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A new approach for constructing subresultants
Applied Mathematics and Computation 183 (2006) 471–476 www.elsevier.com/locate/amc
A new approach for constructing subresultants Yong-Bin Li Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu, Sichuan 610041, China School of Applied Mathematic, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
Abstract Subresultants have many applications including the computation of polynomial GCD and in algebraic cell decomposition. This paper presents a direct method to construct subresultants of two polynomials. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Polynomial; Sylvester matrix; Determinant; Subresultants
1. Introduction and notations The subresultants of two polynomials have been extensively studied, for example, in Brown and Traub [1], Collins [2], Lombardi et al. [3], Abdeljaoued et al. [4], Loos [5] and Mishra [7]. In recent time, Sylvester resultant and subresultants have received much attention both as the elimination theory as well as for the computational efficiency of various constructive algebraic algorithms. It has been widely recognized that forming subresultant chains is one of the most efficient ways to compute polynomial remainder sequence (PRS) [7,9]. By virtue of subresultants, many interesting results and algorithms in resultant-based elimination are developed [8–10]. Let the ring R be restricted to a unique factorization domain (UFD). The ring of univariate polynomials in the indeterminate x with coefficients in R is denoted by R[x]. Let F and G be two univariate polynomials in R[x] and of respective degrees m and n in x with m P n > 0, written as F ¼ a0 xm þ a1 xm1 þ þ am ; G ¼ b0 xn þ b1 xn1 þ þ bn : ð1Þ We form a matrix of dimension m + n by m + n, call the Sylvester matrix of F and G with respect to x, as follows: 19 0 a0 a1 am = .. .. .. C n B C; B . . . C B C9 B a a a 0 1 m C B : M ¼B C> b b b 1 n C= B 0 C m B .. .. .. A> @ . . . ; b 0 b1 bn E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.120
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The determinant of the Sylvester matrix M is called the Sylvester resultant or eliminant of F and G with respect to x, denoted by res(F, G, x). It is well known that if one of a0 and b0 is nonzero, res(F, G, x) = 0 is a necessary and sufficient condition of F and G to have a common zero. Now let Mi,j be the submatrix of M obtained by deleting the last j of n rows of F coefficients, the last j of the m rows of G coefficients, the last 2j + 1 columns, excepting column m + n i j, for 0 6 i 6 j < n. Definition 1.1. The polynomial j X S j ðxÞ ¼ detðMi;j Þxi i¼0
is called the jth subresultant of F and G with respect to x, for 0 6 j < n. Here deg(Sj(x), x) 6 j, and Rj = det(Mj,j) is called the jth principal subresultant coefficient (PSC) or the jth resultant of F and G with respect to x. If m > n + 1, the definition of the jth subresultant Sj(x) and PSC Rj of F and G with respect to x is extended as follows: S n ðxÞ ¼ b0mn1 G; Rn ¼ bmn ; 0
S j ðxÞ ¼ Rj ¼ 0; n < j < m 1:
Sj(x) is said to be defective of degree r if deg(Sj(x), x) = r < j, and regular otherwise. It is easy to see that S0(x) is the resultant of F and G with respect to x. The following proposition provides a direct method for constructing subresultants through computing the determinants of square matrixes [5]. Proposition 1.1. The jth subresultant of F and G can be determined by the determinant of the following square matrix for any 0 6 j < n, 19 0 a0 a1 am xnj1 F > > > B > a0 a1 am xnj2 F C C= B C B .. .. .. .. C>n j B C> B . . . . C> B > C; B a1 amj1 F C B : C9 B B b0 b1 bn xmj1 G C > C> B > > B b0 b1 bn xmj2 G C C= B C B .. .. .. .. C>m j B A> @ . . . . > > ; b1 bnj1 G Definition 1.2. Let F and G be two polynomials in R[x] with m = deg(F, x) P deg(G, x) = n > 0 and m 1 if m > n; l¼ n otherwise: Let Sl+1(x) = F, Sl(x) = G, and Sj (x) be the jth subresultant of F and G with respect to x for 0 6 j < l. The sequence of polynomials S lþ1 ðxÞ; S l ðxÞ; S l1 ðxÞ; . . . ; S 0 ðxÞ in R[x] is called the subresultant chain of F and G with respect to x. It is said to be regular if all Sj(x) are regular, and defective otherwise. Let Rl+1 = 1 and lcðS j ðxÞ; xÞ if S j ðxÞ is regular; Rj ¼ 0 otherwise; where lc(Sj, x) denotes the leading coefficient of Sj (x) in x for 0 6 j 6 l. The sequence of polynomials Rl+1, Rl, . . . , R0 is called the PSC chain of F and G with respect to x. The following theorem provides an effective algorithm for constructing subresultant chains by means of pseudo-division [5,7,9].
