Barnett s Theorems About the Greatest Common Divisor of Several Univariate Polynomials Through Bezout-like Matrices

Transcription

1 J Symbolic Computation (2002) 34, doi:101006/jsco Available online at on Barnett s Theorems About the Greatest Common Divisor of Several Univariate Polynomials Through Bezout-like Matrices GEMA M DIAZ-TOCA AND LAUREANO GONZALEZ-VEGA Dpto de Matematica Aplicada, Universidad de Murcia, Spain Dpto de Matematicas, Estadistica y Comp, Universidad de Cantabria, Spain This article provides a new presentation of Barnett s theorems giving the degree (resp coefficients) of the greatest common divisor of several univariate polynomials with coefficients in an integral domain by means of the rank (resp linear dependencies of the columns) of several Bezout-like matrices This new presentation uses Bezout or hybrid Bezout matrices instead of polynomials evaluated in a companion matrix as in the original Barnett s presentation Moreover, this presentation also allows us to compute the coefficients of the considered greatest common divisor in an easier way than in the original Barnett s theorems c 2002 Elsevier Science Ltd All rights reserved Introduction Let F be a field of characteristic zero Let {A, B 1,, B t } be a family of polynomials in F[x] with A monic and n = deg(a) > deg(b j ) for every j {1,, t} Barnett s theorem (see Barnett, 1971; Barnett, 1983 or Gonzalez-Vega, 1996) assures that the degree of the greatest common divisor of A, B 1,, B t verifies: deg(gcd(a, B 1,, B t )) = n rank(b 1 ( A ), B 2 ( A ),, B t ( A )) where A is the companion matrix of A: a n a n 1 A = a n a 1 and B j ( A ) denotes the evaluation of the polynomial B j in the matrix A Moreover, some linear algebra over the matrix giving the degree of the greatest common divisor provides the coefficients of a greatest common divisor in F[x] of A, B 1,, B t Two proofs are proposed in Barnett (1971) for the result concerning the degree of the greatest common divisor: the first one uses the Jordan form of A and the second one is based on a theorem introduced in Barnett (1970) concerning the degree of the gemadiaz@umes gvega@matescounicanes /02/ $3500/0 c 2002 Elsevier Science Ltd All rights reserved

2 60 G M Diaz-Toca and L Gonzalez-Vega greatest common divisor of two invariant factors for two regular polynomial matrices In Gonzalez-Vega (1996) a more elementary proof is proposed by using a few elementary facts in linear algebra and some easy to prove properties concerning subresultants This new proof generalizes Barnett s results to the case where the considered polynomials have their coefficients in an integral domain This paper is devoted to generalizing Barnett s results by replacing the matrices B j ( A ) with others which correspond to the same linear mapping defined by B j ( A ) One of these matrices is the Bezout matrix of A and B j This version of Barnett s results with Bezout matrices is currently being used to solve several problems in computer algebra (namely, parametric squarefree decomposition of univariate polynomials, Diaz-Toca and Gonzalez-Vega, 2001, and efficient quantifier elimination over the reals, Gonzalez-Vega and Gonzalez-Campos, 1999) due to the good specialization behaviour of the Bezout matrix together with its moderate size (in comparison with the evaluation on the companion matrix) This paper is divided into four sections The first and second sections are devoted to presenting Barnett s theorems as introduced in Gonzalez-Vega (1996) through the corresponding translation to linear mappings on some concrete vector spaces and the right basis for these spaces The third section presents Barnett s theorems by using Bezout matrices, the fourth section by using Hankel matrices and the fifth by using hybrid Bezout matrices Finally the last section shows the computational behaviour of all these matrices in terms of theoretical and practical complexity 1 Barnett s Theorems About the Greatest Common Divisor of Several Univariate Polynomials In this section Barnett s theorems about the greatest common divisor of a finite family of univariate polynomials are to be presented following Gonzalez-Vega (1996) First the definition of the (generalized) companion matrix of a univariate polynomial and its associated linear mapping are introduced Let D be an integral domain of characteristic zero and F be its quotient field Definition 11 Let P (x) be a polynomial in D[x]: P (x) = x n + p 1 x n p n, 0 The (generalized) companion matrix of P (x) is defined by: p n 0 0 p n 1 P = 0 0 p n p 1 If F n [x] is the F-vector space of polynomials in F[x] with degree smaller than n then the matrix P is the matrix associated to the endomorphism Φ P : F n [x] F n [x] U rem( xu, P )

