On Some Subring of Formal Power Series∗

On some subring of formal power series∗

Abdelhafed ELKHADIRI†

Abstract

For each n ∈ IN, we denote by IC[[z1, . . . , zn]] the ring of formal power series with complex coeﬃcients and IC{z1, . . . , zn} the ring of convergent power series. For each n ∈ IN, we consider An ⊂ IC[[z1, . . . , zn]] a subalgebra containing IC{z1, . . . , zn}. In the ﬁrst part we give some conditions under which An possesses good properties as module over IC{z1, . . . , zn}. In the end we give condition under which each An is noetherian algebra and we give some examples.

Introduction

Let IC[[z1, . . . , zn]] be the ring of formal series; we give some conditions under which a subring An of IC[[z1, . . . , zn]] is noetherian. As an application we prove that the subring of IC[[z1, . . . , zn]] deﬁned by the growth of the coeﬃcients is a noetherian ring. This result has been proved by J. Chaumat and A.M. Chollet [2]. Ours method is more simple and general and can be applied in others contexts.

1 background .

We denote by IC[[z1, . . . , zn]] the ring of formal powers series and by On the ring of germs of n holomorphic functions at the origin in IC . Finally IC[z1, . . . , zn] is the ring of polynomial with complex coeﬃcients. p p p p−i If λ = (λ1, . . . , λp) ∈ IC , then P (zn, λ) = zn + Pi=1 λizn is called a generic polynomial in p p zn of degree p. Let σ = (σ1, . . . , σp) : IC → IC be the polynomial map such that each σi is (modulo a sign) the ieme symmetric function and:

P (zn, σ(ξ)) = (zn − ξ1) . . . (zn − ξp), where ξ = (ξ1, . . . , ξp). Let A : ICn × ICp → ICn × ICp be the mapping deﬁned by A(z, ξ) = (z, σ(ξ)). The rank of A, rk(A), is the rank of the jacobian matrix of A considered as a matrix over the ﬁeld [On+p]

(quotient ﬁeld of the ring On+p). We see that rk(A) = n + p; hence the induced mapping: A∗ : IC[[z, ξ]] → IC[[z, ξ]],

∗Mathematics Subject Clasiﬁcation . Primary 32Bxx, 14Pxx, Secondary 26E10 †Key words: noetherian rings, Weierstrass division theorem, implicit function theorem.

1 deﬁned by A∗(ψ(z, ξ)) = ψ(z, σ(ξ)) is injective.

Deﬁnition 1 A weak formal Weierstrass system (w.f.w.s.) is the data, for each n ∈ IN, of a ring An, such that:

1) ∀n, On ⊂ An ⊂ IC[[z1, . . . , zn]] .

n p 2) If ϕ : (IC , 0) → (IC , 0) is a mapping with components in On and if f ∈ Ap, then f ◦ ϕ ∈ An.

3) For each n ∈ IN, the ring An is closed under division by coordinates, this means that, if f ∈ An and f = (zi − α)g, where g ∈ IC[[z]] and α ∈ IC; then g ∈ An.

4) Let σ : (ICp, 0) → (ICn, 0) be a holomorphic mapping. Suppose that rk(σ) = n. Let

f ∈ IC[[z1, . . . , zn]] such that f ◦ σ ∈ Ap; then f ∈ An.

Exemple 1 The system O = (On)n is a weak formal Weierstrass system. The property 4) of the last deﬁnition is the main result of [5].

The aim of this paper is to construct a w.f.w.s. A = (An)n, such that An ⊂ IC[[z]] and An contains strictly On. Before to that, let us give some properties of a w.f.w.s.

2 Generic division theorem

Th´eor`eme 1 Let A = (An)n be a w.f.w.s. and let P (zn, λ) be a generic polynomial in zn of degree p. If f ∈ An, then there exist unique element q(λ, z) ∈ An+p and rj ∈ An−1+p, 1 ≤ j ≤ p, such that p p−j f = P.q + X rjzn . j=1

Proof 0 Let f ∈ An; then by 2), f1 = f ◦ A ∈ An+p; besides, if z = (z1, . . . , zn−1):

0 0 f1(z , zn, ξ) − f1(z , ξ1, ξ) = (zn − ξ1)f2(z, ξ)