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Theorem 1.1 (Subresultant chain). Let Sl+1(x) and Sl(x) be two polynomials in R[x] with deg(Sl+1(x), x) P deg(Sl(x), x) > 0 and let Sl+1(x), Sl(x), . . . , S0(x) be the subresultant chain of Sl+1(x) and Sl(x) with respect to x, with PSC chain Rl+1, Rl, . . . , R0. If both Sj+1(x) and Sj(x) are regular, then R2jþ1 S j1 ðxÞ ¼ premðS jþ1 ðxÞ; S j ðxÞ; xÞ; 1 6 j 6 l: If Sj+1(x) is regular and Sj(x) is defective of degree r < j, then S j1 ðxÞ ¼ S j2 ðxÞ ¼ ¼ S rþ1 ðxÞ ¼ 0; 1 6 r < j < l; Rjr jþ1 S r ðxÞ ¼ lcðS j ðxÞ; xÞ ð1Þ
jr
Rjrþ2 jþ1 S r1 ðxÞ
jr
S j ðxÞ; 0 6 r 6 j < l;
¼ premðS jþ1 ðxÞ; S j ðxÞ; xÞ;
0 < r 6 j < l;
where prem(Sj+1(x), Sj(x), x) denotes the pseudo-remainder of Sj+1(x) with respect to Sj(x) in x. 2. New method An alternative method to compute the pseudo-remainder for polynomials, through computing the determinant of a new matrix, is presented lately [6]. In this section, we also give a conciseness approach to construct subresultants. Given two polynomials F and G in R[x] introduced in form of (1). The following matrix of dimension m + n j by m + n j is formed for 0 6 j < n, 9 1> 0 a0 a1 a2 an = .. .. .. .. C nj B . . . . C> B C; B a 0 a1 a2 an C9 B C> B C= B 1 x C B . . C j B . . : M j ðxÞ ¼ B . . C> C; B 1 x C9 B C Bb b b2 bn C> B 0 1 C= B .. .. .. .. .. A mj @ . . . . . > b 0 b1 b2 bn ; Theorem 2.1. As the above notation, the jth subresultant of F and G with respect to x S j ðxÞ ¼ ð1Þ for 0 6 j < n.
jðmjþ1Þ
detðM j ðxÞÞ;
Proof. It is obvious that S0(x) = det(M0(x)) = res(F, G, x). We proceed to compute det(Mj(x)) for 0 < j < n. By the j times of elementary row operations for interchanging two rows of Mj(x), one can obtain the following matrix: 19 0 1 x > = .. .. C j B . . C> B C; B B 1 x C C9 B C> B a0 a1 a2 an C= B .. .. .. .. C B N 0;j ðxÞ ¼ B C nj ; . . . . C> B C; B a 0 a1 a2 an C9 B C> B b0 b1 b2 bn1 bn C= B C B .. .. .. .. .. A mj @ . . . . . > ; b0 b1 b2 bn1 bn where det(Mj(x)) = (1)j(n j)det(N0,j(x)).
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We can compute det(N0,j(x)) explicitly by expanding the determinant with its first row. It follows that: detðN 0;j ðxÞÞ ¼ ð1Þ
1þmþn2j
detðN 1;j ðxÞÞ þ ðxÞð1Þ
2þmþn2j
detðN 2;j ðxÞÞ;
where N1,j(x) is the submatrix of N0,j(x) obtained by deleting the first row and the m + n 2jth column, and N2,j(x) is the submatrix of N0,j(x) obtained by deleting the first row and the m + n 2j + 1th column. It is easy to see that detðN 2;j ðxÞÞ ¼ ð1Þðmþn2jÞðj1Þ ðxÞj1 detðM j;j Þ; where Mj,j is mentioned in the above section. The matrix N1,j(x) is written as follows: 0 1 B B B B B B B a0 a1 B B .. .. .. .. N 1;j ðxÞ ¼ B . . . . B B B a a 0 1 B Bb b B 0 1 B .. .. .. .. B @ . . . . b0
b1
x .. .. . .