3 Barnett s Theorems 61 with respect to the so-called standard basis of F n [x]: Definition 12 Let B St = {1, x, x 2,, x n 1 } P (x) = x n + p 1 x n p n 1 x + p n, Q(x) = q 1 x n q n 1 x + q n, be two polynomials in D[x] with 0 and m = deg(q) n 1 Then, the polynomial Q(x) in D[x] is defined by: ( ) x Q(x) = p m 0 Q and Let P (x) = x n + p 1 x n p n 1 x + p n Q j (x) = q j,1 x n q j,n 1 x + q j,n, j {1,, t}, be polynomials in D[x] with 0 and m j = deg(q j ) n 1 for every j {1,, t} Next, Barnett s theorems are presented Theorem 11 (Barnett s Theorem I) The degree of the greatest common divisor of P (x), Q 1 (x),, Q t (x) verifies the following formula: deg(gcd(p, Q 1,, Q t )) = n rank(q P (Q 1,, Q t )) where Q 1 ( P ) Q 2 ( P ) Q P (Q 1,, Q t ) = Q t ( P ) and P denotes the (generalized) companion matrix associated to P Note that the matrix Q( P ) is the matrix associated to the endomorphism (m = deg(q)): Φ Q P : F n[x] F n [x] U rem(p m 0 QU, P ) with respect to the standard basis B St Furthermore Q( P ) = p m 0 Q( P/p0 ), (1) where Q( P/p0 ) is the matrix associated to the endomorphism F n [x] F n [x] U rem(qu, P ) with respect to the standard basis B St The next theorem provides information about the linear independence of the columns in the matrix Q P (Q 1,, Q t )

4 62 G M Diaz-Toca and L Gonzalez-Vega Theorem 12 (Barnett s Theorem II) If ρ 1,, ρ n are the columns of the matrix Q P (Q 1,, Q t ) and r is its rank then the columns ρ 1,, ρ r are linearly independent and every ρ r+j (1 j n r) can be written as a linear combination of ρ 1,, ρ r Finally, the next theorem shows how to use the columns of the matrix which gives the degree of the greatest common divisor to obtain its coefficients Theorem 13 (Barnett s Theorem III) If ρ 1,, ρ n are the columns of the matrix Q P (Q 1,, Q t ), r is its rank, δ1 1 δ 1 n r δ = δ1 r δn r r a matrix whose entries (in F) are defined by the relations and then d 0 d 1 d 2 d n r r 1 ρ r+i = δ j i ρ j + δi r ρ r, j=1 p 1 p 2 p 1 = d 0 p n r p n r 1 p n r 2 D(x) = d 0 x n r + d 1 x n r d n r 1 x + d n r 1 δ r 1 δ r 2 δ r n r is a greatest common divisor in F[x] of the polynomials P (x), Q 1 (x),, Q t (x) Remark 11 Barnett s theorems have been presented for a family of polynomials P, Q 1,, Q t such that deg(p ) > deg(q i ) (1 i t) However, the theorems hold for an arbitrary family of polynomials and even if the polynomial with the lowest degree is chosen to be P (x) But if the chosen polynomial P (x) is not the polynomial of the highest degree then the family P, rem(q 1, P ),, rem(q t, P ) is the one actually considered because: mi deg(rem(q,p )) Q i ( P ) = p rem(q i, P )( P ) 0 2 Barnett s Theorems with Respect to Other Basis in F n [x] Let P (x), Q 1 (x),, Q t (x) be a family of polynomials in D[x] with n = deg(p ) and m j = deg(q j ) n 1 for every j {1,, t} Let A and B be two bases of F n [x]: A = {a 0,, a n 1 }, B = {b 0,, b n 1 } If for each j {1,, t}, the linear mapping Φ Qj P is considered taking the basis A in the initial space and the basis B in the final one, the next diagram of linear mappings is

5 Barnett s Theorems 63 obtained: Q j ( P ) where B St B St A S SB A B Q j (Λ P ) A S is the basis change matrix of A to B St, S B is the basis change matrix of B St to B, Λ P is the matrix of Φ P with respect to the basis A and B, Q j (Λ P ) is the matrix of Φ Qj P with respect to the basis A and B Note that the columns of the matrix A S are the coefficients of the polynomials which define the basis A AND that the matrix Q j (Λ P ) could not be the polynomial Qj is evaluated in Λ P, only if the bases A and B are the same, that is, only if the linear mapping Φ Qj P keeps being an endomorphism Proposition 21 The degree of the greatest common divisor of P, Q 1,, Q t verifies the following formula: where deg(gcd(p, Q 1,, Q t )) = n rank(q P (Q 1,, Q t )) Q 1 (Λ P ) Q Q 2 (Λ P ) P (Q 1,, Q t ) = Q t (Λ P ) Proof Since A S and S B are non-singular and then: S B Q P (Q 1,, Q t ) = S B = Thus, it follows from Theorem 11 that S B S B Q 1 ( P ) A S Q t ( P ) Q P (Q 1,, Q t )A S, rank(q P (Q 1,, Q t )) = rank(q P (Q 1,, Q t )) deg(gcd(p, Q 1,, Q t )) = n rank(q P (Q 1,, Q t )) The linear independence of the columns of Q P (Q 1,, Q t ) can also be characterized in some cases Proposition 22 If the polynomials which define the basis A provide a triangular basis arranged in increasing degree order (0 j n 1): a j (x) = a (0) j x j + + a (j) j, a (0) j 0, (2)

