aa r X i v : . [ m a t h . A C ] A p r ARTIN APPROXIMATION OVER BANACH SPACES
GUILLAUME ROND
Abstract.
We give examples showing that the usual Artin Approxima-tion theorems valid for convergent series over a field are no longer truefor convergent series over a commutative Banach algebra. In particularwe construct an example of a commutative integral Banach algebra R such that the ring of formal power series over R is not flat over the ringof convergent power series over R . Introduction
The classical Artin Approximation Theorem is the following:
Theorem 1.1. [1]
Let F ( x, y ) be a vector of convergent power series over C in two sets of variables x and y . Assume given a formal power seriessolution b y ( x ) vanishing at 0, F ( x, b y ( x )) = 0 . Then, for any c ∈ N , there exists a convergent power series solution y ( x ) vanishing at 0, F ( x, y ( x )) = 0 which coincides with b y ( x ) up to degree c , y ( x ) ≡ b y ( x ) modulo ( x ) c . The main tools for proving this theorem are the implicit function theoremand the Weierstrass division theorem. But in the case the equations F ( x, y )are linear in y , this theorem is equivalent to the faithful flatness of themorphism C { x } −→ C J x K (see [13, Example 1.4] for instance or [4, I. 3Proposition 13]). In fact the faithful flatness of this morphism comes fromthe fact that C { x } is a Noetherian local ring. And the Noetherianity of C { x } is usually proved by using the Weierstrass division theorem.Another version of this theorem is the following one: Theorem 1.2. [2][16]
Let F ( x, y ) be a vector of convergent power seriesover C in two sets of variables x and y . Then for any integer c there exists Mathematics Subject Classification.
Key words and phrases.
Banach algebra, flatness, Artin approximation.The author is deeply grateful to the UMI LASOL of the CNRS where this project hasbeen carried out. an integer β such that for any given approximate solution y ( x ) at order β , y (0) = 0 , F ( x, y ( x )) ≡ modulo ( x ) β , there exists a formal power series solution y ( x ) vanishing at 0, F ( x, y ( x )) = 0 which coincides with y ( x ) up to degree c , y ( x ) ≡ y ( x ) modulo ( x ) c . In particular this result implies that, if F ( x, y ) = 0 has approximate so-lutions at any order, then it has a formal (even convergent by Theorem 1.1)power series solution.Let us mention that these results remain valid when we replace C by a com-plete valued field, or when we replace the ring of convergent power seriesover C by the ring of algebraic power series over a field. In fact these resultsremain true in the more general setting of excellent Henselian local rings by[12] (see [13] for a review of all these different results).The aim of this note is to show that these results are no longer true when wereplace C by a commutative Banach algebra over R or C . In the first partwe construct a commutative Banach algebra R such that R { t } −→ R J t K isnot flat, showing that Artin approximation theorem is not true for linearequations with coefficients in R { t } .Let us mention here that R J t K is flat over R , for a commutative ring R , ifand only if R is coherent (indeed R J t K is a direct product of copies of R - see [5, Theorem 2.1]). And there are several known examples of Banachalgebras which are not coherent (in fact most of the known Banach algebrasare not coherent; see for instance [9] or [8] and the references herein). Butthe flatness of R { t } −→ R J t K is a different property that is not related tothe coherence of R .In the second part we provide an example of one polynomial F ( y ) with co-efficients in R [ t ], where R is the Banach algebra of holomorphic functionsover a disc, with the following property: F ( y ) has approximate solutions upto any order but has no solution in R J t K . This shows that Theorem 1.2 doesnot hold for convergent power series over a Banach algebra. Let us mentionthat this example is a slight modification of an example of Spivakovsky re-lated to a similar problem [15].Nevertheless we mention that in the case where R is a complete valua-tion ring of rank one (in particular a non-archimedean Banach algebra),Schoutens and Moret-Bailly proved several extensions of Theorems 1.1 and1.2 (see [14] and [11]).The note has been motivated by questions from Nefton Pali and Wei Xia. RTIN APPROXIMATION OVER BANACH SPACES 3 A Banach algebra R such that R { t } −→ R J t K is not flat Let K = R or C . We begin by the following definition of power series incountable many indeterminates: Definition 2.1.
