DECOMPOSITION OF POLYNOMIALS AND APPROXIMATE ROOTS

Transcription

1 DECOMPOSITION OF POLYNOMIALS AND APPROXIMATE ROOTS ARNAUD BODIN Abstract. We state a kin of Eucliian ivision theorem: given a polynomial P (x) an a ivisor of the egree of P, there exist polynomials h(x), Q(x), R(x) such that P (x) = h Q(x) + R(x), with eg h =. Uner some conitions h, Q, R are unique, an Q is the approximate -root of P. Moreover we give an algorithm to compute such a ecomposition. We apply these results to ecie whether a polynomial in one or several variables is ecomposable or not. 1. Introuction Let A be an integral omain (i.e. a unitary commutative ring without zero ivisors). Our main result is: Theorem 1. Let P A[x] be a monic polynomial. Let such that is a ivisor of eg P an is invertible in A. There exist h, Q, R A[x] such that P (x) = h Q(x) + R(x) with the conitions that (i) h, Q are monic; (ii) eg h =, coeff(h, x 1 ) = 0, eg R < eg P eg P ; (iii) R(x) = i r ix i with (eg Q i r i = 0). Moreover such h, Q, R are unique. The previous theorem has a formulation similar to the Eucliian ivision; but here Q is not given (only its egree is fixe); there is a natural Q (that we will compute, see Corollary ) associate to P an. Notice also that the ecomposition P (x) = h Q(x) + R(x) is not the Q-aic ecomposition, since the coefficients before the powers Q i (x) belong to A an not to A[x]. Date: October 16, Mathematics Subject Classification. 13B5. Key wors an phrases. Decomposable an inecomposable polynomials in one or several variables. 1

2 ARNAUD BODIN Example. Let P (x) = x 6 + 6x 5 + 6x + 1 Q[x]. If = 6 we fin the following ecomposition P (x) = h Q(x) + R(x) with h(x) = x 6 15x 4 +40x 3 45x +30x 10, Q(x) = x+1 an R(x) = 0. If = 3 we have h(x) = x , Q(x) = x + x 4 an R(x) = 40x 3 90x. If = we get h(x) = x 75, Q(x) = 4 x3 + 3x 9x + 7 an R(x) = x + 55x. Theorem 1 will be of special interest when then ring A is itself a polynomial ring. For instance at the en of the paper we give an example of a ecomposition of a polynomial in two variables P (x, y) A[x] for A = K[y]. The polynomial Q that appears in the ecomposition has alreay been introuce in a rather ifferent context. We enote by P the approximate -root of P. It is the polynomial such that ( P ) approximate P in a best way, that is to say P ( P ) has smallest possible egree. The precise efinition will be given in section, but we alreay notice the following: Corollary. Q = P We apply these results to another situation. Let A = K be a fiel an. P K[x] is sai to be -ecomposable in K[x] if there exist h, Q K[x], with eg h = such that P (x) = h Q(x). Corollary 3. Let A = K be a fiel. Suppose that char K oes not ivie. P is -ecomposable in K[x] if an only if R = 0 in the ecomposition of Theorem 1. In particular, if P is -ecomposable, then P = h Q with Q = P. After the first version of this paper, M. Aya an G. Chèze communicate us some references so that we can picture a part of history of the subject. Approximate roots appeare (for = ) in some work of E.D. Rainville [9] to fin polynomial solutions of some Riccati type ifferential equations. An approximate root was seen as the polynomial part of the expansion of P (x) 1 into ecreasing powers of x. The use of approximate roots culminate with S.S. Abhyankar an T.T. Moh who prove the so-calle Abhyankar-Moh-Suzuki theorem in [1] an []. For the latest subject we refer the reaer to an excellent expository article of P. Popescu-Pampu [8]. On the other han Ritt s ecompostion theorems (see [10] for example) have le to several practical algorithms to

3 DECOMPOSITION OF POLYNOMIALS AND APPROXIMATE ROOTS 3 ecompose polynomials in one variable into the form P (x) = h Q(x): for example D. Kozen an S. Lanau in [6] give an algorithm (refine in [5]) that computes a ecomposition in polynomial time. Unification of both subjects starts with P.R. Lazov an A.F. Bearon ([7], [3]) for polynomials in one variable over complex numbers: they notice that the polynomial Q is in fact the approximate -root of P. We efine approximate roots in section an prove uniqueness of the ecomposition of Theorem 1. Then in section 3 we prove the existence of such ecomposition an give an algorithm to compute it. Finally in section 4 we apply these results to ecomposable polynomials in one variable an in section 5 to ecomposable polynomials in several variables.. Approximate roots an proof of the uniqueness The approximate roots of a polynomial are efine by the following property, [1], [8, Proposition 3.1]. Proposition 4. Let P A[x] a monic polynomial an such that is a ivisor of eg P an is invertible in A. There exists a unique monic polynomial Q A[x] such that: eg(p Q ) < eg P eg P. We call Q the approximate -root of P an enote it by P. Let us recall the proof from [8]. Proof. Write P (x) = x n + a 1 x n 1 + a x n a n an we search an equation for Q(x) = x n +b 1 x n 1 +b x n + +bn. We want eg(p Q ) < eg P eg P, that is to say, the coefficients of x n, x n 1,..., x n n in P Q equal zero. By expaning Q we get the following system of equations: a 1 = b 1 a = b + ( ) b 1 (S). a k = b k + c i1...i k 1 b i 1 1 b i k 1 k 1, 1 k n i 1 +i + +(k 1)i k 1 =k where the coefficients c i1...i k 1 are the multinomial coefficients efine by the following formula: ( )! c i1...i k 1 = = i 1,..., i k 1 i 1! i k 1!( i 1 i k 1 )!.

