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An alternative algorithm for computing the pseudo-remainder of multivariate polynomials
Applied Mathematics and Computation 173 (2006) 484–492 www.elsevier.com/locate/amc
An alternative algorithm for computing the pseudo-remainder of multivariate polynomials Yong-Bin Li School of Applied Mathematic, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu, Sichuan 610041, PR China
Abstract This paper presents an alternative method to compute the pseudo-remainder for multivariate polynomials, which plays a very important role in polynomial system solving by many known elimination methods. The efficiency of the new approach dependents heavily on the method for computing the determinant of a matrix. Some examples show that the new algorithm is efficient when one computes the determinants of matrices by using Gaussian elimination in Maple system. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Multivariate polynomial; Pseudo-remainder; Gaussian elimination; Determinant of a matrix
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.04.087
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
485
1. Introduction and notations The computation of the pseudo-remainder of multivariate polynomials plays a crucial role in many methods and their improving revisions for solving polynomial systems by means of triangular sets. These methods are those of Wu [15], Chou [1], Chou and Gao [2,3], Gallo and Mishra [4], Lazard [7], Yang et al. [16], Kalkbrener [5], Wang [10–14] and Li et al. [8,9]. Let the ring R be restricted to a unique factorization domain (UFD). The ring of polynomials in the n indeterminates x1, . . ., xn with coefficients in R is denoted by R[x1, . . ., xn], or R[x] for short. Let G and F be two polynomials in R[x] and xk a fixed variable. While considered as polynomials in xk, G and F can be written as G ¼ G0 xnk þ G1 xm1 þ þ Gm ; k F ¼
F 0 xnk
þ
F 1 xn1 k
þ þ F n;
ð1Þ ð2Þ
where m = deg(G, xk), n = deg(F, xk) and Gi, Fi 2 R[x1, . . ., xk1, xk+1, . . ., xn]. In this expression, F0 is the leading coefficient of F in xk, denoted by lc(F, xk) or I. Namely, degðF ;xk Þ
I ¼ coefðF ; xk
Þ.
Let F 5 0 and m P n as above. For pseudo-dividing G by F, considered as polynomials in xk, we have a division algorithm as follows. Let R = G; repeat the following process until r = deg(R, xk) < n: R
IR R0 xrn k F;
where R0 = lc(R, xk). As r strictly decreases for each iteration, the procedure must terminate. Finally, one obtains two polynomials Q and R in R[x] satisfying the relation I q G ¼ QF þ R;
ð3Þ
where q = max(m n + 1, 0), deg(R, xk) < n, deg(Q, xk) = max(m n, 1). The expression (3) is called a pseudo-remainder formula; Q is called the pseudo-quotient and R, the pseudo-remainder of G with respect to F in xk, denoted by pquo(G, F, xk) and prem(G, F, xk), respectively. Actually, the polynomials Q and R in (3) are uniquely determined by G and F (see [6] for details).
2. Algorithm newprem In this section, one gives an alternative method for computing the pseudoremainder of multivariate polynomials. A new algorithm of pseudo-division is described in details.
486
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
Theorem 2.1. Let G and F be two polynomials in R[x] and xk a fixed variable as above forms (1) and (2). Then prem(G, F, xk) = (1)mdet(A), where 1 0 Gm1 Gm G0 G1 Gn 9 > C BF > C > B 0 F 1 F n1 F n > C > B > = C B F0 F1 F n1 F n C B mnþ2 C B .. .. .. C > B > > C > B . . . > C. > A¼B ; C B F0 F1 F n1 F n C B C 9 B C > B 1 xk C = B C B .. .. C B n1 A > @ . . ; 1 xk Proof. Consider the following equations: 0
G
B xmn F B k B mn1 Bx F B k B . B .. B B B B x0k F B B 0 B B .. B @ .