Let N ( N ) be the submonoid of N N formed by the sequenceswhose all but finitely terms are 0. Let ( x i ) i ∈ N be a countable family ofindeterminates. Then K J x i K i ∈ N is the set of series P α ∈ N ( N ) a α x α where x α = x α · · · x α n n · · · . This former product is finite since α i = 0 for i large enough.This set is a commutative ring since the sum of sequences N ( N ) × N ( N ) −→ N ( N ) has finite fibers (see [3, Chapter III, §
2, 11]). Let us mention that thisring is not the ( x )-adic completion of K [ x ], the ring of polynomials in the x i (see [17] for instance).Let x , y , z and w k for k ∈ N be indeterminates. For simplicity we denoteby w the vector of indeterminates ( w , w , . . . ). We denote by K [ x, y, z, w ]the ring of polynomials in the indeterminates x , y , z , w .For a polynomial p = X k ∈ N ,l ∈ N ,m ∈ N ,α ∈ N ( N ) a k,l,m,α x k y l z m w α ∈ K [ x, y, z, w ] weset k p k := X k,l,m,α | a k,l,m,α | . This is well defined because the sum is finite. This defines a norm on K [ x, y, z, w ].We denote by K { x, y, z, w } the completion of K [ x, y, z, w ] for this norm.This is the following commutative Banach algebra: X k ∈ N ,l ∈ N ,m ∈ N ,α ∈ N ( N ) a k,l,m,α x k y l z m w α | X k ∈ N ,l ∈ N ,m ∈ N ,α ∈ N ( N ) | a k,l,m,α | < ∞ and the norm of an element f := X k ∈ N ,l ∈ N ,m ∈ N α ∈ N ( N ) a k,l,m,α x k y l z m w α is k f k := X k ∈ N ,l ∈ N ,m ∈ N ,α ∈ N ( N ) | a k,l,m,α | . In particular K { x, y, z, w } is a subring of K J x, y, z, w i K i ∈ N .We denote by I the ideal of K [ x, y, z, w ] generated by the polynomials xw − z and yw k − ( k + 1) xw k +1 for all k ≥ . The ideal I K { x, y, z, w } is not closed since it is not finitely generated. Thus,we denote by I its closure. This is the set of sums X k ∈ N f k ( x, y, z, w )such that f k ( x, y, z, w ) ∈ I K { x, y, z, w } and P k k f k ( x, y, z, w ) k < ∞ . GUILLAUME ROND
Definition 2.2.
We denote by R the Banach K -algebra K { x, y, z, w } /I .In order to denote that two series f and g ∈ K { x, y, z, w } have the sameimage in R , we write f ≡ R g . The norm of the image f of an element f ∈ K { x, y, z, w } is k f k = inf g ∈ I k f + g k = inf g ∈ I k f + g k . Now we denote by R { t } the ring of convergent series in the indeterminate t with coefficients in R . We have the following result: Proposition 2.3.
The linear equation (2.1) ( x − yt ) f ( t ) = z has a unique solution f ( t ) in R J t K and this solution is not convergent. From this we will deduce the following result:
Theorem 2.4.
The Banach K -algebra R is an integral domain and themorphism R { t } −→ R J t K is not flat. Proofs of Proposition 2.3 and Theorem 2.4.