1 ..
.
..
.
19 > C= C j1 C> ; x C C C9 C> C= C C nj : C> C; C C9 C> C= C C mj A> ; bn
There are two cases to consider. In the first case when m + n is odd, we have that detðN 2;j ðxÞÞ ¼ xj1 detðM j;j Þ: Hence, detðN 0;j ðxÞÞ ¼ detðN 1;j ðxÞÞ þ xj detðM j;j Þ: By an argument analogous to the above, one can easily compute that detðN 1;j ðxÞÞ ¼ detðN 2;j ðxÞÞ þ xj1 detðM j1;j Þ; where N2,j(x) is the submatrix of N1,j(x) obtained by deleting the first row and the m + n 2(j 1)th column. Hence, detðN 0;j ðxÞÞ ¼ detðN 2;j ðxÞÞ þ xj1 detðM j1;j Þ þ xj detðM j;j Þ: By doing this way successively, we can obtain that detðN 0;j ðxÞÞ ¼
j X
xi detðM i;j Þ ¼ S j ðxÞ:
i¼0
Therefore, detðM j ðxÞÞ ¼ ð1ÞjðnjÞ S j ðxÞ: Now we consider the case when m + n is even. j1
detðN 0;j ðxÞÞ ¼ detðN 1;j ðxÞÞ x detðN 2;j ðxÞÞ; detðN 2;j ðxÞÞ ¼ ðxÞ Hence, j
detðN 0;j ðxÞÞ ¼ detðN 1;j ðxÞÞ þ ð1Þ xj detðM j;j Þ:
detðM j;j Þ:
Y.-B. Li / Applied Mathematics and Computation 183 (2006) 471–476
475
By an argument analogous to the above, one can compute that detðN 1;j ðxÞÞ ¼ detðN 2;j ðxÞÞ þ ð1Þ
j1 j1
x
detðM j1;j Þ;
where N2,j(x) is the submatrix of N1,j(x) obtained by deleting the first row and the m + n 2(j 1)th column. Hence, 2
j
j
detðN 0;j ðxÞÞ ¼ ð1Þ detðN 2;j ðxÞÞ þ ð1Þ xj1 detðM j1;j Þ þ ð1Þ xj1 detðM j;j Þ: Doing this way successively, it follows that: detðN 0;j ðxÞÞ ¼ ð1Þ
j X
j
j
xi detðM i;j Þ ¼ ð1Þ S j ðxÞ:
i¼0
Therefore, jðnjþ1Þ
detðM j ðxÞÞ ¼ ð1Þ
S j ðxÞ:
Colligating the above two cases, we have that detðM j ðxÞÞ ¼ ð1Þjðmjþ1Þ S j ðxÞ: h
This proof is complete.
