IMPLICIT FUNCTION THEOREM FOR FORMAL POWER SERIES

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1 #A9 INTEGERS 8A (208) IMPLICIT FUNCTION THEOREM FOR FORMAL POWER SERIES ining Hu School of Matheatics and Statistics, Huazhong University of Science and Technology, Wuhan, PR China Received: 6/9/7, Accepted: 2/27/8, Published: 3/6/8 Abstract We generalize a result of Furstenberg about the diagonal of bivariate rational fractions and algebraic series, and give a direct proof of an explicit iplicit function theore for foral power series that is valid for all fields, which iplies, in particular, a Lagrange inversion forula and a Flaolet-Soria coe cient extraction forula known for fields of characteristic 0.. Main Result We show that the iplicit function theore for foral power series is a direct consequence of a generalization of Furstenberg s theore about the diagonal of rational functions and algebraic series. Theore. Let K be an arbitrary field. If P (X, ) 2 K[[X, ]] and f(x) 2 K[[X]] are such that f(0) 0, P (X, f(x)) f(x) and P 0 (0, 0) 0, then for all n 2 N [X n ]f X [X n ]( P 0 (X, ))P (X, ). If the characteristic of K is 0, we also have the following for [X n ]f X [Xn ]P (X, ). Reark. The general case can also be deduced fro the 0 characteristic case by P regarding the coe cients of f f i X i as indeterinates and the first forula as an identity in Z[f, f 2,...]. i0 Reark 2. The conditions P (X, f(x)) 0 and f(0) 0 iply that P (0, 0) 0. As P 0 (0, 0) is also 0, the sus in both expressions of [Xn ]f are finite.

2 INTEGERS: 8A (208) 2 Reark 3. Siilar results in characteristic 0 can be found in [6]. Reark 4. When P (X, ) X ( ), where (X) 2 K[[X]] and (0) 6 0, we obtain the Lagrange inversion forula [X n ]f [ n ]( (X) n 0 (X) (X) n ). If the characteristic of K is 0, we also have the following for: [X n ]f n [ n ] ( ) n. For di erent proofs of the Lagrange inversion forula in characteristic 0, see [4]. Reark 5. When P (X, ) is a polynoial in X and, we obtain the Flaolet- Soria coe cient extraction forula. See [], [7] for proofs in characteristic 0. See [5] for a proof for arbitrary fields using Furstenberg s Theore. 2. Proof When K is a field of 0 characteristic, where P (X, ) 2 K[[X, ]] and f(x) 2 K[[X]] with f(0) 0, the Taylor forula takes the for P (X, ) n! ( f(x))n P (n) (X, f(x)). n0 However, because of the n! in the denoinators, we cannot use it in positive characteristic. To ake up for this, we define P [] (X, ) as an alternative to the -th partial derivative of P with respect to that absorbs the factorial. Once this obstacle is circuvented, the Taylor forula works as expected. Definition. Let P (X, ) 2 K[[X, ]], P (X, ) a (X), where a (X) 2 K[[X]]. We define for 2 N. 0 P [] (X, ) X a (X). Proposition. Let K be an arbitrary field. Let P (X, ) 2 K[[X, ]] and f(x) 2 K[[X]] with f(0) 0. Then P (X, ) ( f(x)) P [] (X, f(x)). (*) 0

3 INTEGERS: 8A (208) 3 P Proof. Let P (X, ) a (X) with a (X) 2 K[[X]] for 2 N. We prove that 0 for all k 2 N, the coe cients of k in the left side and the right side of ( ) are equal. Indeed, we have [ k ] ( f(x)) P [] (X, f(x)) 0 k ( f(x)) k X a (X)f(X) k k a (X)f(X) k ( ) k a (X)f(X) k ( ) k, k, k k a k (X). We have the last equality because P, k,k ( ) k if k and 0 if > k. This is because we have the ultinoial expansion (a + b + c) X a b k c k, k, k kappleapple k0, k, k a b k c k. When we take a, b c k0, the identity becoes! ( ) k c k., k, k Seeing this as a polynoial identity in the variable c gives us the desired result. The following corollary is iediate. Corollary. Let K be an arbitrary field. Let Q(X, ) 2 K[[X, ]] and f(x) 2 K[[X]] be such that f(0) 0 and Q(X, f(x)) 0. Then there exists R(X, ) 2 K[[X, ]] such that Q(X, ) ( f(x))r(x, ).

