data can also be used for preservation issues: older resources are more likely to be in an obsolete format.

Statistics on the usageof the archive

The conditions for using the archived resources depend on the national legislations and on the policies of the collecting institution that can restrict direct access to specific locations (e.g. the reading-room of a national library) or to certain parts of the archive. Web archives are called white, grey or dark according to their degree of accessibility (online access / onsite access only / access only for library professionals).

If online access to the archive is possible, usage data can be collected via web analytics (e.g. page views or visits). If access is restricted to a location, unique visitors can be counted and can also be surveyed as to purpose and success of their visit. Usage statistics are crucial for web archives as they can show a direct benefit of the collection.

Preservation data

As with other kind of digital documents, preservation of web archives should be done at two levels: at the basic level,the integrity of the "bitstream"should be ensured, and at a more sophisticated level, the appearance, function, behaviour and even the user experience of digital resources should be preserved, using strategies such as migration and emulation. Statistics are proposed for both kinds of preservation levels: -for bitstream or physical preservation, statistics help in calculating if the data are kept in a safe manner, indicating for example what proportion of the archive gets a replicated storage;

-statistics for logical preservation deal with the nature and the reliability of metadata available for the resources. The most critical information is related to the format of resources; it is necessary to calculate what proportion of the archive is in a format for which a preservation strategy has been defined.

Cost data

The costs of web archiving can be assessed much in the same way as the costs of other library services, namely staff costs, costs of hardware and software and maintenanceof buildings. But web archiving activities are still recent, and some cost factors, especially for long-term preservation, will become more visible over time.

Measuring quality in web archiving

The quality -effectiveness and efficiency -of library activities and services should be measured against the background of a library's mission and goals. The goals of web archiving as defined by legislation and/or libraries' mission statements can be summarised as following:

 to collect and preserve the contents of the web as part of cultural heritage  to organise permanent access to the archived material for research and general information

The Technical Report offers a number of quality indicators or performance indicators that have been tested within the ISO working group. The indicators have been selected according to the criteria for library performance indicators as described in ISO 11620: such indicators should be reliable if used repeatedly, informative for the library and its authorities, yielding results that are comparable between libraries, and practical, using data that the library can collect without high levels of effort (e.g. automated statistics from harvesting software). For measuring the quality of web archiving, another criterion can be added: flexibility. The indicators should be adaptable to quick changes that are to be expected in the development of the web.

Web archiving activities show high quality if they comply with the following requirements:

a. The library has a clear statement of the intended coverage of the collection and succeeds in following it. This criterion is difficult to measure even for the print legal deposit collection. 6 For the web archive, the indicator is -Achieved percentage of mandated scope -Percentage of resources disappeared from the live web during a given period of time

Use of the measures

Not all of the proposed statistics and quality measures should be used for every web archive; some are more adapted for selective harvesting than for bulk harvesting. Though web archiving is a genuine task of national libraries (or national archives), other institutions are also engaged in collecting web resources, mainly for special collections. Most of the measures that have been defined in the Technical report will also be relevant for smaller archives, collected by selective harvesting.

The data will in any case be useful for internal management, for allocation of staff and other resources, for storage and preservation planning. Statistics of usage and cost data will be especially interesting for the library's authorities. Measures demonstrating the size and growth of the archive, the coverage of the legal mandate and the salvage of disappeared websites can interest the public, the media, politicians, and all institutions responsible for cultural heritage.

  For permission-based collecting, the indicator measures: -Percentage of requests for agreements or permissions granted by rights holders b. Care is given to the structuring and indexing of the collection in consideration of searching possibilities. The indicators are: -Percentage of full-text indexed resources -Percentage of catalogued resources c. The library has implemented long-term preservation procedures. There are several indicators for preservation: -Percentage of the resources with at least one replication -Percentage of lost or deteriorated resources -Percentage of resources with identified file format -Percentage of resources whose format has a defined preservation strategy -Percentage of virus-checked resources d. As far as possible, there is free (online)access to the collection. The indicators measure accessibility and accesses. -Percentage of resources accessible to end users -Annual percentage of accessed resources -Percentage of library visits including a visit to the Web archive e. The archiving activity is performed in a cost-effective way. The quality indicators measure: -The costs per collected URL -The percentage of total library staff that is involved in web archiving There is one indicator that underlines thevalue of web archiving for future generations by assessing what part of the web archive does not exist anymore on the live web:

  economiccrisis, Inuit culture) or scholarly importance of the resources.b. Access and description strategiesIndexes in web archives basicallyconsist of the URL of the requested resource, plus the date of its harvest, in order to differentiate between capturesof the same resource.

	Other methods of describing web archives include:
	o cataloguing: a usefulwaytointegratewebarchiveswith alibrary'sexistingcollections,
	so thattheybecomediscoverablethroughcataloguesearch, but resource intensive and
	not suitable for bulk harvest;
	o full-text search: a powerful way to discover resources but
	technicallychallengingtoimplement;
	o automated keyword or metadata extraction
	o data mining techniques (text mining, link analysis…): a
	growingneedexpressedamongresearchcommunitiesthatofferopportunitiestoprovide
	accessto different viewsof a webarchive, unlockingembeddedpatternsandtrends,
	relationshipsandcontexts.

http://www.netpreserve.org/about/index.php

http://archive.org/web/researcher/ArcFileFormat.php

See http://research.library.unt.edu/eotcd/wiki/Main_Pageandespeciallyhttp://research.library.unt.edu/eotcd/wiki/Cate gory:Metrics

ISO/TR 28118: 2009Performance indicators for national libraries, Indicator A.1