Example 2.1. Let F = x4 z3x2 + x2 z4 + 2z2 1 and G = x4 + z2x2 r2x2 + z4 2z2 + 1 which have been considered in [9]. We proceed to compute the subresultant Sj(x) of F and G with respect to x for 0 < j < 4. Mj(x) can be constructed by Theorem 2.1, as follows: 2
1 0 6 6 0 1 6 6 M 3 ðxÞ ¼ 6 6 0 0 6 6 0 0 4
1z
3
0
1 0
6 6 6 6 6 6 6 M 1 ðxÞ ¼ 6 6 6 6 6 6 6 4
0 1
2z 1 z
x
0
0
1
x
0
0
1
x
1 0 r2 þ z2 2
2
4
0 2z2 þ 1 þ z
1 z3
1z
3
1 z3
0
2z2 1 z4
0 1
0
1 z3
0
0
0
0
1
x
0
0
0
0
1
1
0 r2 þ z2
0
2z2 þ 1 þ z4
0
1
r2 þ z2 3
0
1 0
6 6 7 6 7 6 7 6 7 7; M 2 ðxÞ ¼ 6 6 7 6 7 6 7 6 5 6 4 4
2z2 1 z4
0
0
2
3
0
0 2
0
2z 1 z
0 4
0
0
1
0
1 z3
0
0
0
0
0
0
1
1
0 r2 þ z2
0
2z2 þ 1 þ z4
0
0
1
0
r2 þ z2
0
2z2 þ 1 þ z4
0
0
1
0
r2 þ z2
0
7 7 7 7 2 4 7 2z 1 z 7 7 7: x 7 7 7 0 7 7 7 0 5 0
2z2 þ 1 þ z4
We obtain that 6
S 3 ðxÞ ¼ ð1Þ detðM 3 ðxÞÞ ¼ x2 ðr2 z2 1 þ z3 Þ; 6
2
S 2 ðxÞ ¼ ð1Þ detðM 2 ðxÞÞ ¼ x2 ðr2 z2 1 þ z3 Þ ; 4
2
2
3
S 1 ðxÞ ¼ ð1Þ detðM 1 ðxÞÞ ¼ ðz 1Þ ðz þ 1Þ ðr2 z2 1 þ z3 Þ : The following two assertions are obvious. Proposition 2.1. With the same notation in Theorem 2.1, we have that S j ðxÞ ¼ ð1Þj detðN j ðxÞÞ;
0
3
7 2z2 1 z4 7 7 7 7 0 7 7; 7 x 7 7 7 0 5 2z2 þ 1 þ z4
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Y.-B. Li / Applied Mathematics and Computation 183 (2006) 471–476
for 0 6 j < n, where 0 a0 B B B B B B B b0 B B N j ðxÞ ¼ B B B B B B B B B @
a1 .. .
a2 .. .
.. .
b1 .. .
a0 b2 .. .
a 1 a2 .. .. . .
b0
b1
b2
an .. . bn .. .
1
x .. .
19 > C= C nj C> C; an C C9 C> C= C C mj : C> C; bn C C9 C> C= C .. C j A> . ; 1 x
Proposition 2.2. Let G = b0xn + b1xn1 + + bn be a polynomial in R[x] with deg(G, x) = n > 0. It follows that detðM n ðxÞÞ ¼ b0mn1 G for any m > n, where 0 B B B B B M n ðxÞ ¼ B B b0 B B B @
1
b1 b2 .. .. .. . . . b 0 b1
.. . b2
x .. . bn .. .
19 > C= .. C . C>n C; 1 x C C9 : C> C= C C mn A> ; bn
Remark 2.1. Now in the case of m > n + 1 in Definition 1.1, nth subresultant Sn(x) of F and G with respect to x is reasonably extended as S n ðxÞ ¼ b0mn1 G according to the above result. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
W.S. Brown, J.F. Traub, On Euclid’s algorithm and the theory of subresultants, J. ACM 18 (1971) 505–514. G.E. Collins, Subresultants and reduced polynomial remainder sequences, J. ACM 14 (1967) 128–142. H. Lombardi, M.F. Roy, M. Safey, New structure theorem for subresultants, J. Symb. Comput. 29 (2000) 663–689. J. Abdeljaoued, G.M. Diaz-Toca, L. Gonzalez-Vega, Minor of Bezout matrices, subresultants and parameterization of the degree of the polynomial greatest common divisor, Int. J. Comput. Math 81 (10) (2004) 1223–1238. R. Loos, Generalized polynomial remainder sequences, in: B. Buchberger, G.E. Collins, R. Loos (Eds.), Computer Algebra: Symbolic and Algebraic Computation, second ed., Springer, Wien/New York, 1983, pp. 15–137. Y.-B. Li, An alternative algorithm for computing the pseudo-remainder of multivariate polynomials, Appl. Math. Comput. 173 (2006) 484–492. B. Mishra, Algorithmic Algebra, Texts and Monographs in Computer Science, Springer, Berlin/Heidelberg/New York/Tokyo, 1993. D. Wang, Computing triangular systems and regular systems, J. Symb. Comput. 30 (2000) 221–236. D. Wang, Elimination Methods, Springer, Wien/New York, 2001. L. Yang, J.-Z. Zhang, X.-R. Hou, Non-Linear Equation Systems and Automated Theorem Proving, Shanghai Sci. & Tech. Education Publ. House, Shanghai, 1996 (in Chinese).