4 4 ARNAUD BODIN The system (S) being a triangular system, we can inuctively compute the b i for i = 1,,..., n: b 1 = a 1, b = a ( )b 1,... Hence the system (S) amits one an only one solution b 1, b,..., b n. Notice that we nee to be invertible in A to compute b i. Moreover b i epens only on the first coefficients a 1, a,..., a n of P. Proposition 4 enables us to prove Corollary : by conition (ii) of Theorem 1 we know that eg(p Q ) < eg P eg P so that Q is the approximate -root of P. Another way to compute P is to use iterations of Tschirnhausen transformation, see [1] or [8, Proposition 6.3]. We en this section by proving uniqueness of the ecomposition of Theorem 1. Proof. Q is the approximate -root of P so is unique (see Proposition 4 above). In orer to prove the uniqueness of h an R, we argue by contraiction. Suppose h Q + R = h Q + R with R R ; set r i x i to be the highest monomial of R(x) R (x). From one han x i is a monomial of R or R, hence eg Q i by conition (iii) of Theorem 1. From the equality (h h) Q = R R we euce that i = eg(r R ) is a multiple of eg Q ; that yiels a contraiction. Therefore R = R, hence h = h. 3. Algorithm an proof of the existence Here is an algorithm to compute the ecomposition of Theorem 1. Algorithm 5. Input. P A[x], eg P. Output. h, Q, R A[x] such that P = h Q + R. 1st step. Compute Q = P by solving the triangular system (S) of Proposition 4. Set h 1 (x) = x, R 1 (x) = 0. n step. Compute P = P Q = P h 1 (Q) R 1. Look for its highest monomial a i x i. If eg Q i then set h (x) = h 1 (x) + a i x i eg Q, R = R 1. If eg Q i then R (x) = R 1 (x) + a i x i, h = h 1. 3th step. Set P 3 = P h (Q) R, look for its highest monomial a i x i, Final step. P n = P h n 1 (Q) R n 1 = 0 yiels the ecomposition P = h Q + R with h = h n 1 an R = R n 1. The algorithm terminates because the egree of the P i ecreases at each step. It yiels a ecomposition P = h Q + R that verifies all

5 DECOMPOSITION OF POLYNOMIALS AND APPROXIMATE ROOTS 5 the conitions of Theorem 1: in the secon step of the algorithm, an ue to Proposition 4 we know that i < eg P eg P. That implies coeff(h, x 1 ) = 0 an eg R < eg P eg P. Therefore at the en coeff(h, x 1 ) = 0. Of course the algorithm proves the existence of the ecomposition in Theorem Decomposable polynomials in one variable Let K be a fiel an. P K[x] is sai to be -ecomposable in K[x] if there exist h, Q K[x], with eg h = such that P (x) = h Q(x). We refer to [4] for references an recent results on ecomposable polynomials in one an several variables. Proposition 6. Let A = K be a fiel whose characteristic oes not ivie. A monic polynomial P is -ecomposable in K[x] if an only if R = 0 in the ecomposition P = h Q + R. In view of Algorithm 5 we also get an algorithm to ecie whether a polynomial is ecomposable or not an in the positive case give its ecomposition. Proof. If R = 0 then P is -ecomposable. Conversly if P is - ecomposable, then there exist h, Q K[x] such that P = h(q). As P is monic we can suppose h, Q monic. Moreover, up to a linear change of coorinates x x + α, we can suppose that coeff(h, x 1 ) = 0. Therefore P = h(q) is a ecomposition that verifies the conitions of Theorem 1. Remark. Let P (x) = x n + a 1 x n a n, we first consier a 1,..., a n as ineterminates (i.e. P is seen as an element of K(a 1,..., a n )[x]). The coefficients of h(x), Q(x) an R(x) = r 0 x k +r 1 x k 1 + +r k (compute by Proposition 4, the system (S) an Algorithm 5) are polynomials in the a i, in particular r i = r i (a 1,..., a n ) K[a 1,..., a n ], i = 0,..., k. Now we consier a 1,..., a n K as specializations of a 1,..., a n an enote by P the specialization of P at a 1,..., a n. Then, by Proposition 6, P is -ecomposable in K[x] if an only if r i (a 1,..., a n) = 0 for all i = 0,..., k. It expresses the set of -ecomposable monic polynomials of egree n as an affine algebraic variety. We give explicit equations in the following example. Example. Let K be a fiel of characteristic ifferent from. Let P (x) = x 6 +a 1 x 5 +a x 4 +a 3 x 3 +a 4 x +a 5 x+a 6 be a monic polynomial of egree 6 in K[x] (the a i K being ineterminates). Let =. We first look for