1 C C C 0 m 1 C xk C 0 1 C 0 B m1 C C x C B k C B. C C B C¼AB C ¼ B . C. C B ... C @ . A C A @ C 0 C C 1 C C A
0 Eliminating elements G0, . . ., Gmn+1 in the first row of A by the elementary row operations, we can obtain that 1 0 premðG; F ; xk Þ=F 0mnþ1 C B xmn F C B k C B 0 m 1 mn1 C B xk F xk C B 0 1 C B 0 B m1 C C B .. B xk C B C C B . .. C C B C¼BB B ¼B .. C C @ . A; B 0 C B C B xk F @ . A C B 0 C B 0 C B 1 C B .. C B A @ . 0
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
where
0 B BF0 B B B B B B¼B B B B B B @
C0 F1 .. .. . . F0
F n1 .. . F1 1
xk .. .
C n2
C n1
Fn
F n1 ..
Fn
. 1
1 C C C C C C C C. C C C C C A
xk
9 > > > = > > > ; 9 > = > ;
487
mnþ2
n1
It is easy to see that premðG; F ; xk Þ=F 0mnþ1 ¼ C 0 xkn1 þ C 1 xn2 þ þ C n1 ¼ k
n1 X
C i xkn1i .
i¼0
We know that detðBÞ ¼
n1 X ðmnþ2þiÞþ1 ð1Þ C i detðBi Þ; i¼0
where Bi is the submatrix found by deleting the first row and the ith column of A, namely, 1 0 F0 F1 Fn C B .. .. .. C B . . . C B C B B F0 C C B C B 1 xk C B C B C B .. .. Bi ¼ B C. . . C B C B 1 xk C 9 B C B C > B 1 x k C = B C B i . . B .. C .. ; A > @ 1 One can see that detðBi Þ ¼ ðxk Þ This implies that detðBÞ ¼
ni1
F 0mnþ1 for i = 0, , n 1.
n1 X ðmnþ2þiÞþ1 n1i mnþ1 ð1Þ C i ðxk Þ F0 i¼0
¼ ð1Þ
mþ2
F 0mnþ1
n1 X i¼0
C i xkn1i .
488
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
Thus, premðG; F ; xk Þ ¼ ð1Þ The proof is complete.
mþ2
m
detðBÞ ¼ ð1Þ detðAÞ.
h
Example 2.1. Let F = xy2 + 1 and G = 2y3 y2 + x2y. With respect to y, the corresponding prem(G, F, y) can be calculated as follows. Construct matrix A according to Theorem 2.1, 2 3 2 1 x2 0 6x 0 1 0 7 6 7 A¼6 7. 40 x 0 1 5 0
0
1
y
Gaussian elimination on A is used, we obtain that 2 3 2 1 x2 0 60 x 7 0 1 6 7 B¼6 7. 40 0 5 1 y 3 0 0 0 1=2 þ y 1=2yx Then, 3
premðG; F ; yÞ ¼ ð1Þ detðBÞ ¼ x 2xy þ yx4 . The next algorithm newprem follows directly from Theorem 2.1. Algorithm newprem. R prem(G, F, xk). Given two polynomials G, F 2 R[x] and a variable xk (1 6 k 6 n), this algorithm computes the pseudo-remainder R of G with respect to F in xk. P1. Set m deg(G, xk), n deg(F, xk). P2. Construct a m + 1 by m + 1 matrix A according to Theorem 2.1. P3. Gaussian elimination on A is used and return matrix B, set R (1)mdet(B). Moreover, we know that the integer q in form (3) may be determined as small as possible, provided that the division process does not introduce fractions in Q and R. One can take q = 1 instead of 2 in Example 2.1 so that it simplifies to xG ¼ ð2y 1ÞF þ x3 y 2y þ 1. Taking the smallest q is rather crucial for control of the size expansion of the pseudo-remainder in many algorithms (see [14] for the details). Moreover, one
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
489 q
can modify the pseudo-remainder formula (3) by replacing Iq with I 11 I qe e , where I1, . . ., Ie are all the distinct irreducible factors of I, and choosing the smallest q1, . . ., qe so that the corresponding pseudo-remainder formula still holds. Definition 2.1. Let P1, . . ., Ps be polynomials in R[x] which are not all 0. A polynomial G* 2 R[x] is called a greatest common divisor (GCD) of P1, . . ., Ps if G divides P1, . . ., Ps and every common divisor of P1, . . ., Ps divides G*. Now let K be the quotient field of R. A very fundamental problem is to compute GCD(P1, . . ., Ps) in the above definition in K[x]. Now, powerful algorithms have been well developed (see [6] for the details) and implemented in popular computer algebra systems. We can freely use such algorithms and software systems when polynomial factorization over K is necessary. One can modify formula pseudo-remainder (3) as follows: q
I 11 I qe e G ¼ Q F þ R ; q
where R* = R/GCD(Iq, R), I 11 I qe e ¼ I q =GCDðI q ; RÞ and Q* = Q/GCD(Iq, R).