0 and f1(z , ξ1, ξ), f2 are in An+p by 2) and 4). By repeating the processes, we get at the end:

p 0 p−i f1(z, ξ) = g(z, ξ)(zn − ξ1) . . . (zn − ξp) + X gi(z , ξ)zn , i=1 with g ∈ An+p and gi ∈ An−1+p, 1 ≤ i ≤ p. We see that g and gi, 1 ≤ i ≤ p are symmetric with respect (ξ1, . . . , ξp). By the formal Newton’s theorem, there exist q ∈ IC[[z1, . . . , zn, ξ1, . . . , ξp]], ri ∈ IC[[z1, . . . , zn−1, ξ1, . . . , ξp]], 1 ≤ i ≤ p, such that:

g = q ◦ A and ri = gi ◦ A, 1 ≤ i ≤ p.

2 ∗ By assumption 4); q ∈ An+p and ri ∈ An−1+p, 1 ≤ i ≤ p. Since A is injective, we have proved the theorem.

Let h ∈ On − {0}; after making a linear change of coordinates on (z1, . . . , zn), we can ∂h ∂p−1h suppose that h is regular in zn of order p, i.e. h(0, 0) = (0, 0) = . . . = p−1 (0, 0) = 0 while ∂zn ∂zn ∂ph p (0, 0) =6 0. By the Weierstrass division theorem [8]; h = Q.P where Q ∈ On is unit and ∂zn 0 p p 0 p−j P (z , zn) = zn + Pj=1 aj(z )zn , aj ∈ On−1 and aj(0) = 0, 1 ≤ j ≤ p.

Corollaire 1 Let h ∈ On − {0} be regular in zn of order p. If f ∈ An, there are unique q˜ ∈ An and r˜j ∈ An−1, 1 ≤ j ≤ p, such that:

p 0 p−j f = h q˜+ X r˜j(z )zn . j=1

Proof. 0 p p 0 p−j We can suppose f = QP , where Q ∈ On is unit and P (z , zn) = zn + Pj=1 aj(z )zn , aj ∈ On−1, aj(0) = 0, 1 ≤ j ≤ p. By theorem 1, we make division of f by a generic polynomial 0 0 0 in zn of degree p. If we replace a(z ) := (a1(z ), . . . , ap(z )) in λ = (λ1, . . . , λp) and we put −1 0 0 0 0 q˜ = Q q(a(z ), z), r˜j(z ) = rj(z , a(z )), 1 ≤ j ≤ p; the corollary follows.

3 algebraic properties

As consequence of the last corollary, we deduce some ﬂatness properties. For each n ∈ IN, we I put M = C[[z]] . n An

Proposition 1 For each n ∈ IN, Mn is a ﬂat On-module.

Proof. On On By [7 ], we have to show that if I ⊂ On is an ideal, then T or1 ( I , Mn) = 0. From the exact sequence:

0 → An → IC[[z]] → Mn → 0 we deduce the longer exact sequence of ”Tor”.

On On On On An IC[[z]] On . . . → T or1 ( , IC[[z]]) → T or1 ( , Mn) → → → Mn ⊗On → 0. I I IAn IIC[[z]] I

On On Since IC[[z]] is a ﬂat On − module; T or1 ( I , IC[[z]]) = 0; hence, we have the exact sequence:

On On An IC[[z]] 0 → T or1 ( , Mn) → → . I IAn IIC[[z]]

We see then Mn is a ﬂat On-module if and only if, the last arrow is injective i.e IIC[[z]] ∩ An =

IAn. We will prove this by induction on n. Suppose n = 1; then the ideal I is principal and the result holds by property 3) of deﬁnition 1. We suppose n > 1 and the result holds for n−1. Let

3 (h1, . . . , hq) be a system of generators of I. We can suppose h1 is a distinguished polynomial in zn of degree p. By dividing (h2, . . . , hq) with h1, we can suppose that h2, . . . , hq are polynomials q in zn of degree < p. Let f = Pi=1 hifi, fi ∈ IC[[z]], 1 ≤ i ≤ q, and f ∈ An. After making a Weierstrass division of f2, . . . , fq and f by h1, we can suppose that f, f2, . . . , fq are polynomials in zn of degree < p ( f has its coeﬃcients in An−1). We deduce then that f1 is also a polynomial of degree < p − 1. Let us identify the subset of On of elements in On−1[zn] of degree less or 2p−1 equal to 2p−1 with (On−1) . In this identiﬁcation we can replace the ideal I by a submodule 2p−1 N ⊂ (On−1) . In order to prove the assumption for n, we have to show:

0 2p−1 0 NIC[[z ]] ∩ An−1 = NAn−1, z = (z1, . . . , zn−1), but this equality follows by the inductive hypothesis, hence the proposition.