We begin by givingthe following key result:
Lemma 2.5. x is not a zero divisor in R .Proof. First of all, we will determine a subset of K { x, y, z, w } such that everyelement of K { x, y, z, w } is equal modulo I to a unique series of this subset.First we remark that(2.2) M := yw i w j ≡ R ( i + 1) xw i +1 w j ≡ R i + 1 j yw i +1 w j − =: M for all integers i and j with i < j . If j = i +1 these two monomials are equal,otherwise the largest index of a monomial w j appearing in the expression of M is strictly less than for M .Now we have, for i > z w i ≡ R xw w i ≡ R i yw w i − . A well chosen composition of these operations transforms any monomial ofthe form Cx a z k y l w n · · · w n i i into a monomial of the form rCx a ′ z k ′ y l w n ′ · · · w n ′ j j where j is minimal and r ∈ (0 , z ε x a y l w n with a > , l, n ≥ ε ∈ { , } ,z ε y l w n i i with l > , i > , n i > ε ∈ { , } ,z ε y l w n i i w n i +1 i +1 with l > , i ≥ , n i , n i +1 > ε ∈ { , } ,z ε w n . . . w n i i with n i > ε ∈ { , } , We denote by E the subset of K [ x, y, z, w ] of polynomials that are sums ofmonomials of (2.4) (up to multiplicative constants), and by E the closure RTIN APPROXIMATION OVER BANACH SPACES 5 of E in K { x, y, z, w } , that is the set of convergent power series whose nonzero monomials are those of (2.4) (up to multiplicative constants). We haveshown that every polynomial is equivalent to a polynomial of E modulo I .To prove the unicity we proceed as follows.We set F := xw − z , F k +1 := yw k − ( k + 1) xw k +1 for k ≥ G k,l := ( l + 1) yw k w l +1 − ( k + 1) yw l w k +1 for all k < l. Then we consider the following monomial order: We define x a y k z l w α · · · w α n n > x a ′ y k ′ z l ′ w α ′ · · · w α ′ n n if a + k + l + X i α i > a ′ + k ′ + l ′ + X i α ′ i , or a + k + l + X i α i = a ′ + k ′ + l ′ + X i α ′ i and ( l, a, k, α n , . . . , α ) > lex ( l ′ , a ′ , k ′ , α ′ n , . . . , α ′ )where > lex denotes the lexicographic order. That is, we first compare thetotal degree of two monomials, then we order the indeterminates as z > x > y > w l > w k for all l > k. We claim that { F j , G k,l } j,k,l ∈ N , l>k is a Gr¨obner basis of I for this order.In order to prove this, we only need to compute the S-polynomials of theelements of this set of polynomials, and then their reduction (see [6] for theterminology). This is Buchberger’s Algorithm which is very classical in theNoetherian case. The case of polynomial rings in countably many indeter-minates works identically, cf. [7, Proposition 1.13] for instance. The onlyS-polynomials we have to consider are those of polynomials whose leadingterms are not coprime, that is, for l > k , S ( F k +1 , F l +1 ); S ( G k,l , F l +1 ); S ( G k,l , F k ) . We have S ( F k +1 , F l +1 ) = G k,l . Moreover S ( G k,l , F l +1 ) = y ( yw k w l − ( k + 1) xw l w k +1 ) . This leading term of S ( G k,l , F l +1 ) is − ( k + 1) xyw l w k +1 , and it is equal to y ( F k +1 w l − yw k w l ). Therefore S ( G k,l , F l +1 ) = F k +1 yw l . Finally we have S ( G k,l , F k ) = kx (( l +1) yw k w l +1 − ( k +1) yw l w k +1 )+( l +1) yw l +1 ( yw k − − kxw k )= ( l + 1) y w k − w l +1 − k ( k + 1) xyw l w k +1 . Its leading term is − k ( k + 1) xyw l w k +1 and it is divisible by the leading termof F k +1 . The remainder of the division of S ( G k,l , F k ) by F k +1 is( l + 1) y w k − w l +1 − ky w k w l = yG k − ,l GUILLAUME ROND
