model of computation, based on the notion of Locally Smooth Systems (LSS), and we showed that these systems have good computational properties: stan-.
On the complexity of algebraic power series. M.E.Alonso
T.Moray
M.Raimondoy
1 Introduction. A classical computational model do deal with "computable" power series consists in giving an algorithm to compute their coecients and to consider suitable truncations in order to perform the required operations. In the case of algebraic power series, i.e. when the series f(X) is given by a polynomial G(X1 ; ; Xn; T) s.t.G(X1; ; Xn; f(X)) = 0, these coecients can be computed for instance using [K-T]. Since there is in general more than one series vanishing at the origin and satisfying the above identity, one must compute enough terms of the Taylor expansion of f, which, at least, permit to distinguish it from the other roots of G. It is clear that computational problems arise naturally in case the series f should be used for further calculations, e.g. to determine the solution h of a polynomial depending on the X variables and on f. In order to avoid these problems, in [AMR], we introduced a purely symbolic model of computation, based on the notion of Locally Smooth Systems (LSS), and we showed that these systems have good computational properties: standard bases and normal forms can be calculated in the ring of algebraic power series and it is possible to give e ective versions of classical theorems like the Weierstrass Preparation Theorem and the Noether Normalization Lemma. The aim of this paper is to look for suitable measures for the complexity of algebraic power series. In [R1], [R2] R.Ramanakoraisina de nes the complexity of an analytic function f satisfying a polynomial equation to be the degree c(f) of its minimal polynomial; and he shows that this de nition satis es all the required properties of complexities. Since we are interested in the local properties of f (at the origin) we consider here a notion which takes care of Departamento de Algebra, Facultad de Ciencias Matem aticas, Universidad Complutense, Madrid, SPAIN - Partially supported by CICYT-PB860062 (Spain) and by Accion Integrada Espana-Italia y Dipartimento di Matematica, Universit a di Genova, ITALY - Partially supported by MPI Funds(40% and 60%) , by C.N.R. (progetto strategico 'Matematica Computazionale') and by Azione Integrata Italia-Spagna
1
both the degree and the multiplicity de ned as the minimum order e(f) of de ning polynomials. We introduce then a notion of complexity for algebraic series represented in model of computation of LSS's. To do this we introduce suitable costs for Locally Smooth Systems by means of the length (f)(i.e. the number of extra variables) and the degree (f) (i.e. the product of the degrees of the involved polynomials). We de ne then the complexity of f : (f) = ((f); (f)) as the minimum of the costs of all Locally Smooth Systems representing f . The main result of this note shows how these costs can be estimated in terms of degree and multiplicity : (f) e(f) and (f) c(f)(f ) c(f)e(f ) . Conversely, we have that o(f) c(f) (f), where o(f) is the order of f at the origin. Then we introduce the complexity 0 as the minimum cost of Standard Locally Smooth Systems de ning f, and we nd bounds for 0 in terms of and of the maximum degree of involved polynomials. Using this we show that the cost of representing the Weierstrass form of a distinguished polynomial of order b can be estimated by (b(r + 1); Db(r+1) ) where r is the lenght of the involved Locally Smooth Sytem and with D depending on the degrees of the data. Finally, using the above estimates, we nd a test to check whether an algebraic function is indeed a rational function.