6 64 G M Diaz-Toca and L Gonzalez-Vega c 1,, c n are the columns of the matrix Q P (Q 1,, Q t ) and r is its rank, then the columns c 1,, c r are linearly independent and every c r+j (1 j n r) can be written as a linear combination of c 1,, c r Proof Since A = {a 0,, a n 1 } with a j (x) = a (0) j upper triangular: So, A S = S B Q P (Q 1,, Q t ) = By hypothesis, which implies that a (0) 0 a (0) n 1 a (n 1) n 1 S B x j + + a (j) j, the matrix A S is a (0) 0 a (0) n 1 Q P (Q 1,, Q t ) a (n 1) n 1 rank(q P (Q 1,, Q t )) = r, rank(q P (Q 1,, Q t )) = r Thus, if ρ 1,, ρ n are the columns of Q P (Q 1,, Q t ), ρ 1,, ρ r are linearly independent then, since A S is an upper triangular matrix, the first r columns of Q P (Q 1,, Q t )A S are linearly independent In other words, multiplying times A S to the right keeps the linear independence between the columns The proof is finished by noting that if the first r columns of Q P (Q 1,, Q t )A S are linearly independent then the first r columns of Q P (Q 1,, Q t ) are also linearly independent Proposition 23 If the polynomials which define the basis A provide a triangular basis arranged in decreasing degree order (0 j n 1): a j (x) = a (0) j x n j a (n j 1) j, a (0) j 0, c 1,, c n are the columns of the matrix Q P (Q 1,, Q t ) and r is its rank then the last r columns c n r+1,, c n are linearly independent and each c i (1 i n r) can be written as a linear combination of c n r+1,, c n Proof Since A = {a 0,, a n 1 } with a j (x) = a (0) j x n j a (n j 1) j matrix A S is triangular with shape: So, A S = S B Q P (Q 1,, Q t ) = a (0) 0 a (0) a (n 1) 0 S B n 1 Q P (Q 1,, Q t ) a (0) 0 a (0) a (n 1) 0 n 1 then the

7 By hypothesis, which implies that rank(q P (Q 1,, Q t )) = r, rank(q P (Q 1,, Q t )) = r Barnett s Theorems 65 Thus, if ρ 1,, ρ n are the columns of Q P (Q 1,, Q t ) and ρ 1,, ρ r are linearly independent then, because of the shape of A S which implies that multiplying times A S to the right reverses the relation of linear independence between the columns, the last r columns of Q P (Q 1,, Q t )A S are linearly independent The proof is finished by noting that if the last r columns of Q P (Q 1,, Q t )A S are linearly independent then the last r columns of Q P (Q 1,, Q t ) are also linearly independent In order to generalize Barnett s theorem III it is required to have a more concrete knowledge (and control) of the considered basis which is done into the next sections 3 Using Bezout Matrices to Formulate Barnett s Theorems Given a polynomial P (x) = x n + p 1 x n p n 1 x + p n, 0, the Horner basis of F n [x] associated to P (x), denoted by B Ho and also called control basis, is defined by with: B Ho = {α 1 (x),, α n (x)} α i (x) = x n i + + p n i 1 x + p n i Following the notation introduced in Section 2, if then diagram (2) specializes to: where A = B Ho, B = B St, Q j ( P ) B St B St H S SS B Ho B St Q j (Λ P ) the basis change matrix of B Ho to B St is p n 1 p n 2 p 1 p 0 p n 2 p n 3 0 H S =, p S S = I n,

8 66 G M Diaz-Toca and L Gonzalez-Vega the matrix of Φ P with respect to the basis B Ho and B St is the matrix of Φ Qj P Note that t P Λ P = P H S, with respect to the basis B Ho and B St is Q j (Λ P ) = Q j ( P ) H S (3) is the matrix associated to the endomorphism Φ P : F n [x] F n [x] U rem( xu, P ) with respect to the Horner basis B Ho which implies that Q j ( t P ) is the matrix associated to the endomorphism Φ Qj P with respect to the basis B Ho Moreover, Q j ( t P ) = p m 0 Q j ( t P/ ), (4) where t P/ is the matrix associated to the endomorphism F n [x] F n [x] U rem(q j U, P ) with respect to the Horner basis B Ho Next, following Helmke and Fuhrmann (1989) or Krein and Naimark (1981), the most general definition of a Bezout matrix of two univariate polynomials is introduced (see Heinig and Rost, 1984 for a definition for four univariate polynomials with one additional hypothesis): Definition 31 If P (x) and Q(x) are polynomials in D[x] such that d = max{deg(p ), deg(q)} then the Bezout matrix associated to P (x) and Q(x) is: c 0,0 c 0,d 1 Bez(P, Q) = c d 1,0 c d 1,d 1 where the c i,j are defined by the formula: P (x)q(y) P (y)q(x) x y = d 1 i,j=0 c i,j x i y j The Bezoutian associated to P (x) and Q(x) is defined as the determinant of the matrix Bez(P, Q) and it will be denoted by bez(p, Q) For instance, if Q(x) = 1 then the matrix Bez(P, 1) is: p n 1 p n 2 p 1 p 0 p n 2 p n 3 0 Bez(P, 1) = p and note that Bez(P, 1) is the basis change matrix of B Ho to B St