A new approach for constructing subresultants Yong-Bin Li Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu, Sichuan 610041, China School of Applied Mathematic, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
Abstract Subresultants have many applications including the computation of polynomial GCD and in algebraic cell decomposition. This paper presents a direct method to construct subresultants of two polynomials. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Polynomial; Sylvester matrix; Determinant; Subresultants
1. Introduction and notations The subresultants of two polynomials have been extensively studied, for example, in Brown and Traub [1], Collins [2], Lombardi et al. [3], Abdeljaoued et al. [4], Loos [5] and Mishra [7]. In recent time, Sylvester resultant and subresultants have received much attention both as the elimination theory as well as for the computational efficiency of various constructive algebraic algorithms. It has been widely recognized that forming subresultant chains is one of the most efficient ways to compute polynomial remainder sequence (PRS) [7,9]. By virtue of subresultants, many interesting results and algorithms in resultant-based elimination are developed [8–10]. Let the ring R be restricted to a unique factorization domain (UFD). The ring of univariate polynomials in the indeterminate x with coefficients in R is denoted by R[x]. Let F and G be two univariate polynomials in R[x] and of respective degrees m and n in x with m P n > 0, written as F ¼ a0 xm þ a1 xm1 þ þ am ; G ¼ b0 xn þ b1 xn1 þ þ bn : ð1Þ We form a matrix of dimension m + n by m + n, call the Sylvester matrix of F and G with respect to x, as follows: 19 0 a0 a1 am = .. .. .. C n B C; B . . . C B C9 B a a a 0 1 m C B : M ¼B C> b b b 1 n C= B 0 C m B .. .. .. A> @ . . . ; b 0 b1 bn E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.120
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Y.-B. Li / Applied Mathematics and Computation 183 (2006) 471–476
The determinant of the Sylvester matrix M is called the Sylvester resultant or eliminant of F and G with respect to x, denoted by res(F, G, x). It is well known that if one of a0 and b0 is nonzero, res(F, G, x) = 0 is a necessary and sufficient condition of F and G to have a common zero. Now let Mi,j be the submatrix of M obtained by deleting the last j of n rows of F coefficients, the last j of the m rows of G coefficients, the last 2j + 1 columns, excepting column m + n i j, for 0 6 i 6 j < n. Definition 1.1. The polynomial j X S j ðxÞ ¼ detðMi;j Þxi i¼0
is called the jth subresultant of F and G with respect to x, for 0 6 j < n. Here deg(Sj(x), x) 6 j, and Rj = det(Mj,j) is called the jth principal subresultant coefficient (PSC) or the jth resultant of F and G with respect to x. If m > n + 1, the definition of the jth subresultant Sj(x) and PSC Rj of F and G with respect to x is extended as follows: S n ðxÞ ¼ b0mn1 G; Rn ¼ bmn ; 0
S j ðxÞ ¼ Rj ¼ 0; n < j < m 1:
Sj(x) is said to be defective of degree r if deg(Sj(x), x) = r < j, and regular otherwise. It is easy to see that S0(x) is the resultant of F and G with respect to x. The following proposition provides a direct method for constructing subresultants through computing the determinants of square matrixes [5]. Proposition 1.1. The jth subresultant of F and G can be determined by the determinant of the following square matrix for any 0 6 j < n, 19 0 a0 a1 am xnj1 F > > > B > a0 a1 am xnj2 F C C= B C B .. .. .. .. C>n j B C> B . . . . C> B > C; B a1 amj1 F C B : C9 B B b0 b1 bn xmj1 G C > C> B > > B b0 b1 bn xmj2 G C C= B C B .. .. .. .. C>m j B A> @ . . . . > > ; b1 bnj1 G Definition 1.2. Let F and G be two polynomials in R[x] with m = deg(F, x) P deg(G, x) = n > 0 and m 1 if m > n; l¼ n otherwise: Let Sl+1(x) = F, Sl(x) = G, and Sj (x) be the jth subresultant of F and G with respect to x for 0 6 j < l. The sequence of polynomials S lþ1 ðxÞ; S l ðxÞ; S l1 ðxÞ; . . . ; S 0 ðxÞ in R[x] is called the subresultant chain of F and G with respect to x. It is said to be regular if all Sj(x) are regular, and defective otherwise. Let Rl+1 = 1 and lcðS j ðxÞ; xÞ if S j ðxÞ is regular; Rj ¼ 0 otherwise; where lc(Sj, x) denotes the leading coefficient of Sj (x) in x for 0 6 j 6 l. The sequence of polynomials Rl+1, Rl, . . . , R0 is called the PSC chain of F and G with respect to x. The following theorem provides an effective algorithm for constructing subresultant chains by means of pseudo-division [5,7,9].