In the following section we will give an example of w.f.w.s. that we are interested.

4 Formal series of class M

4.1 notations and deﬁnitions

∞ µ(n) Fix a sequence (mn)n=0 with mn = e , where µ is a nonnegative, increasing, convex function µ(t) deﬁned in {t ∈ IR / t ≥ 0 }, µ(0) = 0, t → ∞ as t → ∞. Since µ is convex; for each t ≥ 0 0 0 the derivative of µ at the right of t exists, say µd(t), and the function t → µd(t) is increasing. 0 We suppose that there exists a > 0 such that µd(t) ≤ at. We put Mn = n!mn. The sequence M = (Mn)n∈IN will be called the class M.

4.2 formal series of class M

ω1 ωn If ϕ ∈ IC[[z1, . . . , zn]], ϕ = Pω∈IN n ϕωz1 . . . zn , and C > 0; we put:

| ϕω | kϕkM,C = Supm Sup|ω|=m m ∈ [0, ∞]. C Mm

Deﬁnition 2 A formal power series ϕ ∈ IC[[z1, . . . , zn]] is said to be in the class M, if there exists a constant C > 0 such that kϕkM,C < ∞.

In the following, IC[[z]]M,n, z = (z1, . . . , zn), denotes the collection of all ϕ ∈ IC[[z]] in the class M.

µ˜(t) Remarque 1 If a, b ∈ IR, let µ˜(t) = µ(t) + at + b and put m˜n = e . We can then deﬁne an ˜ other class M. We can easily see that IC[[z]]M,n = IC[[z]]M˜ ,n; hence the class does not change when µ is replaced by µ˜.

4 Since the function t → µ(t) is convex, we can prove :

MjMn−j ≤ Mn, for 0 ≤ j ≤ n.

Using this inequality, it is easy to show that IC[[z]]M,n is a ring for each n ∈ IN.

p Since Mp ≥ p ; IC[[z]]M,n contains On, ∀n ∈ IN.

4.3 One dimensional characterization of formal series of class M

If ϕ ∈ IC[[z]]; we put Hj(ϕ) ∈ IC[z1, . . . , zn] the homogeneous polynomial of degree j in the ∞ n−1 n expansion of ϕ; we have then ϕ = Pj=0 Hj(ϕ)(z). Let Ω be an open set of the sphere S ⊂ IC ; ∞ j if ξ ∈ Ω; ϕ|ξ denotes the formal series Pj=1 Hj(ϕ)(ξ)t ∈ IC[[t]] called the restriction of ϕ to the line t → ξt. For each ξ ∈ Ω and m ∈ IN; we put: | H (ϕ)(ξ) | θ (ξ) = m , m M(m) and

θ(ξ) = Supmθm(ξ). Proposition 2 Let Ω be a nonempty open subset of Sn−1 and ϕ ∈ IC[[z]]; we assume that for each ξ ∈ Ω, there is a constant Cξ > 0 such that θ(ξ) ≤ Cξ; then ϕ ∈ IC[[z]]M . Proof

The function θ is lower semicontinuous; by Baire’s theorem, there exists Ω1 ⊂ Ω an open subset, Ω1 =6 ∅, and a constant C1 > 0, such that:

∀ξ ∈ Ω1, θ(ξ) ≤ C1.

By a result in [6]; there is a constant C2,Ω1 , such that:

m Sup|ξ|=1 | Hm(ϕ)(ξ) |≤ C2,Ω1 Supξ∈Ω1 | Hm(ϕ)(ξ) |, ∀m ∈ IN.

In view of Bernstein inequality; there exists a constant C3, such that:

m C3 | ϕω |≤ Sup|ξ|=1 | Hm(ϕ)(ξ) |, ∀m ∈ IN, where ω ∈ IN n, | ω |= m. C We put ρ = 2,Ω1 ; we have then kϕk < ∞, hence the result. C3 M,ρ ∞ j µ ∞ ν Lemme 1 Let ϕ = Pj=0 ϕjt ∈ IC[[t]] and µ ∈ IN. We put h = ϕ ; then h = Pν=0 hνt , where:

µ! k1 kl hν = X ϕµ1 . . . ϕµl , k1! . . . kl! l and µ = k1 + . . . + kl. The sum is taken over all sets {µ1, . . . , µl} of distinct elements in IN l and (k1, . . . , kl) ∈ (IN − {0}) , l = 1, 2, 3, . . . such that k1µ1 + . . . + klµl = ν.