3. Implementation The algorithm newprem has been implemented in Maple system. The efficiency of algorithm newprem depends heavily on the one of Gaussian elimination. Our experiments indicate that it has not any advantage when computing the pseudo-remainder of univariate polynomials, but it is rather efficient in some cases of multivariate polynomials. Here the set of eight examples is taken to show the performance of our draft implementation. The following timings in CPU seconds are obtained in Maple 9.5 on Pentium III PC. Timings
Ex. 1
Ex. 2
Ex. 3
Ex. 4
Ex. 5
Ex. 6
Ex. 7
Ex. 8
prem 41.820 24.134 48.610 234.296 P500 92.372 28.391 55.911 newprem 23.644 8.832 3.013 9.774 55.810 57.972 10.995 10.525
Example 3.1 G1 ¼ 25y 7 x27 uz49 53y 27 x18 uz31 þ 57x20 u11 z25 þ 5y 17 x16 u37 z7 þ 81y 52 x16 u4 z3 37y 30 x7 u16 z; F 1 ¼ 57yu7 z5 2y 2 x4 uz5 31y 5 x4 z3 34x8 uz2 þ 24y 4 x5 u2 z þ 89yx5 u2 .
490
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
Example 3.2 G2 ¼ 65y 8 x30 u2 z39 þ 34y 2 x41 u4 z26 þ 59y 24 x20 u23 z16 32y 19 x2 z11 þ 13y 47 u13 z9 32y 4 x19 u53 z2 ; F 2 ¼ 34u2 z3 þ 77xu4 z3 49y 3 x2 u3 z þ 83y 6 u5 z þ 32y 3 xu 55y 2 x7 u3 . Example 3.3 G3 ¼78y 4 x47 z77 þ 68y 65 x43 z68 þ 94y 47 x85 z21 66y 14 x24 z16 þ 39y 94 x29 z14 32y 47 x9 z5 ; F 3 ¼ 88y 2 z6 59y 3 x7 z6 þ 25y 4 x5 z6 73x2 y 10 z2 þ 4y 3 x10 z2 43y 4 x6 z2 . Example 3.4 G4 ¼ 75y 7 x22 z119 þ 12y 25 x19 z78 71xy 59 z68 49y 46 x7 z59 þ 23y 63 x72 z7 46y 62 x53 z4 ; F 4 ¼ 79y 5 z7 53y 6 z7 80y 2 x11 z þ 26y 2 x5 z 85y 3 þ 32y 13 x. Example 3.5 G5 ¼ 45y 5 x6 z86 þ 8y 24 x48 z31 þ 22y 5 x44 z30 99y 62 x55 z14 þ 60y 72 x47 z14 þ 11y 17 x38 z8 ; F 5 ¼ 50y 2 z6 63y 3 x5 z6 þ 98y 8 x2 z6 21yx9 z6 þ x9 z3 24x15 z. Example 3.6 G6 ¼ 38yx5 u35 w7 z40 50y 14 x13 u36 v5 w54 z17 þ 50y 21 x4 u55 v16 w8 z17 99y 19 x13 u5 v7 w55 z11 þ 77y 20 x8 u12 v52 w35 z9 þ 11y 38 x58 uv13 w16 z5 ; F 6 ¼ 81uv4 wz þ 81x7 uv2 z 35y 3 xuv2 þ 53y 5 x4 u þ 23y 3 u3 w9 23y 2 x7 v7 .