Therefore the reductions of these S-polynomials is always zero, hence thefamily { F j , G k,l } j,k,l ∈ N , l>k is a Gr¨obner basis of I . Thus, the initial ideal of I is generated by the monomials z , xw k +1 , yw k w l +1 for 0 ≤ k < l. Therefore every polynomial of K [ x, y, z, w ] is equivalent modulo I to a uniquepolynomial of E .Now let f ∈ K { x, y, z, w } . We can write f = P n ∈ N C n x a n y b n z c n w α n wherethe C n are in K ∗ . In particular P n | C n | < ∞ . For every n ∈ N , there is aunique ( a ′ n , b ′ n , c ′ n , α ′ n ) and a unique r n ∈ (0 ,
1] such that C n x a n y b n z c n w α n − r n C n x a ′ n y b ′ n z c ′ n w α ′ n ∈ I and x a n y b n z c n w α n has one the forms given in (2.4). Now, for every n ∈ N ,we set g n := n − X k =0 r k C k x a ′ k y b ′ k z c ′ k w α ′ k + X k ≥ n C k x a k y b k z c k w α k . In particular we have that P n := f − g n ∈ I and the sequence ( g n ) n convergesin K { x, y, z, w } to the series g = P k ∈ N r k C k x a ′ k y b ′ k z c ′ k w α ′ k ∈ K { x, y, z, w } .Therefore the sequence ( P n ) n converges in K { x, y, z, w } , and its limits is in I .Therefore, every power series of K { x, y, z, w } can be written as a sum of apower series in I and a convergent power series whose monomials are as in(2.4) (up to multiplicative constants).We remark that, by repeating (2.2) ⌊ j − i ⌋ times, we have yw i w j ≡ R ryw i + ⌊ j − i ⌋ w j −⌊ j − i ⌋ for some constant r ∈ (0 , z of a monomial. Therefore, a monomial of the form Cx a y b z c w α · · · w α j j of total degree d = a + b + c + P k α k , is not equal to a monomial involvingonly the indeterminates x, y, z, and w i for i < j − c d . Moreover (2.2) and (2.3) transforms monomials into monomials of the samedegree since I is generated by homogeneous binomials. Therefore, given amonomial M among those of (2.4) (up to some multiplicative constant),there is finitely many monomials that are equal to M modulo I .Now let f ∈ E ∩ I , f = P ( a,b,c,α ) f ( a,b,c,α ) x a y b z c w α . Let us fix such ( a, b, c, α )such that x a y b z c w α is one of the monic monomials of (2.4). There is onlya finite number of distinct monomials that are equal to f ( a,b,c,α ) x a y b z c w α modulo I . Let us denote them by C x a y b z c w α , . . . , C N x a N y b N z c N w α N . RTIN APPROXIMATION OVER BANACH SPACES 7
We can remark that there is only a finite number of F l that have a monomialthat divides at least one of the following monic monomials(2.5) x a y b z c w α , x a y b z c w α , . . . , x a N y b N z c N w α N . We denote them by F l , . . . , F l p . Because f ∈ I , we can write f = P l ∈ N f l F l where the f l are in K J x, y, z, w K . For every i ∈ { , . . . , p } we remove from f l i all the monomials that do not divide one of the monomials (2.5), and wedenote by f ′ l i the resulting polynomial. Then we have that P := p X i =1 f ′ l i F l i ∈ I. By construction the coefficients of the monomials (2.5) in the expansion of P are the corresponding coefficients in the expansion of f , that is f ( a,b,c,α ) , , . . . , x a y b z c w α in the expansion of theunique Q ∈ E such that Q ≡ R P , is equal to f ( a,b,c,α ) because no othermonomial than those listed in (2.5) (up to some multilplicative constants) isequivalent to a monomial of the form Cx a y b z c w α where C ∈ K ∗ . But Q = 0since P ∈ I , thus f ( a,b,c,α ) = 0. Hence f = 0 and E ∩ I = 0.Therefore every series of K { x, y, z, w } is equivalent modulo I to a uniqueseries of E .Now take f ∈ K { x, y, z, w } such that x ≡ R