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Theorem 1.1 (Subresultant chain). Let Sl+1(x) and Sl(x) be two polynomials in R[x] with deg(Sl+1(x), x) P deg(Sl(x), x) > 0 and let Sl+1(x), Sl(x), . . . , S0(x) be the subresultant chain of Sl+1(x) and Sl(x) with respect to x, with PSC chain Rl+1, Rl, . . . , R0. If both Sj+1(x) and Sj(x) are regular, then R2jþ1 S j1 ðxÞ ¼ premðS jþ1 ðxÞ; S j ðxÞ; xÞ; 1 6 j 6 l: If Sj+1(x) is regular and Sj(x) is defective of degree r < j, then S j1 ðxÞ ¼ S j2 ðxÞ ¼ ¼ S rþ1 ðxÞ ¼ 0; 1 6 r < j < l; Rjr jþ1 S r ðxÞ ¼ lcðS j ðxÞ; xÞ ð1Þ
jr
Rjrþ2 jþ1 S r1 ðxÞ
jr
S j ðxÞ; 0 6 r 6 j < l;
¼ premðS jþ1 ðxÞ; S j ðxÞ; xÞ;
0 < r 6 j < l;
where prem(Sj+1(x), Sj(x), x) denotes the pseudo-remainder of Sj+1(x) with respect to Sj(x) in x. 2. New method An alternative method to compute the pseudo-remainder for polynomials, through computing the determinant of a new matrix, is presented lately [6]. In this section, we also give a conciseness approach to construct subresultants. Given two polynomials F and G in R[x] introduced in form of (1). The following matrix of dimension m + n j by m + n j is formed for 0 6 j < n, 9 1> 0 a0 a1 a2 an = .. .. .. .. C nj B . . . . C> B C; B a 0 a1 a2 an C9 B C> B C= B 1 x C B . . C j B . . : M j ðxÞ ¼ B . . C> C; B 1 x C9 B C Bb b b2 bn C> B 0 1 C= B .. .. .. .. .. A mj @ . . . . . > b 0 b1 b2 bn ; Theorem 2.1. As the above notation, the jth subresultant of F and G with respect to x S j ðxÞ ¼ ð1Þ for 0 6 j < n.
jðmjþ1Þ
detðM j ðxÞÞ;
Proof. It is obvious that S0(x) = det(M0(x)) = res(F, G, x). We proceed to compute det(Mj(x)) for 0 < j < n. By the j times of elementary row operations for interchanging two rows of Mj(x), one can obtain the following matrix: 19 0 1 x > = .. .. C j B . . C> B C; B B 1 x C C9 B C> B a0 a1 a2 an C= B .. .. .. .. C B N 0;j ðxÞ ¼ B C nj ; . . . . C> B C; B a 0 a1 a2 an C9 B C> B b0 b1 b2 bn1 bn C= B C B .. .. .. .. .. A mj @ . . . . . > ; b0 b1 b2 bn1 bn where det(Mj(x)) = (1)j(n j)det(N0,j(x)).
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We can compute det(N0,j(x)) explicitly by expanding the determinant with its first row. It follows that: detðN 0;j ðxÞÞ ¼ ð1Þ
1þmþn2j
detðN 1;j ðxÞÞ þ ðxÞð1Þ
2þmþn2j
detðN 2;j ðxÞÞ;
where N1,j(x) is the submatrix of N0,j(x) obtained by deleting the first row and the m + n 2jth column, and N2,j(x) is the submatrix of N0,j(x) obtained by deleting the first row and the m + n 2j + 1th column. It is easy to see that detðN 2;j ðxÞÞ ¼ ð1Þðmþn2jÞðj1Þ ðxÞj1 detðM j;j Þ; where Mj,j is mentioned in the above section. The matrix N1,j(x) is written as follows: 0 1 B B B B B B B a0 a1 B B .. .. .. .. N 1;j ðxÞ ¼ B . . . . B B B a a 0 1 B Bb b B 0 1 B .. .. .. .. B @ . . . . b0
b1
x .. .. . .