5 Proof. The lemma is an easy consequence of the following: if a1, . . . , aq ∈ IR and ν ∈ IN; then

ν ν! k1 kq (a1 + . . . + aq) = X a1 . . . aq , k1! . . . kq!

q the sum is taken over all (k1, . . . , kq) ∈ IN such that k1 + . . . + kq = ν.

p ∞ ν Remarque 2 1) Let ϕ = (ϕ1, . . . , ϕp) ∈ IC[[t]] , ϕj = Pν=0 ϕj,νt , 1 ≤ j ≤ p; and for p p each ν ∈ IN, we put ϕν = (ϕ1,ν, . . . , ϕp,ν) ∈ IC . Let α = (α1, . . . , αp) ∈ IN and put α α1 αp ν h = ϕ = ϕ1 . . . ϕp . As in the previous lemma, we have h = Pν hνt , where:

α! k1 kl hν = X ϕµ1 . . . ϕµl , k1! . . . kl!

α = k1 + . . . + kl and the sum is taken over all sets {µ1, . . . , µl} of distinct elements in p l IN and (k1, . . . , kl) ∈ (IN − {0}) , l = 1, 2, 3, . . . such that µ1 | k1 | + . . . + µl | kl |= ν.

Mp µ(p) 2) Since the sequence p! = e is logarithmically convex we have, by [3], :

Ms Mµ1 Mµ M1 Mν ( )t1 . . . ( l )tl ≤ ( )s , s! µ1! µl! 1! ν!

for every t1, . . . , tl ∈ IN and µ1, . . . , µl ∈ IN, such that µ1t1 +. . . µltl = ν and k1 +. . .+kl = s. p If k1, . . . , kl ∈ IN , µ1, . . . , µl ∈ IN such that µ1 | k1 | + . . . µl | kl |= ν and k1+. . .+kl = α; then M|α| Mµ1 Mµ M1 Mν ( )|k1| . . . ( l )|kl| ≤ ( )|α| | α |! µ1! µl! 1! ν! .

p 1 ν 3) Let a > 0, put g(z , . . . , zp) = , ψ (t) = . . . = ψp(t) = t and ψ(t) = 1 Qj=1 (1−azj ) 1 Pν>0 (ψ1(t), . . . , ψp(t)). The function g(ψ(t)) is holomorphic in a neighborhood of the origin. ν We have g(ψ(t)) = Pν qνt , where, by 1),

α! |α| qν = X a , k1! . . . kl! the sum is taken as in 1).

∞ ν Proposition 3 Let f ∈ IC[[z]]M,p and suppose that ϕj = Pν=0 ϕj,νt inIC[[t]]M,1, ϕj,0 = 0, 1 ≤ j ≤ p; then f(ϕ1, . . . , ϕp) ∈ IC[[t]]M,1.

6 Proof µ1 µp ∞ ν Put f = Pµ∈IN p fµz1 . . . zp ; then f(ϕ1, . . . , ϕp) = Pν=1 hνt , where :

α! k1 kl hν = X ϕµ1 . . . ϕµl , k1! . . . kl! and the sum as in 1) of remark 2. There are constants c, ρ > 0 such that:

|α| p | fα |≤ ρc M|α|, ∀α ∈ IN ,

ν | ϕj,ν |≤ ρc Mν, ∀ν ∈ IN, ∀j = 1, . . . , p. We have then:

ν α! |α| |k1| |kl| M|α| Mµ1 |k1| Mµl |kl| | hν |≤ ρc X (cρ) (µ1!) . . . (µl!) (| α |!) ( ) . . . ( ) . k1! . . . kl! | α |! µ1! µl!

|k | |k | (µ1!) 1 ...(µl!) l (|α|!) By 2) of remark 2 and using the trivial inequality ν! ≤ 1, we have:

ν α! |α| | hν |≤ ρc Mν X (cM1ρ) . k1! . . . kl! By 3) of remark 2, if we put α! |α| qν = X (cM1ρ) , k1! . . . kl! ν ν there are c1 > 0, and ρ1 > 0, such that,| qν |≤ c1ρ1; hence, | hν |≤ ρρ1(cc1) Mν, which proves the lemma.