0. We can write f = xp ( x, y, z, w )+ q ( y, z, w ) and assume that the monomials in the expansion of xp ( x, y, z, w )+ q ( y, z, w ) are only those of (2.4). Then x p ( x, y, z, w ) + xq ( y, z, w ) ≡ R . The representation of x p ( x, y, z, w ) + xq ( y, z, w ) as a sum of monomials asin (2.4) has the form(2.6) x p ( x, y, z, w ) + xq ( y, z, w ,
0) + q ( y, z, w ) = 0where q ( y, z, w ) is the series obtained from xq ( y, z, w ) − xq ( y, z, w ,
0) byreplacing the monomials as follows (using the two previous operations (2.2)and (2.3)):(2.7) xz ε y l w n i i i z ε y l +1 w i − w n i − i , if i > xz ε y l w n i i w n i +1 i +1 i +1 z ε y l +1 w n i +1 i w n i +1 − i +1 , if i > xz ε w n . . . w n i i Cz ε yw m j j w m j +1 j +1 or Cz ε yw m j j for i > n i > C ∈ K , | C | ≤ , j ≥ xz ε w n . . . w n i i ≡ R i + 1 z ε yw n · · · w n i − +1 i − w n i − i and this monomial on the right side can be transformed into a monomial ofthe form Cz ε yw m j j w m i +1 j +1 or Cz ε yw m j j for some C ∈ K , | C | ≤
1, and j ≥ GUILLAUME ROND by using the two operations (2.2) and (2.3) on monomials.This shows that the three types of monomials that we obtain after multipli-cation by x are all distinct, that is the map defined by (2.7) is injective. By(2.6) we have q ( y, z, w ) = 0, therefore q ( y, z, w ) − q ( y, z, w ,
0) = 0.Moreover, again by (2.6), we have x p ( x, y, z, w ) + xq ( y, z, w ,
0) = 0 . This shows that x p ( x, y, z, w ) + xq ( y, z, w ) = 0. Therefore x is not a zerodivisor in R . (cid:3) Proof of Proposition 2.3.
Let f ( t ) ∈ R J t K such that( x − yt ) f ( t ) = z . By writing f = P ∞ k =0 f k t k with f k ∈ R for every k , we have xf = z xf k − yf k − = 0 ∀ k ≥ . Thus xf = z = xw so x ( f − w ) = 0 and f = w by Lemma 2.5. Then we will prove byinduction on k that f k = k ! w k for every k . Assume that this is true for aninteger k ≥
0. Then we have xf k +1 = yf k = k ! yw k = ( k + 1)! xw k +1 . Hence x ( f k +1 − ( k + 1)! w k +1 ) = 0 and f k +1 = ( k + 1)! w k +1 by Lemma 2.5.Therefore the only solution of( x − yt ) f ( t ) = z is the series P ∞ k =0 k ! w k t k , and this one is divergent because k w k k = 1. Thisholds because in every element of I , the monomial w k has coefficient 0. (cid:3) Now we can give the proof of Theorem 2.4:
Proof of Theorem 2.4.
Since x is not a zero divisor in R by Lemma 2.5, thelocalization morphism R −→ R /x is injective. But R /x is isomorphic to K { x, y, z } /x since in R /x we have w = z /x and ∀ k ≥ , w k = 1 k ! y k z x k +1 . But K { x, y, z } /x is an integral domain (this is a localization of the integraldomain K { x, y, z } ), therefore so is R .Now assume that the morphism R { t } −→ R J t K is flat. By [10, Theorem RTIN APPROXIMATION OVER BANACH SPACES 9 x − yt ) F − z G = 0, there exist aninteger s ≥
1, and convergent series a ( t ) , . . . , a s ( t ) , b ( t ) , . . . , b s ( t ) ∈ R { t } such that(2.8) ( x − yt ) a i ( t ) − z b i ( t ) = 0 for every i, and formal power series h ( t ) , . . . , h s ( t ) ∈ R J t K such that f ( t ) = s X i =1 a i ( t ) h i ( t ) , s X i =1 b i ( t ) h i ( t ) . Indeed the vector ( f ( t ) ,