1 ..
.
..
.
19 > C= C j1 C> ; x C C C9 C> C= C C nj : C> C; C C9 C> C= C C mj A> ; bn
There are two cases to consider. In the first case when m + n is odd, we have that detðN 2;j ðxÞÞ ¼ xj1 detðM j;j Þ: Hence, detðN 0;j ðxÞÞ ¼ detðN 1;j ðxÞÞ þ xj detðM j;j Þ: By an argument analogous to the above, one can easily compute that detðN 1;j ðxÞÞ ¼ detðN 2;j ðxÞÞ þ xj1 detðM j1;j Þ; where N2,j(x) is the submatrix of N1,j(x) obtained by deleting the first row and the m + n 2(j 1)th column. Hence, detðN 0;j ðxÞÞ ¼ detðN 2;j ðxÞÞ þ xj1 detðM j1;j Þ þ xj detðM j;j Þ: By doing this way successively, we can obtain that detðN 0;j ðxÞÞ ¼
j X
xi detðM i;j Þ ¼ S j ðxÞ:
i¼0
Therefore, detðM j ðxÞÞ ¼ ð1ÞjðnjÞ S j ðxÞ: Now we consider the case when m + n is even. j1
detðN 0;j ðxÞÞ ¼ detðN 1;j ðxÞÞ x detðN 2;j ðxÞÞ; detðN 2;j ðxÞÞ ¼ ðxÞ Hence, j
detðN 0;j ðxÞÞ ¼ detðN 1;j ðxÞÞ þ ð1Þ xj detðM j;j Þ:
detðM j;j Þ:
Y.-B. Li / Applied Mathematics and Computation 183 (2006) 471–476
475
By an argument analogous to the above, one can compute that detðN 1;j ðxÞÞ ¼ detðN 2;j ðxÞÞ þ ð1Þ
j1 j1
x
detðM j1;j Þ;
where N2,j(x) is the submatrix of N1,j(x) obtained by deleting the first row and the m + n 2(j 1)th column. Hence, 2
j
j
detðN 0;j ðxÞÞ ¼ ð1Þ detðN 2;j ðxÞÞ þ ð1Þ xj1 detðM j1;j Þ þ ð1Þ xj1 detðM j;j Þ: Doing this way successively, it follows that: detðN 0;j ðxÞÞ ¼ ð1Þ
j X
j
j
xi detðM i;j Þ ¼ ð1Þ S j ðxÞ:
i¼0
Therefore, jðnjþ1Þ
detðM j ðxÞÞ ¼ ð1Þ
S j ðxÞ:
Colligating the above two cases, we have that detðM j ðxÞÞ ¼ ð1Þjðmjþ1Þ S j ðxÞ: h
This proof is complete.
