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On Some Subring of Formal Power Series∗ On some subring of formal power series∗ Abdelhafed ELKHADIRIy Abstract For each n 2 IN, we denote by IC[[z1; : : : ; zn]] the ring of formal power series with complex coefficients and ICfz1; : : : ; zng the ring of convergent power series. For each n 2 IN, we consider An ⊂ IC[[z1; : : : ; zn]] a subalgebra containing ICfz1; : : : ; zng. In the first part we give some conditions under which An possesses good properties as module over ICfz1; : : : ; zng. 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) 2 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 yKey words: noetherian rings, Weierstrass division theorem, implicit function theorem. 1 defined by A∗( (z; ξ)) = (z; σ(ξ)) is injective. Definition 1 A weak formal Weierstrass system (w.f.w.s.) is the data, for each n 2 IN, of a ring An, such that: 1) 8n, On ⊂ An ⊂ IC[[z1; : : : ; zn]] . n p 2) If ' : (IC ; 0) ! (IC ; 0) is a mapping with components in On and if f 2 Ap, then f ◦ ' 2 An. 3) For each n 2 IN, the ring An is closed under division by coordinates, this means that, if f 2 An and f = (zi − α)g, where g 2 IC[[z]] and α 2 IC; then g 2 An. 4) Let σ : (ICp; 0) ! (ICn; 0) be a holomorphic mapping. Suppose that rk(σ) = n. Let f 2 IC[[z1; : : : ; zn]] such that f ◦ σ 2 Ap; then f 2 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 2 An, then there exist unique element q(λ, z) 2 An+p and rj 2 An−1+p, 1 ≤ j ≤ p, such that p p−j f = P:q + X rjzn : j=1 Proof 0 Let f 2 An; then by 2), f1 = f ◦ A 2 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 2 An+p and gi 2 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 2 IC[[z1; : : : ; zn; ξ1; : : : ; ξp]]; ri 2 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 2 An+p and ri 2 An−1+p; 1 ≤ i ≤ p. Since A is injective, we have proved the theorem. Let h 2 On − f0g; 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 2 On is unit and @zn 0 p p 0 p−j P (z ; zn) = zn + Pj=1 aj(z )zn ; aj 2 On−1 and aj(0) = 0; 1 ≤ j ≤ p: Corollaire 1 Let h 2 On − f0g be regular in zn of order p. If f 2 An, there are unique q~ 2 An and r~j 2 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 2 On is unit and P (z ; zn) = zn + Pj=1 aj(z )zn ; aj 2 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 2 IN, we I put M = C[[z]] . n An Proposition 1 For each n 2 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 2 IC[[z]]; 1 ≤ i ≤ q, and f 2 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 1 µ(n) Fix a sequence (mn)n=0 with mn = e , where µ is a nonnegative, increasing, convex function µ(t) defined in ft 2 IR = t ≥ 0 g, µ(0) = 0, t ! 1 as t ! 1. 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)n2IN will be called the class M. 4.2 formal series of class M !1 !n If ' 2 IC[[z1; : : : ; zn]], ' = P!2IN n '!z1 : : : zn , and C > 0; we put: j '! j k'kM;C = Supm Supj!j=m m 2 [0; 1]: C Mm Definition 2 A formal power series ' 2 IC[[z1; : : : ; zn]] is said to be in the class M, if there exists a constant C > 0 such that k'kM;C < 1.
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