Example 3.7 G7 ¼ 53y 31 x10 u5 vw21 z50 þ 72y 28 x9 u5 v42 w21 z33 15y 48 x28 u5 w29 z24 þ 85y 59 x11 u3 v43 w5 z8 64y 20 x25 u19 v8 w27 z6 þ 31y 39 x6 u5 v31 w14 z3 ; F 7 ¼ 89y 4 x2 u3 vwz5 64x4 v3 w4 z5 62x4 u2 v7 z2 þ 28y 2 x6 u2 z2 þ 80y 4 u6 v3 z 66y 8 v3 wz.
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
491
Example 3.8 G8 ¼ 31y 2 xu27 v31 w23 z52 þ 6y 48 x6 u13 v11 w10 z49 þ 50y 10 x5 u8 v82 w6 z29 þ 90y 9 x40 u6 v5 w34 z10 þ 82y 16 x2 u6 v14 w21 z3 þ 80y 10 x18 u34 v63 z2 ; F 8 ¼F 7 .
Acknowledgement The author thanks specially his friend Prof. Mingsheng Wang who mentioned this interest topic on a walk in Chengdu, January, 2005.
References [1] S. Chou, Mechanical Geometry Theorem Proving, Reidel, Dordrecht, 1988. [2] S. Chou, X. Gao, Ritt-WuÕs decomposition algorithm and geometry theorem proving, in: Proceedings CADE-10, Kaiserslautern, Germany, LNCS, vol. 449, Springer Verlag, 1990, pp. 207–220. [3] S. Chou, X. Gao, Solving parametric algebraic systems, in: Proceedings ISAACÕ92, Berkeley, CA, ACM Press, New York, 1992, pp. 335–341. [4] G. Gallo, B. Mishra, Efficient algorithms and bounds for Wu-Ritt characteristic sets, in: Proceedings MEGAÕ90, Lavorno, Italy, Progress in mathematics 94, Birkhau¨ser, 1990, pp. 119–142. [5] M. Kalkbrener, A generalized Euclidean algorithm for computing triangular representations of algebraic varieties, J. Symb. Comput. 15 (1993) 143–167. [6] D.E. Knuth, second ed., The Art of Computer Programming, vol. 2, Addison-Wesley, Reading, Mass, 1981. [7] D. Lazard, A new method for solving algebraic systems of positive dimension, Discrete Appl. Math. 33 (1991) 147–160. [8] Y.-B. Li, J.-Z. Zhang, L. Yang, Decomposing polynomial systems into strong regular sets, in: A.M. Cohen, X.-S. Gao, N. Takayama (Eds.), Proceedings ICMS 2002, Beijing, 17–19 August 2002, World Scientific, Singapore, 2002, pp. 361–371. [9] Y.-B. Li, Applications of the theory of weakly nondegenerate conditions to zero decomposition for polynomial systems, J. Symb. Comput. 38 (1) (2004) 815–832. [10] D. Wang, Some improvements on WuÕs method for solving systems of algebraic equations, in: W.-T. Wu, M.-D. Cheng, (Eds.), Proceedings of the Int. Workshop on Math. Mechanisation, Beijing, China. Institute of Systems Science, Academia Sinica, 1992. [11] D. Wang, An implementation of the characteristic set method in Maple, in: J. Pfalzgraf, D.M. Wang (Eds.), Automated Practical Reasoning: Algebraic Approaches, Springer, Wien, 1995, pp. 187–201. [12] D. Wang, Decomposing triangular systems into simple systems, J. Symb. Comput. 25 (1998) 295–314. [13] D. Wang, Decomposing triangular systems and regular systems, J. Symb. Comput. 30 (2000) 221–236. [14] D. Wang, Elimination Methods, 107–109, Springer, Wien, New York, 2001, pp. 191–203.
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[15] W.-T. Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, Sci. Sinica 21 (1978) 159–172. [16] L. Yang, J.-Z. Zhang, X.-R. Hou, Non-linear Equation Systems and Automated Theorem Proving, Shanghai Sci. Tech. Education Publ., Shanghai, 1996 (in Chinese).