Example 2.1. Let F = x4 z3x2 + x2 z4 + 2z2 1 and G = x4 + z2x2 r2x2 + z4 2z2 + 1 which have been considered in [9]. We proceed to compute the subresultant Sj(x) of F and G with respect to x for 0 < j < 4. Mj(x) can be constructed by Theorem 2.1, as follows: 2
1 0 6 6 0 1 6 6 M 3 ðxÞ ¼ 6 6 0 0 6 6 0 0 4
1z
3
0
1 0
6 6 6 6 6 6 6 M 1 ðxÞ ¼ 6 6 6 6 6 6 6 4
0 1
2z 1 z
x
0
0
1
x
0
0
1
x
1 0 r2 þ z2 2
2
4
0 2z2 þ 1 þ z
1 z3
1z
3
1 z3
0
2z2 1 z4
0 1
0
1 z3
0
0
0
0
1
x
0
0
0
0
1
1
0 r2 þ z2
0
2z2 þ 1 þ z4
0
1
r2 þ z2 3
0
1 0
6 6 7 6 7 6 7 6 7 7; M 2 ðxÞ ¼ 6 6 7 6 7 6 7 6 5 6 4 4
2z2 1 z4
0
0
2
3
0
0 2
0
2z 1 z
0 4
0
0
1
0
1 z3
0
0
0
0
0
0
1
1
0 r2 þ z2
0
2z2 þ 1 þ z4
0
0
1
0
r2 þ z2
0
2z2 þ 1 þ z4
0
0
1
0
r2 þ z2
0
7 7 7 7 2 4 7 2z 1 z 7 7 7: x 7 7 7 0 7 7 7 0 5 0
2z2 þ 1 þ z4
We obtain that 6
S 3 ðxÞ ¼ ð1Þ detðM 3 ðxÞÞ ¼ x2 ðr2 z2 1 þ z3 Þ; 6
2
S 2 ðxÞ ¼ ð1Þ detðM 2 ðxÞÞ ¼ x2 ðr2 z2 1 þ z3 Þ ; 4
2
2
3
S 1 ðxÞ ¼ ð1Þ detðM 1 ðxÞÞ ¼ ðz 1Þ ðz þ 1Þ ðr2 z2 1 þ z3 Þ : The following two assertions are obvious. Proposition 2.1. With the same notation in Theorem 2.1, we have that S j ðxÞ ¼ ð1Þj detðN j ðxÞÞ;
0
3
7 2z2 1 z4 7 7 7 7 0 7 7; 7 x 7 7 7 0 5 2z2 þ 1 þ z4
476
Y.-B. Li / Applied Mathematics and Computation 183 (2006) 471–476
for 0 6 j < n, where 0 a0 B B B B B B B b0 B B N j ðxÞ ¼ B B B B B B B B B @
a1 .. .
a2 .. .
.. .
b1 .. .
a0 b2 .. .
a 1 a2 .. .. . .
b0
b1
b2
an .. . bn .. .
1
x .. .
19 > C= C nj C> C; an C C9 C> C= C C mj : C> C; bn C C9 C> C= C .. C j A> . ; 1 x
Proposition 2.2. Let G = b0xn + b1xn1 + + bn be a polynomial in R[x] with deg(G, x) = n > 0. It follows that detðM n ðxÞÞ ¼ b0mn1 G for any m > n, where 0 B B B B B M n ðxÞ ¼ B B b0 B B B @
1
b1 b2 .. .. .. . . . b 0 b1
.. . b2
x .. . bn .. .
19 > C= .. C . C>n C; 1 x C C9 : C> C= C C mn A> ; bn
Remark 2.1. Now in the case of m > n + 1 in Definition 1.1, nth subresultant Sn(x) of F and G with respect to x is reasonably extended as S n ðxÞ ¼ b0mn1 G according to the above result. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
W.S. Brown, J.F. Traub, On Euclid’s algorithm and the theory of subresultants, J. ACM 18 (1971) 505–514. G.E. Collins, Subresultants and reduced polynomial remainder sequences, J. ACM 14 (1967) 128–142. H. Lombardi, M.F. Roy, M. Safey, New structure theorem for subresultants, J. Symb. Comput. 29 (2000) 663–689. J. Abdeljaoued, G.M. Diaz-Toca, L. Gonzalez-Vega, Minor of Bezout matrices, subresultants and parameterization of the degree of the polynomial greatest common divisor, Int. J. Comput. Math 81 (10) (2004) 1223–1238. R. Loos, Generalized polynomial remainder sequences, in: B. Buchberger, G.E. Collins, R. Loos (Eds.), Computer Algebra: Symbolic and Algebraic Computation, second ed., Springer, Wien/New York, 1983, pp. 15–137. Y.-B. Li, An alternative algorithm for computing the pseudo-remainder of multivariate polynomials, Appl. Math. Comput. 173 (2006) 484–492. B. Mishra, Algorithmic Algebra, Texts and Monographs in Computer Science, Springer, Berlin/Heidelberg/New York/Tokyo, 1993. D. Wang, Computing triangular systems and regular systems, J. Symb. Comput. 30 (2000) 221–236. D. Wang, Elimination Methods, Springer, Wien/New York, 2001. L. Yang, J.-Z. Zhang, X.-R. Hou, Non-Linear Equation Systems and Automated Theorem Proving, Shanghai Sci. & Tech. Education Publ. House, Shanghai, 1996 (in Chinese).