n p Proposition 4 Let ϕ : (IC , 0) → (IC , 0) be a holomorphic mapping and let f ∈ IC[[y1, . . . , yp]]M,p; then ψ := f(ϕ) ∈ IC[[z1, . . . , zn]]M,n. Proof

By lemma 1, we have to prove that the restriction of ψ to the line t → ξt is in IC[[t]]M,1, for n−1 n−1 each ξ ∈ S . Let ξ ∈ S , since ϕ|ξ = (ϕ1|ξ, . . . , ϕn|ξ) ∈ IC[[z]]M,1; by proposition 3, we have the result.

Proposition 5 Let f ∈ IC[[z]]M,n and α ∈ IC; suppose that f = (zi − α)g(z), where g ∈ IC[[z]]; then g ∈ IC[[z]]M,n. proof ω µ n Put f = Pω∈IN n fωz and g = Pµ∈IN n gµz . Suppose α = 0. For each µ = (µ1, . . . , µn) ∈ IN such that gµ =6 0; we have gµ = fωµ , where ωµ = (µ1 . . . , µi−1, µi + 1, µi+1, . . . , µn); hence |µ|+1 | gµ |≤ Cρ M|µ|+1. By (*) of paragraph 6, there exists a constant c1 > 0 such that |µ|+1 M|µ|+1 ≤ c1 M|µ|; hence the result. Now if α =6 0, then the function 1 is holomorphic in a neighborhood of the origin in IC, hence zi−α 1 ∈ O ⊂ On, and the result follows, since IC[[z]]M,n is a ring containing On. zi−α 1

7 Proposition 6 IC[[t]]M,1 is a local algebra; its maximal ideal is

mM,1 = {f ∈ IC[[z]]M,1; f(0) = 0}. Proof −1 Let ϕ ∈ IC[[t]]M,1 with a non-zero constant term ϕ0. Considering ϕ0 ϕ instead of ϕ, we can suppose ϕ0 = 1. We put ϕ = 1 − ψ; ψ ∈ IC[[t]]M,1 with constant term equal to zero. ∞ n Let h(τ) = Pn=0 τ ∈ IC{τ} (the ring of convergent power series). Since IC{τ} ⊂ IC[[t]]M,1; by proposition 2, we have h(ψ) ∈ IC[[t]]M,1. We see that ϕh(ψ) = 1, so ϕ has an inverse in IC[[t]]M,1; hence the proposition. The following corollary was announced in [2] paragraph 2. The authors use a result of E.M. Dynkin [4] for functions. But for a general class we have not a version of

Borel extension theorem for the ring IC[[z]]M,n (for example if the class is quasianalytic, that is the series Mn is not convergent). We give here,in the following corollary, a direct proof of Pn mn+1 this fact.

Corollaire 2 IC[[z]]M,n is a local algebra; its maximal ideal is

mM,n = {f ∈ IC[[z]]M,n; f(0) = 0}.

Proof

Let ϕ ∈ IC[[z]]M,n with a non-zero constant term; then ϕ admits an inverse ψ ∈ IC[[z]]. We will n−1 show that ψ ∈ C[[z]]M,n. Let ξ ∈ S , then the restriction of ψ to the line t → ξt, ψ|ξ, is the n−1 inverse of ϕ|ξ ∈ IC[[t]]M,1; by the last proposition, ψ|ξ ∈ IC[[t]]M,1, for all ξ ∈ S ; hence the corollary, by proposition 2.

By proposition 5, the maximal ideal, mM,n, is generated by (z1, . . . , zn). The proof of the following theorem is the same as the proof given in [5] for convergent power series instead of formal series of class M. For completeness we will outline the proof in the last section. We put y = (y1, . . . , yp). Th´eor`eme 2 Let ϕ : (ICp, 0) → (ICn, 0) be a holomorphic mapping with rk(ϕ) = n. For each f ∈ IC[[z]] such that f(ϕ) ∈ IC[[y]]M,p, we have f ∈ IC[[z]]M,n.

On the whole, we have proved that, if M is a class as in 4.1; the system (IC[[z]]M,n)n satisﬁes the properties 1), 2), 3) and 4) of deﬁnition 1, hence (IC[[z]]M,n)n is a w.f.w.s.