4 INTEGERS: 8A (208) 4 Definition 2. For the foral power series in K((X,..., X )) P (X, X 2,..., X ) X a nn 2...n X n Xn2 2...Xn n i> µ its (principal) diagonal Df(t) is defined as the eleent in K((T )) DP (T ) X a nn...n T n. In [3], Furstenberg proved the following result about algebraic series and the diagonal of rational fractions. Proposition 2. (Furstenberg) Let K be an arbitrary field. Let Q(X, ) 2 K[X, ] and f(x) 2 K[[X]] be such that f(0) 0, Q(X, f(x)) 0 and Q 0 (0, 0) 6 0, then f(x) D 2 Q0 (X, ). Q(X, ) Proposition 2 can be used to deduce a coe cient extraction forula for algebraic series with coe cients in an arbitrary field (see Reark 5) and can be used to copute e ciently the n-th ter of an algebraic series with coe cients in a field of positive characteristic [2]. The following proposition is a generalization of Proposition 2, where Q(X, ) can be any foral power series and not ust a polynoial. The only di erence in the proof is in the first step where we use Corollary to factorize Q(X, ). Proposition 3. Let K be an arbitrary field. Let Q(X, ) 2 K[[X, ]] and f(x) 2 K[[X]] be such that f(0) 0, Q(X, f(x)) 0 and Q 0 (0, 0) 6 0. Then f(x) D 2 Q0 (X, ). Q(X, ) Proof. Using Corollary we can write Q(X, ) ( f(x))r(x, ) with R(X, ) 2 K[[X, ]]. We have R(0, 0) 6 0 because f(0) 0 and Q 0 (0, 0) 6 0. Then Q(X, ) Q0 (X, ) f(x) + R0 (X, ) R(X, ). Replacing X by X and ultiplying by 2 we get D 2 Q0 (X, ) 2 D Q(X, ) f(x ) + D 2 R0 (X, ). ( ) R(X, )

5 INTEGERS: 8A (208) 5 For the first ter on the right side of ( ) we have 2 D f(x ) D f(x )! D n+ f(x ) n n0 D (f(x )) f(x). For the second ter, as R(0, 0) 6 0, R0 (X, ) R(X, ) is a power series in X and, so when we ultiply this by 2 there is no diagonal ter. Proof of Theore. Let the power series Q(X, ) be defined as Q(X, ) P (X, ), then Q 0 (0, 0) P 0 (0, 0) 6 0, and Q(X, f(x)) P (X, f(x)) f(x) 0. According to Proposition 3, f D{ 2 Q 0 (X, )/Q(X, )} D{ 2 (P 0 (X, ) )/(P (X, ) )} D{ ( P 0 P (X, ) (X, ))/( )} D{ ( P 0 (X, ))( + X ( P (X, ) ) )}. We have the last equality due to the fact that 0 P (X, ) (0, 0) 0, has no constant P (X, ) ter, and therefore /( ) + P ( P (X, ) ). As in each ter of ( P 0 (X, )) the power of is larger than that of X, it cannot contribute to the diagonal. Therefore, f n [X n n ] ( P 0 (X, ))( + X ( P (X, ) ) ) [X n n ] ( P 0 (X, ))( X ( P (X, ) ) ) X [X n ]( P 0 (X, ))P (X, ).

6 INTEGERS: 8A (208) 6 References [] C. Banderier and M. Drota, Forulae and asyptotics for coe cients of algebraic functions, Cobin. Probab. Coput. 24, (205), No., -53. [2] A. Bostan, G. Christol, P. Duas, Fast coputation of the nth ter of an algebraic series over a finite prie field, Proceedings ISSAC 6, pp. 9-26, ACM Press, 206. [3] H. Furstenberg, Algebraic functions over finite fields, J. Algebra 7, (967) [4] I. Gessel, Lagrange inversion, J. Cobin. Theory Ser. A 44 (206), [5]. Hu, Coe cient extraction forula and Furstenberg s theores, arxiv: [6] A. Sokal, A ridiculously siple and explicit iplicit function theore, Sé. Lothar. Cobin. 6A (2009/). [7] M. Soria, Thèse d habilitation (990), LRI, Orsay.