9 Barnett s Theorems 67 The following theorem summarizes the main properties of the Bezout matrix and its determinant (see Barnett, 1983; Helmke and Fuhrmann, 1989 or Mignotte, 1992 for a proof) Theorem 31 Let P and Q be two polynomials in D[x] with n = deg(p ) m = deg(q) and Then: P = x n + p 1 x n p n 1 x + p n bez(p, Q) = 0 if and only if deg(gcd(p, Q)) 1 Barnett factorization: p m 0 Bez(P, Q) = Q( P ) Bez(P, 1) = p m 0 Q( P/p0 ) Bez(P, 1) bez(p, Q) = ( 1) n(n 1)/2 p n m 0 Resultant(P, Q) = Bez(P, 1) Q( t P ) = p m 0 Bez(P, 1) Q( t P/ ) (5) Notice that Barnett factorization and equation (3) imply that p m 0 Bez(P, Q) = Q(Λ P ) In other words, p m 0 Bez(P, Q) is the matrix associated to the linear mapping Φ Q P considering the Horner basis in the initial vector space and the standard basis in the final one 31 barnett s theorems through bezout matrices Since p m 0 Bez(P, Q) is the matrix associated to the linear mapping Φ Q P considering the Horner basis in the initial vector space and the standard basis in the final one, following Section 2, Barnett s theorems I and II (Theorems 11 and 12) can be rewritten with Bezout matrices (note that the polynomials which define B Ho are arranged in decreasing degree order) Theorem 32 The degree of the greatest common divisor of P (x), Q 1 (x),, Q t (x) verifies the following formula: where deg(gcd(p, Q 1,, Q t )) = n rank(b P (Q 1,, Q t )) Bez(P, Q 1 ) B P (Q 1,, Q t ) = Bez(P, Q t ) Theorem 33 If c 1,, c n are the columns of the matrix B P (Q 1,, Q t ) and its rank is n k then the last n k columns c k+1,, c n are linearly independent and each c i (1 i k) can be written as a linear combination of c k+1,, c n Finally it is shown how to use the matrix B P (Q 1,, Q t ) in order to get the coefficients of the greatest common divisor of P (x), Q 1 (x),, Q t (x) The resulting formula is simpler than the one presented in Theorem 13 (note moreover that the entries in B P (Q 1,, Q t ) are simpler than those in Q P (Q 1,, Q t ))

10 68 G M Diaz-Toca and L Gonzalez-Vega Theorem 34 If c 1,, c n are the columns of the matrix B P (Q 1,, Q t ), n k is its rank, n c k i = h k+1 k i c k+1 + h j k i c j, i = 0,, k 1, {d 1,, d k } given by and d 0 F then: j=k+2 d j = d 0 h k+1 k j+1 D(x) = d 0 x k + d 1 x k d k 1 x + d k is a greatest common divisor for the polynomials P (x), Q 1 (x),, Q t (x) Proof The desired equality is d 0 d 1 = d 0 d k 1 h k+1 k h k+1 1 (6) By Barnett factorization, it follows that the matrix BH P (Q 1,, Q t ) is equal to: Bez(P, Q 1 ) 1/p m1 0 Q 1 ( P ) = Bez(P, 1) Bez(P, Q t ) 1/p mt 0 Q t ( P ) and since Q j ( P ) = p mj 0 Q j( P/p0 ), we have: Bez(P, Q 1 ) Q 1 ( P/p0 ) = Bez(P, Q t ) Q t ( P/p0 ) Bez(P, 1) Hence, if ρ 1,, ρ n denote the columns of Q P/p0 (Q 1,, Q t ) then its first n k columns are linearly independent and, by Barnett s theorem III (Theorem 13), the coefficients of D(x) = d 0 x k + d 1 x k d k 1 x + d k are given by where d 0 1 d 1 = d 0 p 1 δ n k 1, d k p k p k 1 δ n k k ρ n k+i = n k 1 j=1 δ j i ρ j + δ n k i ρ n k

11 Thus if the equality Barnett s Theorems h k+1 k = p 1 δ n k 1 (7) h k+1 1 p k p k 1 δ n k k holds, the equality (6) will follow directly But proving (7) requires proving: h k+1 j = p k j+1 + p k j δ n k 1 + p k j 1 δ n k δ n k k j+1, for every k j 1 The relations between the columns of Q P/p0 (Q 1,, Q t ) and B P (Q 1,, Q t ) are given by: or more precisely: Hence, if n k ρ j p n k+1 j + j=1 Bez(P, Q 1 ) Bez(P, Q t ) = Q 1 ( P/p0 ) Q t ( P/p0 ) Bez(P, 1) (c 1,, c k, c k+1,, c n ) = (ρ 1,, ρ n k, ρ n k+1,, ρ n )Bez(P, 1) c n = ρ 1 c n 1 = ρ 1 p 1 + ρ 2 c k+1 = ρ 1 p n k ρ n k c k = ρ 1 p n k + + ρ n k p 1 + ρ n k+1 c 1 = ρ 1 p n ρ n 1 p 1 + ρ n ( n k 1 j=1 n k+1 j=1 c k = h k+1 k c k+1 + ρ j p n k+1 j = h k+1 k ) δ j 1 ρ j + δ1 n k ρ n k = h k+1 k ( n k n j=k+2 h j k c j ) ρ j p n k j + j=1 ( n k ) ρ j p n k j + then (looking at the coefficients of ρ n k at both sides of the equality): p 1 + δ n k 1 = h k+1 k Using the same argument, for an arbitrary j with j < k: c j = h k+1 j c k+1 + n i=k+2 h i jc i = n j+1 i=1 j=1 ρ i p n j+1 i n j=k+2 n j=k+2 h j k c j h j k c j