1) is a solution of the linear equation( x − yt ) f ( t ) − z g ( t ) = 0with f ( t ) := P ∞ k =0 k ! w k t k .Then e g ( t ) := s X i =1 b i ( t ) h i (0) = 1 + tε ( t )for some ε ( t ) ∈ R { t } . Since 1 is a unit of R , 1 + tε ( t ) is a unit in R { t } .Set e f ( t ) := P i a i ( t ) h i (0). By (2.8), ( e f ( t ) , e g ( t )) is a solution of the equation( x − yt ) e f ( t ) − z e g ( t ) = 0 . Since e g ( t ) is a unit in R { t } we have( x − yt ) e f ( t ) e g ( t ) − = z . This contradicts Theorem 2.3. Therefore R { t } −→ R J t K is not flat. (cid:3) An Example concerning the strong Artin approximationtheorem
Let n be a positive integer, x = ( x , . . . , x n ) and ρ >
0. We set K = R or C . Then B nρ := ( f = X α ∈ N n a α x α | || f || ρ := X α ∈ N n | a α | ρ | α | < ∞ ) is a Banach space equipped with the norm || · || ρ . Of course K [ x ] ⊂ B nρ . Remark . We do not have B nρ J t K ∩ K { x, t } = B nρ { t } . For instance, the power series f = X k ∈ N x k !1 t k is a convergent power series in ( x, t ), belongs to B n J t K , but X k k x k !1 k τ k = X k k ! τ k = ∞ for every τ >
0. Therefore f / ∈ B n { t } .We provide two examples based on an example of Spivakovsky concerningthe extension of Theorem 1.2 to the nested case (see [15]). Example . Let n = 1 and set F ( x, t, y , y ) := xy − ( x + t ) y ∈ B ρ { t } [ y , y ] . Let √ t = 1 + X n ≥ a n t n ∈ Q { t } be the unique power series such that ( √ t ) = 1 + t and whose valueat the origin is 1. For every c ∈ N we set y ( c )2 ( t ) := x c and y ( c )1 ( t ) := x c + P cn =1 a n x c − n t n ∈ B ρ { t } . Then F ( x, t, y ( c )1 ( t ) , y ( c )2 ( t )) ∈ ( t ) c +1 . On the other hand the equation f ( x, t, y ( t ) , y ( t )) = 0 has no solution ( y ( t ), y ( t )) ∈ B ρ { t } but (0 , T the Taylor map at 0: T : B ρ { t } −→ K J x, t K . If f ( x, t, y ( t ) , y ( t )) = 0 then xT ( y ( t )) − ( x + t ) T ( y ( t )) = 0 . But since K J x, t K is a unique factorization domain, this equality implies that T ( y ( t )) = T ( y ( t )) = 0, hence y ( t ) = y ( t ) = 0.This shows that there is no β : N −→ N such that for every y ( t ) ∈ B ρ { t } and every k ∈ N with F ( x, t, y ( t )) ∈ ( t ) β ( k ) there exists e y ( t ) ∈ B ρ { t } such that F ( x, t, e y ( t )) = 0and e y ( t ) − y ( t ) ∈ ( t ) k . RTIN APPROXIMATION OVER BANACH SPACES 11
Example . We can modify a little bit the previous example to constructa F as before that does not depend on t . We set G ( x, y , y , y ) := xy − ( x + y ) y ∈ B ρ [ y , y , y ] . For every c ∈ N we set y ( c )2 ( t ) := x c , y ( c )1 ( t ) := x c + P cn =1 a n x c − n t n and y ( c )3 ( t ) := t ∈ B ρ { t } . Then G ( x, y ( c )1 ( t ) , y ( c )2 ( t ) , y ( c )3 ( t )) ∈ ( t ) c . Now if e y ( t ) ∈ B ρ { t } satisfies G ( x, e y ( t )) = 0 and e y ( t ) − y ( t ) ∈ ( t ) then e y ( t ) = x + t + ε ( t ) with ε ( t ) ∈ ( t ). Thus x + e y ( t ) is an irreduciblepower series in x and t , and it is coprime with x . By the same argumentbased on the Taylor map as in Example 3.2, the relation x e y ( t ) − ( x + t + ε ( t )) e y ( t ) = 0implies that e y ( t ) = e y ( t ) = 0.This shows that there is no β : N −→ N such that for every y ( t ) ∈ B ρ { t } and every k ∈ N with G ( x, y ( t )) ∈ ( t ) β ( k ) there exists e y ( t ) ∈ B ρ { t } such that G ( x, e y ( t )) = 0and e y ( t ) − y ( t ) ∈ ( t ) k . References [1] M. Artin, On the solutions of analytic equations,
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