5 Formal Weierstrass system

Deﬁnition 3 Let B = (Bn)n be a w.f.w.s.; we said that B is a formal Weierstrass system (f.w.s.), if the following conditions are satisﬁed:

1) ∀n ∈ IN, Bn is a local ring with maximal ideal denoted by mn generated by (z1, . . . , zn).

2) Put y = (y1, . . . , yp) and let f1(z, y), . . . , fp(z, y) ∈ Bn+p such that f1(0, 0) = . . . = f (0, 0) = 0 and the jacobien D(f1,...,fp) (0, 0) =6 0; then there are ϕ , . . . , ϕ ∈ B , p D(y1,...,yp) 1 p n ϕ1(0) = . . . = ϕp(0) = 0 such that f(z1, ϕ1(z)) = . . . = fp(z, ϕ(z)) = 0.

8 Proposition 7 Let B = (Bn)n be a formal Weierstrass system; then, for each n ∈ IN, Bn is a local regular ring of dimension n. Besides, the Weierstrass division theorem holds in the system

B = (Bn)n, n ∈ IN.

We can deduct then the following:

I Corollaire 3 Let B = (B ) be a formal Weierstrass system; then C[[z1,...,zn]] and IC[[z , . . . , z ]] n n Bn 1 n are ﬂat modules over On, ∀n ∈ IN.

As in the analytic case, the Artin’s theorem [1] is also true in this situation, more precisely:

Let f1, . . . , fq ∈ Bn+p and consider the system of implicit equations:

f1(z, y) = . . . = fq(z, y) = 0, with f1(0, 0) = . . . = fq(0, 0) = 0. p Let ψ(z) = (ψ1(z), . . . , ψp(z)) ∈ (IC[[z]]) , ψ(0) = 0, be a formal solution of this system. Then p ν p for each ν ∈ IN, there exists a solution ψν ∈ (Bn) , ψν(0) = 0, such that ψ − ψν ∈ mn(IC[[z]]) .

Proof of proposition 7

Let f ∈ Bn, we suppose that f is regular of order p with respect zn. By theorem 1 we can make p p p−j division of f by the generic polynomial P (zn, λ) = zn + Pj=1 λjzn :

p 0 p−j f = qP + X rj(z , λ)zn , j=1 where q ∈ Bn+p; rj ∈ Bn−1+p, 1 ≤ j ≤ p. Since f is regular of order p with respect zn, we can easily see that:

q(0, 0) = 0; rj(0, 0) = 0, 1 ≤ j ≤ p, and D(r , . . . , r ) 1 p (0, 0) =6 0. D(λ1, . . . , λp) 0 0 By condition 2) in deﬁnition 4, there are ψ1(z ), . . . , ψp(z ) ∈ Bn−1, ψj(0) = 0, 1 ≤ j ≤ p, such that: 0 0 f = q(z, ψ(z ))P (zn, ψ(z )),

0 0 0 where ψ(z ) = (ψ1(z ), . . . , ψp(z )). We see then that f is equivalent, in Bn, to the distinguished p p 0 p−j polynomial : zn + Pj=1 ψj(z )zn ∈ Bn−1[zn]. Let h ∈ Bn; we can make division of h by the generic polynomial P (zn, λ), and hence by 0 0 P (zn, ψ(z )), so by f, after the substitution λ → ψ(z ):

p 0 p−j h = fQ + X hj(z )zn , j=1

9 with Q ∈ Bn; hj ∈ Bn − 1, ,1 ≤ j ≤ p, and this decomposition is unique. Since the Weierstrass theorem is true in Bn, ∀n; we deduce that Bn is a noetherian ring for all n ∈ IN. We have the inclusions:

On ⊂ Bn ⊂ IC[[z1, . . . , zn]], which implies Bˆn = IC[[z1, . . . , zn]]; Bˆn is the completion of Bn with respect the topology deﬁned by the maximal ideal mn. Since the completion of the local noetherian ring Bn is the ring of formal series C[[z1, . . . , zn]], we deduce that Bn is a regular ring of dimension n; and the proposition is proved.

Th´eor`eme 3 Let M be a class as in 4.1, the weak formal Weierstrass system (IC[[z]]M,n)n is a formal Weierstrass system.