2 Notation and preliminaries Let K be a sub eld of the eld of complex numbers, let X = (X1 ; :::; Xn) a set of variables and K[[X]]alg the algebraic closure of K[X] in K[[X]], which is the set of algebraic formal power series, and which is also the henselization of the ring of polynomials with respect to the maximal ideal corresponding to the origin. Let us recall from [AMR] the notion of Locally Smooth Systems. We say that a system of polynomials F = (F1 ; :::; Fr) is a LSS if the Fi 's are polynomials in K[X1 ; :::; Xn; Y1; :::; Yr] vanishing at the origin and s.t. the Jacobian of the Fi's with respect to the Yj 's at the origin is a lower triangular non singular matrix, i.e. we can write: r X Fi(X; Y1 ; :::; Yr) = cij Yj + Hi(X; Y1 ; :::; Yr) j =1
with Hi 2 (X; (Y1; ::::; Yr) ) and (cij ) a non-singular lower triangular r by r matrix. Under this assumption, by the Implicit Function Theorem, there are unique algebraic series f1; :::; fr 2 K[[X]]alg s.t. fj (0) = 0 8 j, and Fi (X; f1 ; :::; fr) = 0 8 i. We will also say that F = (F1; :::; Fr) is a LSS for the fi 's ( or de ning the fi 's; or that the fi0 s are given by the LSS F etc.). The key point of our approach in [AMR] was to look for results in K[[X]]alg by working with suitable, and computable, extensions of K[X]. Namely, given 2
2
a LSS F , we then consider the rings K[X1 ; ; Xn; f1 ; ::::; fr] := K[X,F] and K[X1; ; Xn; f1 ; ::::; fr]loc := K[X; F ]loc viewed as a subring of K[[X]]alg and the evaluation map F : K[X1 ; ; Xn; Y1 ; ::::; Yr]loc ! K[[X]] de ned by F (Yi ) = fi . We have: ::::; Yr]loc K[X; F ]loc := K[X; f1 ; ::::; fr]loc ' K[X;(FY1; ;:::; Fr ) 1 ( Where, for any K-algebra A, with K[Z] A K[[Z]] , we denote: Aloc = f 1+a b , a, b 2 A , b(0) = 0 g ). The classical approach to compute with algebraic series (cf [K-T]) consists in representing them as solutions of polynomial equations, i.e. a series f(X) 2 K[[X]]alg is given by a polynomial G(X1 ; ; Xn; T) s.t. G(X; f(X)) = 0, and, since there is in general (also in case G is irreducible) more than one series vanishing at the origin and satisfying G, by an algorithm which computes the Taylor expansion of f up to order d, 8 d, or, at least, enough terms in order to distinguish f from the other roots of G. In this paper we introduce suitable measures of the complexities for algebraic power series in both representations and we will compare them. We rst recall from [AMR] how the two computational models are compatible (cf [AMR] Appendix and Proposition 2.3). Theorem 1 (a)(Artin-Mazur) Let f 2 K[[X]]alg , G 2 K[X; T] such that G(X; f(X)) = 0 and assume that an algorithm to compute the Taylor expansion of f up to order d, 8 d, is given . Then it is possible to compute a locally smooth system (F1 ; :::; Fr) de ning algebraic series f1 ; :::; fr, with f1 = f . (b) Conversely, let F = (F1 ; :::; Fr) be a LSS in K[X; YQ1; :::; Yr] de ning the series f1 ; :::; fr 2 K[[X]alg , and let di = deg(Fi ) and d = ri=1 di. Then: given h(X) = H(X; f1 (X); ::; fr (X)) 2 K[X; F ]loc represented by H 2 K[X; Y1; :::; Yr]loc with H = 1+HH0 1 and deg(H0 ) and deg(H1 ) bounded by m, there exist a polynomial Q 2 K[X; T] with deg(Q) (m + 1)d s.t. Q(X; h(X)) = 0. (Note: h = F (H) 2 K[X; F ]loc ).
Remark 1 In our computational model all the ring operations with series turn out almost automatically, since our computational tool are the "rings" [X; F ]loc . Suppose that f and g are given by distinct LSS's (respectively F and G then it is enough to merge them. More precisely, let f = F (F) 2 [X; F ]loc with F 2 K[X; Y ; :::; Yr]loc and g = G(G) 2 [X; G]loc with G 2 K[X; Y ; :::; Ys]loc then any rational function h = H(f; g) 2 K[X]alg with H = HH0 1 , H (0) = H (0) = 0 can be represented by the LSS H := (F, G) = (F ; :::; Fr; Fr ; :::; Fr s) K[X; Y ; :::; Yr; Yr ; :::; Yr s] with Fr i = Gi (Yr ; :::; Yr s) via the evaluation h = H (H(F; G)) 2 [X; F ]loc . 1
1
0
1+
1
1
+1
+
+
+1
+1
1
+
+
Of course, in many cases it will be not necessary to add so many extra variables: we will propose a test for this in the last section.