12 70 G M Diaz-Toca and L Gonzalez-Vega provides: h k+1 j obtaining ( n k as desired ) ρ i p n k i + i=1 n i=k+2 n k = ρ i p n j+1 i + i=1 ( n k 1 + i=1 h k+1 j h i j(ρ 1 p n i + + ρ n i+1 ) ( n k 1 i=1 δ i k jρ i + δ n k k j ρ n k δ i 1ρ i + δ n k 1 ρ n k )p k j + ) p 1 + ( n k 1 i=1 δ i k j+1ρ i + δ n k k j+1 ρ n k = p k j+1 + p k j δ n k p 1 δ n k k j + δ n k k j+1 ) 4 Using Hankel Matrices to Formulate Barnett s Theorems Following the notation of Section 2, if then diagram (2) specializes to: where: A = B St, B = B Ho, Q j ( P ) B St B St I n SH B St B Ho Q j (Λ P ) the basis change matrix of B St to B Ho is Bez(P, 1) 1, the matrix of Φ Qj P with respect to the basis B Ho and B St is and by Barnett factorization it follows: 1 p m 0 Q j (Λ P ) = Bez(P, 1) 1 Qj ( P ), Q j (Λ P ) = Bez(P, 1) 1 Bez(P, Q j )Bez(P, 1) 1 (8) There is a well-known matrix verifying property (8): the Hankel matrix of P (x) and Q j (x) which is presented in the next definition Definition 41 Given P (x) and Q(x) in D[x] such that deg(q) = m < n = deg(p ), the Hankel matrix of P (x) and Q(x) is defined by: h 1 h 2 h n h 2 h 3 h n+1 H(P, Q) = h n h n+1 h 2n 1

13 Barnett s Theorems 71 where the entries (in F) are given by the power series expansion of the rational function at infinity, ie Q(x) P (x) R(x) = Q(x) P (x) = h i x i This matrix and its main properties can be found, for example, in Bini and Pan (1994) and Helmke and Fuhrmann (1989) Theorem 41 Given P (x) and Q(x) in D[x] such that n = deg(p ) > m = deg(q) and Then: i=1 P (x) = x n + p 1 x n p n 1 x + p n 1 H(P, Q) = Bez(P, 1) 1 Bez(P, Q)Bez(P, 1) 1 2 rank(bez(p, Q)) = rank(h(p, Q)) 3 deg(gcd(p, Q)) = i i = n rank(h(p, Q)) Since p m 0 H(P, Q) is exactly the matrix of Φ Q P with respect to B St and B Ho, Barnett s theorems can be rewritten with Hankel matrices 41 barnett s theorems through hankel matrices Following Section 2, Barnett s theorems I and II (Theorems 11 and 12) can be rewritten with Hankel matrices (note that the polynomials which define B St are arranged in increasing degree order) Theorem 42 The degree of the greatest common divisor of P (x), Q 1 (x),, Q t (x) verifies the following formula: where deg(gcd(p, Q 1,, Q t )) = n rank(h P (Q 1,, Q t )) H(P, Q 1 ) H P (Q 1,, Q t ) = H(P, Q t ) Theorem 43 If c 1,, c n are the columns of the matrix H P (Q 1,, Q t ) and its rank is n k then the first n k columns c 1,, c n k are linearly independent and each c n k+i (1 i k) can be written as a linear combination of c 1,, c n k Finally it is shown how to use the matrix H P (Q 1,, Q t ) in order to get the coefficients of the greatest common divisor of P (x), Q 1 (x),, Q t (x) Theorem 44 If c 1,, c n are the columns of the matrix H P (Q 1,, Q t ), n k is its rank, h 1 1 h 1 k h = h n k 1 h n k k

14 72 G M Diaz-Toca and L Gonzalez-Vega a matrix whose entries (in F) are defined by the relations c n k+i = n k 1 j=1 h j i c j + h n k i c n k, and then: d 0 1 d 1 = d p 1 h n k 0 1 d k p k p k 1 h n k k D(x) = d 0 x k + d 1 x k d k 1 x + d k is a greatest common divisor in F[x] of the polynomials P (x), Q 1 (x),, Q t (x) Proof Let {ρ 1,, ρ n } denote the columns of Q P/p0 (Q 1,, Q t ) The equation H P (Q 1,, Q t ) = implies that (i = 1,, k): Bez(P, 1) 1 Bez(P, 1) 1 n k n k c n k+i = h j i c j ρ n k+i = h j i ρ j j=1 j=1 Q P/p0 (Q 1,, Q t ), (9) and, by Barnett s theorem III (Theorem 13), the desired result is obtained 5 Using Hybrid Bezout Matrices to Formulate Barnett s Theorems Given two polynomials P (x) = x n + p 1 x n p n 1 x + p n, 0, Q(x) = q 0 x m + q 1 x m q m 1 x + q m, q 0 0, with n m, the reversed Horner basis of F n [x] associated to P (x) and denoted by B Ho, is defined by: B Ho and the basis T n,m of F n [x] is defined by: B Tn,m Following the notation of Section 2, if = {α n (x),, α 1 (x)} = {1, x, x 2,, x m 1, α n m (x),, α 1 (x)} A = B Ho, B = B Tn,m,