Proof

By corollary 2, condition 1) of definition is satisfied. By remark 1, we can suppose M1 = 1. Since µ is convex, we have, for all n ∈ IN, nµ(n − 1) ≤ (n − 1)µ(n). Applying this repeatedly, we get: (p − 1)µ(q) ≤ C(q − 1)µ(p − 1), ∀p ≥ q ≥ 2 where C is a constant. Hence the class satisfies the following:

Mq 1 Mp 1 ( ) q−1 ≤ C( ) p−1 , 2 ≤ q ≤ p. q! p!

By a result of [3], the implicit function theorem holds in the ring IC[[z]]M,n; hence the theorem.

6 Outline of the proof of theorem 2

In the following, all the considered morphisms between rings of formal series are induced by holomorphic functions. We keep the notations of theorem 2 and put y = (y1, . . . , yp), z = ∗ (z1, . . . , zn); let ϕ : IC[[z]] → IC[[y]] be the homomorphism induced by ϕ. Recall that Mn = µ(n) n µ(n) n!e where µ is as in 4.1. Let M˜ = (M˜ n)n where M˜ n = n e . We can easily see that n µ(n) IC[[z]]M˜ ,n = IC[[z]]M,n, so we suppose that Mn = n e . We put, for t ≥ 1, m(t) = t log t + µ(t) − µ(1) and m(t) = 0 for 0 ≤ t ≤ 1. We see that m is convex and there exists b > 0 such 0 that md(t) ≤ bt + 1, ∀t ≥ 0. For all p ∈ IN and j ∈ IN, we have:

m(p + j) − m(j) ≤ j(b(p + j) + 1).

jb j We put Aj = e and ρj = e ; then we have:

(p+j) (∗) Mp+j ≤ ρjAj Mj.

10 Deﬁnition 4 We say that ϕ is strongly M-injective, if for each f ∈ IC[[z]] such that ϕ(f) ∈

IC[[y]]M,p; then f ∈ IC[[z]]M,n.

n n Exemples 1 1) Let ϕ : (IC , 0) → (IC , 0) be a mapping such that ϕ(z1, . . . , zn) = (z1, . . . , zi−1, zizj, zi+1 . . . , zn); then ϕ is strongly M-injective. ω ∗ n Indeed, if f = Pω fωz ∈ IC[[z]] such that ϕ (f) ∈ IC[[y]]M,p; then, for each ω ∈ IN , |ω|+ωj | fω |≤ cρ M|ω|+ωj . So the result by inequality (*). ,

q 2) Suppose that ϕ(z1, . . . , zn) = (z1, . . . , zi−1, zi , zi+1 . . . , zn), q ∈ IN. We can easily see that ϕ is strongly M-injective.

∗ ∗ 3) Put w = (w1, . . . , ws); if ϕ : IC[[z]] → IC[[y]] and ψ : IC[[y]] → IC[[w]] are homomorphisms such that ϕ∗ and ψ∗ are strongly M-injective; then ψ∗ ◦ ϕ∗ is strongly M-injective. Proof of the theorem 2 The proof uses an algorithm, introduced in [5], which consists of modifying ϕ∗ by a ﬁnite number of steps. Each step preserves the rank and it is strongly M-injective.

References

[1] M.Artin. On the solutions of analytic equations. Invent.Math. 5, 277-291 (1968).

[2] J. Chaumat et A.M. Chollet. Caractérisation des anneaux noethériens de séries formelles a` croissance controlée. Application a` la synthèse spectrale. Publications Mathématiques, Vol. 41 (1997), 545-561.

[3] C.L Childress, Weierstrass division in a quasianalytic local rings. Can. J. Math., Vol. XXVIII, N.5, 1976,pp.938-953.

[4] E. M. Dyn’kin, Pseudoanalytic extention of smooth functions, Amer.Math. Soc. Transl. (2), 115 (1980), pp. 33-58.

[5] P.M. Eakin and G.A. Harris. When φ(f) convergent implies f is convergent. Math.Ann. 229, 201-210 (1977).

[6] M. Klimek. Pluripotentiel theory. London Math. Soc. Monographs.

[7] D.G. Northcott. An introduction to homological algebra. Cambridge: Univ. Press 1960.

[8] J.-Cl. Tougeron. Ideaux de fonctions diﬀ´erentiables. Springer Verlag, Ergebnisse der Math- ematik (1971). abdelhafed elkhadiri faculty of fciences department of mathematics. b.p 133. kenitra,´ 14000, morocco. E.mail:[email protected]