3
3 The complexity of algebraic power series. We recall a notion of complexity which has been recently introduced for Nash functions by R. Ramanakoraisina (cf [R1] and [R2]). Let f 2 K[[X]]alg; we de ne the complexity c(f) of f as c(f) = minfdegP; where P 2 K[X; T] and P(X; f(X)) = 0g In [R1],[R2] it is shown that: c(f + g) c(f)c(g) , c(fg) 2c(f)c(g) , c(f 2 ) @f ) c(f)2 2c(f) and that c( @X i Let us remark that, if P is irreducible, then c(f) = deg(P). Take in fact the Zariski closure in Cn+1 of the analytic germ (X; f(X)), then W = f(x; t) 2 C n+1 : P(x; t) = 0g. If Q is s.t. Q(X; f(X)) = 0 then Q vanishes on W, hence degQ degP. Let o(g) denote the order of vanishing of a function g at the origin of the coordinates, de ned as the lowest degree for which, in the Taylor expansion of g, there is a non-zero coecient. We introduce the representative multiplicity of f 2 K[[X]]alg (e(f)) as e(f) = minfo(P); whereP 2 K[X; T]andP(X; f(X)) = 0g: Recall that, for a local ring R, e(R) denotes its multiplicity (cf [Z-S] Ch.VIII ]loc 10). Let P(X; T) an irreducible polynomial de ning f, then e(f) = e(QKP[X;T ): (X;T ) r Let F = (F1; :::; Fr) be a LSS and let di = deg(Fi ) and d = i=1 di we will say that d is the degree of F, deg(F) = d and that r, i.e. the number of new variables, is its lenght, l(F) = r. Moreover we introduce the cost of F as: cost(F) = ( l(F), deg(F)), and we order costs lexicographically: cost(F ) < cost(G) , l(F ) < l(G) or l(F ) = l(G) and deg(F ) < deg(G): Moreover we will denote by VF C n+r the algebraic variety de ned by the ideal (F1 ; :::; Fr). By de nition VF is non-singular at the origin of the coordinates. Nevertheless VF can be reducible, therefore we will consider also its irreducible component WF through the origin. We also remark that, in general, WF may be not de nable by r polynomial equations; i.e. it may be not a complete intersection.
Example 1 Let F = 3Y ? 4X + Y ? 2XY + 2Y ? 2XY ? X ? X Y F = Y ? 2Y + X + XY ? Y 1
2
2 2
1
2
1
2 1
2
2
1
2
2 1
2 1
Then (F1; F2) is not a prime ideal in K[X; Y1 ; Y2] and the associated prime at the origin is
} = (F1 ; F2; X 3 + 3X 2 ? Y1Y2 + 3X ? Y 1 ? Y 2) On the other hand we have:
4
c(f1 ) = 4 de ned by: X 4 + 4X 3 ? T 3 + 6X 2 ? 3T 2 + 4X ? 3T . c(f2 ) = 5 de ned by: X 5 + 5X 4 + 10X 3 ? 3T 2 + 5X ? 3T . Moreover we have that if W = V (}) then degW = 5, but for every de ning LSS G , deg( G) 6. We are going now to introduce complexities of algebraic power series in the LSS model; it turns out convenient to introduce this notion for a set of such functions. We say that f is de ned via the LSS H if f 2 fh1; :::; hrg, where the h0i s are the algebraic power series de ned by H. Similarly, we say that ff1 ; :::; fsg are de ned via H if ff1; :::; fsg fh1; :::; hrg as a set.