6 6 ARNAUD BODIN the approximate -root of P (x). P (x) = Q(x) = x 3 +b 1 x +b x +b 3. In view of the triangular system (S) we get b 1 = a 1, b = a b 1, b 3 = a 3 b 1 b. Once we have compute Q, we get h(x) = x + a 6 b 3. Therefore R(x) = (a 4 b 1 b 3 b )x + (a 5 b b 3 )x. Now P (x) is -ecomposable in K[x] if an only if R(x) = 0 in K[x] that is to say if an only if (a 1,..., a 6 ) satifies the polynomial system of equations in a 1,..., a 5 : { a 4 b 1 b 3 b = 0, a 5 b b 3 = Decomposable polynomials in several variables Again K is a fiel an. Set n. P K[x 1,..., x n ] is sai to be -ecomposable in K[x 1,..., x n ] if there exist Q K[x 1,..., x n ], an h K[t] with eg h =, such that P (x 1,..., x n ) = h Q(x 1,..., x n ). Proposition 7. Let A = K[x,..., x n ], P A[x 1 ] = K[x 1,..., x n ] monic in x 1. Fix that ivies eg x1 P, such that char K oes not ivie. P is -ecomposable in K[x 1,..., x n ] if an only if the ecomposition P = h Q + R of Theorem 1 in A[x 1 ] verifies R = 0 an h K[t] (instea of h K[t, x,..., x n ]). Proof. If P amits a ecomposition as in Theorem 1 with R = 0 an h K[t] then P = h Q is -ecomposable. Conversly if P is -ecomposable in K[x 1,..., x n ] then P = h Q with h K[t], Q K[x 1,..., x n ]. As P is monic in x 1 we may suppose that h is monic an Q is monic in x 1. We can also suppose coeff(h, t 1 ) = 0. Therefore h, Q an R := 0 verify the conitions of Theorem 1 in A[x]. As such a ecomposition is unique, it ens the proof. Example. Set A = K[y] an let P (x) = x 6 + a 1 x 5 + a x 4 + a 3 x 3 + a 4 x + a 5 x + a 6 be a monic polynomial of egree 6 in A[x] = K[x, y], with coefficients a i = a i (y) A = K[y]. In the example of section 4 we have compute the ecomposition P = h Q + R for = an set b 1 = a 1, b = a b 1, b 3 = a 3 b 1 b. We foun h(t) = t +a 6 b 3 A[t]

7 DECOMPOSITION OF POLYNOMIALS AND APPROXIMATE ROOTS 7 an R(x) = (a 4 b 1 b 3 b )x +(a 5 b b 3 )x A[x]. By Proposition 7 above, we get that P is -ecomposable in K[x, y] if an only a 6 b 3 K, a 4 b 1 b 3 b = 0 in K[y], a 5 b b 3 = 0 in K[y]. Each line yiels a system of polynomial equations in the coefficients a ij K of P (x, y) = a ij x i y j K[x, y]. In particular the set of -ecomposable monic polynomials of egree 6 in K[x, y] is an affine algebraic variety. References [1] S.S. Abhyankar, T.T. Moh, Newton-Puiseux expansion an generalize Tschirnhausen transformation. J. Reine Angew. Math. 60 (1973), an 61 (1973), [] S.S. Abhyankar, T.T. Moh, Embeings of the line in the plane. J. Reine Angew. Math. 76 (1975), [3] A.F. Bearon, Composition factors of polynomials. The Chuang special issue. Complex Variables Theory Appl. 43 (001), [4] A. Boin, P. Dèbes, S. Najib, Inecomposable polynomials an their spectrum. Acta Arith. 139 (009), [5] J.v.z. Gathen, Functional ecomposition of polynomials: the tame case. J. Symb. Comp. 9 (1990), [6] D. Kozen, S. Lanau, Polynomial ecomposition algorithms. J. Symb. Comp. 7 (1989), [7] P.R. Lazov, A criterion for polynomial ecomposition. Mat. Bilten 45 (1995), [8] P. Popescu-Pampu, Approximate roots. Valuation theory an its applications, Vol. II (Saskatoon, SK, 1999), Fiels Inst. Commun. 33, Amer. Math. Soc. (003), [9] E.D. Rainville, Necessary conitions for polynomial solutions of certain Riccati equations. Amer. Math. Monthly 43 (1936), [10] A. Schinzel, Polynomials with special regar to reucibility. Encyclopeia of Mathematics an its Applications, 77. Cambrige University Press, Cambrige, 000. Laboratoire Paul Painlevé, Mathématiques, Université e Lille 1, Villeneuve Ascq, France. aress: Arnau.Boin@math.univ-lille1.fr