15 Barnett s Theorems 73 then diagram (2) specializes to: H S Q j ( P ) B St B St S Tn,m B Ho B Tn,m Q j (Λ P ) where the basis change matrix of B Ho to B St is p 1 p n 2 p n 1 0 p n 3 p n 2 HS =, 0 0 p S Tn,m = 1 p m p n 1 1 p 1 p n m p n m 1 the matrix of Φ P with respect to the basis B Ho and B Tn,m is 1 Λ P = S Tn,m P H S, the matrix of Φ Qj P with respect to the basis B Ho and B T n,m is, Q j (Λ P ) = S Tn,m Qj ( P ) H S (10) Next the definition of a hybrid Bezout matrix is introduced This matrix is defined as a Bezout matrix in some books and articles, see for example Griss (1978), Wang (2000) and Zippel (1952) The Computer Algebra System Maple also calls this matrix a Bezout matrix (see LinearAlgebra[BezoutMatrix] or linalg[bezout]) Definition 51 Given P (x) and Q(x) in D[x], deg(p ) = n deg(b) = m, the hybrid Bezout matrix associated to P (x) and Q(x), denoted by Hbez(P, Q), is a square matrix of size n whose entries are defined by: for 1 i m, 1 j n, the (i, j)-entry is the coefficient of x n j in the polynomial K m i+1 = ( x m i + + p m i )(q m i+1 x n m+i q m x n m ) (p m i+1 x n m+i p n )(q 0 x m i + + q m i ); for m + 1 i n, 1 j n, the (i, j)-entry is the coefficient of x n j in the polynomial x n i Q(x)

16 74 G M Diaz-Toca and L Gonzalez-Vega The next proposition and its corollary provide a factorization of Hbez(P, Q) which makes it possible to rewrite Barnett s theorems with hybrid Bezout matrices Proposition 51 Let P (x), Q(x) D[x] such that n = deg(p ) m = deg(q) and P (x) = x n + p 1 x n p n, Q(x) = q 0 x m + q 1 x m q m Let the Sylvester matrix of P (x) and Q(x), be represented in the form where and Then: Sylv(P, Q) = n+m { }} { p n p n q m q 0 q m q 0 ( T1 T Sylv(P, Q) = 2 T 3 T 4 ), T 1 M m (D), T 2 M m,n (D), T 3 M n,m (D), T 4 M n (D), Proof Since T 1 is equal to: ( T1 0 T = m,n m 0 n m,m I n m ) Hbez(P, Q) = T T 4 T T 3 T 1 1 T 2 T 1 = p m 1 and since 0, T 1 is non-singular Furthermore, the rest of the blocks in the partition of Sylv(P, Q) are: q p m q m 1 n T 2 =, T 3 = q, p 1 n 0 n m,n q ṃ q T 4 = 1 q m q 0 q m 1 q m q 0 q m n m,

17 Thus T T 4 T T 3 T1 1 T 2 is equal to: q ṃ T 1 0 m,n m q 1 q m T 1 q 0 q m Barnett s Theorems 75 q 0 q m 1 q 0 q m Then the last n m rows of Hbez(P, Q) are exactly the last n m rows of T T 4 T T 3 T 1 1 T 2 Regarding the first m rows of Hbez(P, Q), since q 0 q m 1 p m 1 T 1 = we have: = q 0 q 0 q m 1 q 0 p m 1 = q 0 0 n m,n q 0 q m 1 q 0 T 1 q 0 q m 1 q 0 q m 0 0 q 0 q m 1 T 1 T 1 q 1 q m 0 0 q 0 q m 0 0 q 0 q m 1 = T 1 q 1 q m 0 0 q 0 p m 1 q m 0 0 = q 1 q m 0 0 q 0 q m 1 p m p n q 0 p 1 p n 1 T 2 T 1, T 1 1 T 2 T 2 So the ith row (1 i m) is: q m 0 0 ( 0 0 p m i ) q 1 q m 0 0 p m p n ( 0 0 q 0 q m i ), p 1 p n

18 76 G M Diaz-Toca and L Gonzalez-Vega a matrix whose entries are the coefficients of the polynomial K m i+1 defined by: K m i+1 = ( x m i + + p m i )(q m i+1 x n (m i+1) + + q m x n m ) Thus we can conclude that: Note that the matrix (q 0 x m i + + q m i )(p m i+1 x n (m i+1) + + p n ) Hbez(P, Q) = T T 4 T T 3 T 1 1 T 2 p m T = is the basis change matrix of B Ho to B Tn,m Lemma 51 Following the notation of Proposition 51, ( ) ( ) ( ) Im 0 T1 T T 3 T1 1 2 T1 T 2 = I n T 3 T 4 0 J n Q( t, (11) P/ )J n where J n = 1 1 Proof See Barnett (1983) or Mignotte (1992) Note that equality (11) is equivalent to: J n Q( t P/ ) J n = T 3 T 1 1 T 2 + T 4 Corollary 51 Hbez(P, Q) = T J n Q( t P/ ) J n Proof By Proposition 51 and Lemma 51, it follows that as desired Hbez(P, Q) = T T 4 T T 3 T 1 1 T 2 = T J n Q( t P/ ) J n The last results provide the following properties of Hbez(P, Q) Corollary 52 Let P (x), Q(x) D[x] such that n = deg(p ) m = deg(q) and P (x) = x n + p 1 x n p n 1 x + p n