De nition 1 We call cost of ff ; :::; fsg the minimum of cost(H) where f ; :::; fs are de ned via H . We write: (f ; :::; fs) = ((f ; :::; fs); (f ; :::; fs)) = (length(H ); deg(H )) = cost(H ) where H reaches the minimum. Moreover, we set : d(f ; :::; fs) = deg(Z) where Z is the Zariski closure of the germ (X; f (X); :::; fs(X)) in C n+s . Remark 2 Notice that our de nition does not strictly satisfy the notion of 1
1
1
1
1
1
1
complexity given by Benedetti and Risler (cf[BR]), while it is straightforward to verify the following formulas, (the proofs come out easily by the merging procedure described in Remark 1): 1. (f; g) ((f) + (g); (f)(g)) 2. (f + g) ((f) + (g) + 1; (f)(g)) 3. (fg) ((f) + (g) + 1; 2(f)(g)) 4. (f 2 ) ((f) + 1; 2(f)) @f )) (2(f); (f)2 ) 5. ( @X i
Proposition 1 1) If F and G de ne the same algebraic functions g ; :::; gr then 1
WF = W G 2) d(f1 ; :::; fs) deg(WF ), where f1 ; :::; fs are de ned via F 3) d(f1 ; :::; fs) (f1 ; :::; fs) Proof : 1) It is clear, since WF and WG are irreducible algebraic varieties with the same germ at the origin of the coordinates. 2) Let U (resp. Z) be the Zariski closure of (X; g1(X); :::; gr (X)) in Cn+r ( resp of (X; f1 (X); :::; fs(X)) in Cn+s . Then f1 ; :::; fs g1; :::; gr as sets, and w.l.o.g. assume that f1 = gr?s+1 ; :::; fs = gr . We consider the projection : Cn+r ?! Cn+s to the rst n and the last s factors (x; y1; :::; yr?s; yr?s+1 ; :::; yr) 7! (x; yr?s+1; :::; yr). Then (U) Z and: 5
C n+r # C n+s # Cn
U
# Z #
VF
Cn
By 1) we obtain that U = WF , moreover Z is the Zariski closure of (U) and then: deg(U) = deg(Z)deg() and hence deg(WF ) deg(Z) = d(f1; :::; fs). 3) is clear.
4 Estimating complexities The following theorem shows us that the measures of complexity for algebraic power series we have introduced in the two models are compatible, i.e. we can estimate (f) in term of c(f) and e(f). We will furthermore assume that the base eld K is algebraically closed. Theorem 2 Let f 2 K[[X]]alg , then: A) (f) e(f) B) (f) c(f)(f ) c(f)e(f ) Proof :A) For a local ring R let e(R) denote its multiplicity. ]loc ): Let P(X; T) an irreducible polynomialde ning f, such that e(f) = e( KP[X;T (X;T ) If e := e(f) = 1 we have nished.By induction suppose that there exist k variables Y1 ; :::; Yk and an ideal Ik such that K [X;YI1k;:::;Yk] is an integral extension X;T ] and such that e := e( K [X;Y1 ;:::;Yk]loc ) e + 1 ? k. Let R := of PK([X;T k k ) Ik K [X;Y1 ;:::;Yk]loc . If ek > 1, since we know by hypothesis that there is an analytic Ik smooth branch through the origin, the local ring Rk is not unibranche (cf. the proof of Artin-Mazur as in [AMR]) so there exists an integer function h in the normalization of it such that h assumes two di erent values at two distinct K [X;Y1 ;:::;Yk]loc [Yk+1 ] k ]loc [h] = branches of it . Let us consider R0 := K [X;Y1I;:::;Y Ik+1 k k ;Yk+1 ]loc . By the projection formula ([Z-S] ) we and let Rk+1 := K [X;Y1 ;:::;Y Ik+1 P have e(Rk ) = e(Rk+1) + [R0=}i : K]e(}i ) and hence e(Rk ) > e(Rk+1) . The procedure then halts when ek = 1, then k = (f) and 1 e ? k+1 = e ? (f)+1. B) Let Z the Zariski closure of (X; f(X)) as above and let WH be the irreducible component through the origin of VH where H is a LSS de ning f and such that WH is dominated by the normalization Z 0 of Z. Then there exists a dominating morphism : WH ?! K n and the degree of WH is given by [K(WH ) : K(X1 ; :::; Xn)] = [K(WH ) : K(Z)][K(Z) : K(X1 ; :::; Xn)] = [K(Z) : K(X1 ; :::; Xn)] = c(f). Hence deg(WH ) = c(f) . Suppose now that H is minimal, as constructed in A), and that r = . By Heintz' results on de nability (cf. [H] Prop.3), there exist n + r + 1 = n + (f) + 1 polynomials gj0 s of degree bounded by c(f) such that WH = g1 = ::: = gn+(f )+1 = 0 as a set . By means 6
of a generalization of Heintz proof (it is enough to choose projections which distinguish not only points but also the tangents at the origin), we obtain that the gj0 s can be chosen in order that the variety g1 = ::: = gn+(f )+1 = 0 is non singular at the origin of coordinates, then WH is actually its irreducible component through the origin. Computing the Jacobian determinant of the gi0 s w.r.t. the Yj0s variables we know that it has rank (f) and therefore we can choose the corresponding polynomials g10 ; :::; g0 (f ) which give a LSS H0 de ning the fi0 s and with deg(H0 ) c(f)(f ) c(f)e(f ) . On the other side we know that the complexity of f can be estimated by the degree of any LSS de ning f (cf Theorem 1 b)). We will give an improvement of this result and will also show how to compute c(f) in term of the LSS, for this we need the following lemma. Lemma 1 Let G(X; T); F(X; T) 2 K[X; T] be polynomials of degree d and m respectively s.t. F is a factor of G and let h(X) 2 K[[X]]alg be s.t. G(X; h(X)) = 0. Then if the Taylor expansion of F(X; h(X)) vanishes up to order dm we have that F(X; h(X)) = 0. Proof : In fact take an irreducible factor G1 of G with G1(X; h(X)) = 0, if fF = 0g and fG1 = 0g do not have a common component there exists a set of linear forms Hj through the origin such that fF = 0g \ fG1 = 0g \ fH1 = ::: = Hn?1=0g is a nite set of points, whose multiplicity at the origin is greater than dm, in contradiction with Bezout theorem. Proposition 2 Let F a LSS de ning f1; :::; fr . Then we have: a) c(fi ) deg(F)Q8 i b) d(f1; :::; fr ) ri c(fi ) And there is an algorithm to compute c(fi ) 8 i . Proof : Let VF = V (F1; :::; Fr) K n+r , Wi = Zar:cl:f(X; fi(X)); X 2 K ng K n+1, W = Zar:clf(X; f1 (X); :::; fr (X)); X 2 K ng K n+r , and i : K n+r ?! K n+1 the projection (X; Y1 ; :::; Yr) 7! (X; Yi ) . Then W VF K n+r
#
#
Wi i (VF ) K n+1 So, c(fi ) deg(i (VF ) deg(VF )deg(F), by Bezout inequality. As for b) let Vi = Wi K r?1 Kn+r , then we observe that deg(Vi ) = c(fi ) and that W is an irreducible component of W 0 = V1 \ ::: \ Vr , and apply Bezout inequality. Moreover: i(VF ) is de ned by the ideal J = (F1; :::; Fr) \ K[X; Yi] which can be calculated by an elimination Groebner basis computation. Now, J is a principal ideal, say J = (Q)K[X; Yi ] . Therefore we only have to compute the irreducible factor Q0 of Q such that Q0(X; fi (X)) = 0. This can be done using Lemma 4. 7
Corollary 1 Let F a LSS de ning f ; :::; fr , h = F (H) with H 2 K[X; Y ] , d = deg(F) and m = deg(H), then : i) o(h) c(h) md ii) o(fi ) c(fi ) d8i provided fi 6= 0. Remark 3 In the Example above, we have: e(f ) = e(f ) = 1 c(f ) = 4 ; 1
1
2
1
(f1 ) = (1; 4); c(f2 ) = 5 ; (f2 ) = (1; 5); d(f1 ; f2) = 5 ; (f1 ; f2) = (2; 6) ; (f1 ; f2) = 6. p p Moreover,let us consider: f(X) = X 1 ? X ; g(X) = 1 ? 1 ? X . Then c(f) = 3 , e(f) = 2 and (f) = (f; g) = (2; 4).