19 Barnett s Theorems 77 Then: det(hbez(p, Q)) = Resultante(P, Q) If n = m then Bez(P, Q) = Hbez(P, Q) J n Corollary 53 The matrix p m 0 HBez(P, Q) is the matrix associated to the linear mapping Φ Q P considering the reversed Horner basis in the initial vector space and the basis T n,m in the final one Proof According to equality (10), the matrix Q(Λ P ) represents Φ Q P considering B Ho in the initial vector space and B Tn,m in the final one and it is given by: This equality implies that if Q(Λ P ) = S Tn,m Q( P ) H S p m 0 HBez(P, Q) = S Tn,m Q( P ) H S, (12) then the statement of the corollary is satisfied Since the matrix T is the basis change matrix of B Ho to B Tn,m, according to the diagram: HS B Ho B St T S Tn,m B Tn,m the matrix T can be factorized as T = S Tn,m H S Hence, equality (12) follows from p m 0 HBez(P, Q) = p m 0 T J n Q( t P/ )J n = S Tn,m H SJ n Q( t P )J n = S Tn,m Bez(P, 1) Q( t P )J n = S Tn,m Q( P )Bez(P, 1)J n = S Tn,m Q( P )H SJ n J n = S Tn,m Q( P )H S The main consequence of this corollary is that it is possible to rewrite Barnett s theorems with the hybrid Bezout matrix 51 barnett s theorems through hybrid bezout matrices Since p m 0 Hbez(P, Q) is the matrix associated to the linear mapping Φ Q P considering the reversed Horner basis in the initial vector space and the basis T n,m in the final one, following Section 2, Barnett s theorems I and II (Theorems 11 and 12) can be rewritten with hybrid Bezout matrices (note that the polynomials which define B Ho are arranged in increasing degree) Theorem 51 (Barnett s Theorem I) The degree of the greatest common divisor of P (x), Q 1 (x),, Q t (x) verifies the following formula: deg(gcd(p, Q 1,, Q t )) = n rank(bh P (Q 1,, Q t ))

20 78 G M Diaz-Toca and L Gonzalez-Vega where Hbez(P, Q 1 ) BH P (Q 1,, Q t ) = Hbez(P, Q t ) Theorem 52 If c 1,, c n are the columns of the matrix BH P (Q 1,, Q t ) and its rank is r then the first r columns c 1,, c r are linearly independent and each c r+i (1 i n r) can be written as a linear combination of c 1,, c r Finally, the next theorem shows how to use the columns of BH P (Q 1,, Q t ) in order to obtain the coefficients of the greatest common divisor Theorem 53 If c 1,, c n are the columns of the matrix BH P (Q 1,, Q t ), r is its rank, r 1 c r+i = h j i c j + h r i c r, i = 1,, n r, j=1 and {d 1,, d k } given by d j = d 0 h r j, d 0 F, then: D(x) = d 0 x n r + d 1 x n r d n r 1 x + d n r is a greatest common divisor of the polynomials P (x), Q 1 (x),, Q t (x) Proof Note that Hbez(P, Q j ) = S Tn,mj Q j ( P/p0 )Bez(P, 1)J n and so BH P (Q 1,, Q t ) = S Tn,m1 B P (Q 1,, Q t )J n S Tn,mt Moreover, multiplying B P (Q 1,, Q t ) times J n to the right implies that the columns of B P (Q 1,, Q t ) are reversed Hence, the relations between the columns of B P (Q 1,, Q t ) are the same relations between the columns of BH P (Q 1,, Q t ) in reverse order Therefore the assertion follows from Theorem 34 6 Theoretical and Practical Complexity Analysis This section shows the algorithm GCD for computing the greatest common divisor of a finite family of polynomials by using Barnett s theorems through the different matrices introduced in the previous sections together with its theoretical and practical complexity