5 Standard Locally Smooth Systems In [AMR] an important role in order to perform computations in the ring K[[X]]alg has been played by the notion of standard locally smooth system (SLSS) We recall (in a simpli ed version) this de nition: G is a standard locally smooth system (SLSS) if: 1) G = (G1; :::; Gr) is a LSS for the functions f1 ; :::; fr 2) fi 6= 08i 3) Gi = Yi (1 + Qi ) ? Ri with Qi , Ri 2 (X; Y ), Ri 2 K[X; Y1; :::; Yi?1; Yi+1; :::; Yr] and in(Ri ) = in(fi ) 2 K[X]. The introduced notion turns out to be quite important both for its own sake, since it directly gives explicit information on the fi0 s, and because it is a computational tool for standard bases, Weierstrass Preparation Algorithm and elimination algorithms (cf [AMR]). In this section we look for bounds for the costs of SLSS's. To do this, we introduce the complexity 0 of ff1; :::; fsg as the minimum of the cost(H) where the fi0 s are de ned via a SLSS H, and we write: 0 (f1 ; :::; fs) = (0 (f1 ; :::; fs); 0(f1 ; :::; fs)) = (length(H ); deg(H )); 0
0
where H0 reaches the minimum. We further assume that fi 6= 08 i and that f1; :::; fs are linearly independent. We also introduce the natural numbers = Sup Q fdi; di = deg(Fi )g
= si=1 o(fi ) . Lemma 2 Let F = (F1 ; :::; Fr) be a LSS de ning f1 ; :::; fr , where fi 6= 0 for every i, and let G = (G1 ; :::; Gr) be a SLSS obtained by F , then o(fi ) deg(Gi ) + o(fi ) ? 1 Proof : Gi is obtained by the corresponding Fi, as follows: while there is a term t, deg(t) o(fi ) depending on some Yj , we substitute an occurence of Yj by Yj ? Fj . Each such substitution can at worst introduce terms of degree o(fi ) + ? 1. 8
Proposition 3 With the above notation, we have: a) 0 = b) Supf ; (f ; :::; fs)g 0 (f ; :::; fs) Inf f2 ; (f ; :::; fs)g. Proof : Let us take F minimal and G obtained from F as inQthe above Lemma. Let di = degGi . Then di o(fi ) + ? 1 di + ? 1 and h (dh + ? 1) Inf f2 ; (f ; :::; fs)g: 1
1
1
=1
1
6 Weierstrass Preparation Theorem
In [AMR] x5 we gave constructive versions of Weierstrass Preparation and Division theorems, we are going now to bound the complexity of these constructions. Let X 0 = (X1 ; :::; Xn?1) so X = (X 0 ; Xn ) and let g be an algebraic series distinguished in Xn , say g(0; Xn ) = Xnb + higher degree terms, P then there exist a unit v and series hi 2 K[[X 0]]alg such that g = v(Xnb + i Xnb and a new G2 2
0
0
such that F2 (G) = g. (2) Let F the SLSS costructed by F2 and G such that F (G) = g. (3) Let U = (U1;0; :::; Ur;0; U0; U1;1; :::; Ur;1; U1; :::; U1;b?1; :::; Ur;b?1; Ub?1) a new set of variables and let P := Xnb ? Pbj?=01 Uj Xnj Pi := Yi ? Pbj?=01 Uij Xnj 8 i = 1; :::; r: (4) Apply Buchberger reduction (with respect to a suitable term ordering) to G; F1; :::; Fr, we obtain polynomials H0; :::; Hd?1; H1;0; :::; H1;b?1; :::; Hr;0; :::; Hr;b?1 2 (X1 ; :::; Xn?1; U)K[X1 ; :::; Xn?1; U] = K[X 0 ; U] s.t.: 9
G ? Pbj?=01 Uj Xnj 2 (P; P1; :::; Pr) Fi ? Pbj?=01 Uij Xnj 2 (P; P1; :::; Pr) 8 i
Let us now examinate the costs of these constructions: (1) length(F2 ) r ; deg(F2;i) bd0;i ; 2 = supfdeg(F2;i)g b0 and deg(G2) bm. (2) length(F) r ; deg(Fi ) 2b0 ;G = G2 and := supfdeg(Fi ); deg(G)g supfbm; 2b0g (4) length(H) b(r + 1) , deg(Hi ) and deg(Hij ) are bounded by + b and therefore degH) ( + b)b(r+1) Db(r+1) . We conclude this section remarking that, by the above result we obtain a single exponential complexity also for the Weierstrass Division Theorem, however the main drawback consists in the large number of extra variables needed to represent the Weierstrass polynomials: by an initial form computation we can cancel those which are zero, in the next section we propose a test to check if they are in fact rational functions.
7 Applications We apply now the above results, and precisely the Proposition 2 and the Corollary 1, in order to obtain some informations on the rationality of power series. This is clearly of great interest since, as we have remarked in the previous section, the cost grows exponentially on thePnumber of involved functions. Let f be P a power series in X, f = P1a2IN n fa X a , with fa 2 K; then we write f(i) := jaj=i fa X a , so that f = i=0 f(i). Proposition 4 Let h 2 K[[X]]alg such that c(h) t, then: h 2 K[X] () h(j ) = 0 8j : t < j t2: Proof . The same proof of Prop. 2.7 of [AMR] works. Corollary 2 It is possible to check whether h 2 K[X]alg is a rational function. Proof : Let Q 2 K[X; T] be an irreducible polynomial such that Q(X; h(X)) = 0. Let s = degT(Q) c(h) = deg(Q) t and write Q(X; T) = a0(X)T s + a1(X)T s?1 + ::: + as?1 (X)T + as (X) 2 K[X; T] Let us further introduce the following polynomial Q 2 K[X; T]: Q (X; T) = as0Q(X; aT0 ) = T s +a1T s?1 +a0a2 T s?2 +:::+a0s?2as?1 T +a0s?1as = 2 P T s + si=1 ai0?1ai T s?i Then we obtain that deg(Q) s(t ? s + 1) (t+1) 4 .Let us consider k = ha0 2 K[[X]]alg . Now, if h(X) = fg((XX)) 2 K[X]loc , it is easy to see that g is a factor of a0 and therefore we obtain that k(X) = a0(X ) f(X) 2 K[X]. It is straightforward to see that k is a root of Q. Then g(X ) 10
2
. Conversely, suppose that k is a polynomial root of Q, then h = c(k) (t+1) 4 k 2 K(X) \ K[[X]] = K[X] . In order to apply Proposition 4 , we need to alg loc a0 4 (t+1)2 check whether k(j) = 0 for 4 j (t+1) , this can be done using suitable 16 linear systems (with the coecients of4 a0 as unknowns), once we know the Taylor expansion of h up to degree (t+1) and we know that deg(a0 ) t ? s t. 16 More precisely we write a0 as a polynomial of degree t ? 1 (or if is possible we use a better estimate) with unknown coecients Ua with a =2(a1 ; :::; an). We then @ k where b = jbj = [ (t+1) ] an enough number consider starting from k(b) := @X 4 t + n ? 1 2 of equations (at least max( ; b ? b)) involving the unknowns U n and the Taylor coecients of h up to degree d2 . The problem is then reduced to compute whether this system has a non-zero solution, i.e. to compute the vanishing of a suitable determinant
Remark 4 If we dispose of an ecient factorization modulus, we can test
whether an algebraic function is polynomial (resp. rational) in the following way. 1. Given f produce a polynomial Q(X; T) over which it vanishes. 2. Factorize Q to check whether it has a polynomial factor which is linear in T. Then check whether it is of the form: T ? f(x). 3. If not, construct Q and factorize it, etc.as in 2).
References
[AMR] M.E.Alonso, T.Mora, M.Raimondo.A computational model for algebraic power series. J. Pure and Appl. Algebra, to appear. [B-R] R.Benedetti, J.J.Risler.Real Algebraic and Semialgebraic sets. Hermann,Paris 1990. [H] J.Heintz. De nability and fast quanti er elimination in algebraically closed elds. Theoretical Computer Science 24 (1983). [K-T] H.T.Kung, J.F.Traub.All Algebraic Functions can Be Computed Fast. J. ACM 25 (1978). [R1] R.Ramanakoraisina. Complexite des fonctions de Nash. Comm. Algebra 17 (1989). [R2] R.Ramanakoraisina.Bezout Theorem for Nash functions. Preprint U.E.R. Math.Univ. Rennes (1989). 11
[Z-S] O.Zariski, P.Samuel.Commutative Algebra Vol II. Van Nostrand 1960.
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