21 Barnett s Theorems 79 Algorithm (GCD) Input: {P (x), Q 1 (x),, Q t (x)} in D[x] with n = deg(p ) > deg(q i ) = m i Output: gcd(p, Q 1,, Q t ) in F[x] (GCD1): Compute A P (Q 1,, Q t ), where A P (Q 1,, Q t ) is one of the following matrices: Q P (Q 1,, Q t ) B P (Q 1,, Q t ) H P (Q 1,, Q t ) BH P (Q 1,, Q t ) (GCD2): Compute the rank of A P (Q 1,, Q t ) (GCD3): Compute the coefficients of the gcd using either Theorem 13 or its different versions, depending on the chosen matrix A P (Q 1,, Q t ) 61 theoretical analysis Table 1 shows an estimation of the complexity of the algorithm GCD For the case of integer coefficients, the size of an integer number is defined as the base 2 logarithm of its absolute value Let M be a bound for the size of the coefficients of the polynomials in the case of integer coefficients 62 practical analysis Table 2 shows the computing time (in seconds) and the memory (in megabytes) required to compute the greatest common divisor of 12 families of random polynomials, in Z[x], with Barnett s theorems and by using the different matrices introduced in the previous sections Furthermore this table shows the maximum of the degrees of the considered polynomials and the degree of the greatest common divisor The algorithm GCD has been implemented in the Computer Algebra System Maple providing a greatest common divisor of P (x), Q 1 (x),, Q t (x) A 266 MHz Pentium II PC with 64 MB Ram has been used to perform the computations 63 conclusions Tables 1 and 2 show the bad behaviour of the matrix Q P (Q 1,, Q t ) because of the size of its entries when the polynomials are in Z[x] As for the other matrices, the matrix BH P (Q 1,, Q t ) presents the best behaviour The computational behaviour of the matrices BH P (Q 1,, Q t ) and B P (Q 1,, Q t ) is very close, but when the difference between the degrees of used polynomials is appreciable, computing BH P (Q 1,, Q t ) is much faster than computing B P (Q 1,, Q t )

22 80 G M Diaz-Toca and L Gonzalez-Vega Table 1 Required arithmetic operations (GCD1) In Entries sizes when D = Z (GCD2) (GCD3) Q P O(tn 2 ) D O(Mn) O(tn(n k) 2 ) O(kn 2 ) B P O(tn 2 ) D O(log 2 n + M) O(tn(n k) 2 ) O(kn 2 ) BH P O(tn 2 ) D O(log 2 n + M) O(tn(n k) 2 ) O(kn 2 ) H P O(tn 2 ) F O(n(log 2 n + M)) O(tn(n k) 2 ) O(kn 2 ) Table 2 Practical analysis Ej Q P B P BH P H P Max degree = 10 1 t = deg(gcd) = MB 1789 MB 1834 MB 1834 MB 2 t = deg(gcd) = MB 1789 MB 1834 MB 1834 MB 3 t = deg(gcd) = MB 1789 MB 1834 MB 1834 MB Max degree = 20 1 t = deg(gcd) = MB 2162 MB 2162 MB 2293 MB 2 t = deg(gcd) = MB 2162 MB 2227 MB 2358 MB 3 t = deg(gcd) = MB 2358 MB 2358 MB 2489 MB Max degree = 30 1 t = deg(gcd) = MB 2883 MB 3014 MB 2883 MB 2 t = deg(gcd) = MB 3014 MB 3145 MB 3472 MB 3 t = 7 1 h gcd = MB 3276 MB 2752 MB 2948 MB Max degree = 40 1 t = deg(gcd) = MB 2752 MB 2817 MB 3079 MB 3 t = deg(gcd) = MB 4390 MB 4445 MB 5045 MB 2 t = 7 3 h deg(gcd) = MB 4128 MB 4259 MB 5372 MB Finally, when polynomials P, Q 1,, Q t are in Z[a, b][x], computing H P (Q 1,, Q t ) involves working in Q(a, b) and this implies a high cost Acknowledgement This work was partially supported by DGESIC PB C02-02 References Barnett, S (1970) Degrees of greatest common divisors of invariant factors of two regular polynomial matrices Proc Camb Phil Soc, 66, Barnett, S (1971) Greatest common divisor of several polynomials Proc Camb Phil Soc, 70,

23 Barnett s Theorems 81 Barnett, S (1983) Polynomials and Linear Control Systems, Marcel Dekker Bini, D, Pan, V (1994) Polynomial and matrix computations Fundamental Algorithms, volume 1 Progress in Theoretical Computer Science, Birkhäuser Diaz-Toca, G M, Gonzalez-Vega, L (2001) Squarefree decomposition of univariate polynomials depending on a parameter J Symb Comput, 32, Gonzalez-Vega, L (1996) An elementary proof of Barnett s theorem about the greatest common divisor of several univariate polynomials Linear Algebr Appl, 247, Gonzalez-Vega, L, Gonzalez-Campos, N (1999) Simultaneous elimination by using several tools from real algebraic geometry J Symb Comput, 28, Griss, M L (1978) Using an efficient sparse minor expansion Algorithm to compute polynomial subresultants and the greatest common denominator IEEE Trans Comput, c-27, Heinig, G, Rost, K (1984) Algebraic methods for Toeplitz-like matrices and operators, Operator Theory, volume 13, Birkäuser Helmke, U, Fuhrmann, P A (1989) Bezoutians Linear Algebr Appl, 122/123/124, Krein, M G, Naimark, M A (1981) The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations Linear Multilinear Algebr, 10, (the original Russian version was published in 1936) Mignotte, M (1992) Mathematics for Computer Algebra, Universitext, Springer Wang, D (2000) Subresultants with the Bezout Matrix In Gao, X-S, Wang, D eds, Computer Mathematics: Proceedings of the Fourth Asian Symposium (ASCM 2000), pp Zippel, R (1952) Effective Polynomial Computation, Kluwer Academic Publishers Group Received 31 December 2001 Accepted 13 March 2002