An alternative algorithm for computing the pseudo-remainder of multivariate polynomials Yong-Bin Li School of Applied Mathematic, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu, Sichuan 610041, PR China
Abstract This paper presents an alternative method to compute the pseudo-remainder for multivariate polynomials, which plays a very important role in polynomial system solving by many known elimination methods. The efficiency of the new approach dependents heavily on the method for computing the determinant of a matrix. Some examples show that the new algorithm is efficient when one computes the determinants of matrices by using Gaussian elimination in Maple system. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Multivariate polynomial; Pseudo-remainder; Gaussian elimination; Determinant of a matrix
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.04.087
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
485
1. Introduction and notations The computation of the pseudo-remainder of multivariate polynomials plays a crucial role in many methods and their improving revisions for solving polynomial systems by means of triangular sets. These methods are those of Wu [15], Chou [1], Chou and Gao [2,3], Gallo and Mishra [4], Lazard [7], Yang et al. [16], Kalkbrener [5], Wang [10–14] and Li et al. [8,9]. Let the ring R be restricted to a unique factorization domain (UFD). The ring of polynomials in the n indeterminates x1, . . ., xn with coefficients in R is denoted by R[x1, . . ., xn], or R[x] for short. Let G and F be two polynomials in R[x] and xk a fixed variable. While considered as polynomials in xk, G and F can be written as G ¼ G0 xnk þ G1 xm1 þ þ Gm ; k F ¼
F 0 xnk
þ
F 1 xn1 k
þ þ F n;
ð1Þ ð2Þ
where m = deg(G, xk), n = deg(F, xk) and Gi, Fi 2 R[x1, . . ., xk1, xk+1, . . ., xn]. In this expression, F0 is the leading coefficient of F in xk, denoted by lc(F, xk) or I. Namely, degðF ;xk Þ
I ¼ coefðF ; xk
Þ.
Let F 5 0 and m P n as above. For pseudo-dividing G by F, considered as polynomials in xk, we have a division algorithm as follows. Let R = G; repeat the following process until r = deg(R, xk) < n: R
IR R0 xrn k F;
where R0 = lc(R, xk). As r strictly decreases for each iteration, the procedure must terminate. Finally, one obtains two polynomials Q and R in R[x] satisfying the relation I q G ¼ QF þ R;
ð3Þ
where q = max(m n + 1, 0), deg(R, xk) < n, deg(Q, xk) = max(m n, 1). The expression (3) is called a pseudo-remainder formula; Q is called the pseudo-quotient and R, the pseudo-remainder of G with respect to F in xk, denoted by pquo(G, F, xk) and prem(G, F, xk), respectively. Actually, the polynomials Q and R in (3) are uniquely determined by G and F (see [6] for details).
2. Algorithm newprem In this section, one gives an alternative method for computing the pseudoremainder of multivariate polynomials. A new algorithm of pseudo-division is described in details.
486
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
Theorem 2.1. Let G and F be two polynomials in R[x] and xk a fixed variable as above forms (1) and (2). Then prem(G, F, xk) = (1)mdet(A), where 1 0 Gm1 Gm G0 G1 Gn 9 > C BF > C > B 0 F 1 F n1 F n > C > B > = C B F0 F1 F n1 F n C B mnþ2 C B .. .. .. C > B > > C > B . . . > C. > A¼B ; C B F0 F1 F n1 F n C B C 9 B C > B 1 xk C = B C B .. .. C B n1 A > @ . . ; 1 xk Proof. Consider the following equations: 0
G
B xmn F B k B mn1 Bx F B k B . B .. B B B B x0k F B B 0 B B .. B @ .
1 C C C 0 m 1 C xk C 0 1 C 0 B m1 C C x C B k C B. C C B C¼AB C ¼ B . C. C B ... C @ . A C A @ C 0 C C 1 C C A
0 Eliminating elements G0, . . ., Gmn+1 in the first row of A by the elementary row operations, we can obtain that 1 0 premðG; F ; xk Þ=F 0mnþ1 C B xmn F C B k C B 0 m 1 mn1 C B xk F xk C B 0 1 C B 0 B m1 C C B .. B xk C B C C B . .. C C B C¼BB B ¼B .. C C @ . A; B 0 C B C B xk F @ . A C B 0 C B 0 C B 1 C B .. C B A @ . 0
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
where
0 B BF0 B B B B B B¼B B B B B B @
C0 F1 .. .. . . F0
F n1 .. . F1 1
xk .. .
C n2
C n1
Fn
F n1 ..
Fn
. 1
1 C C C C C C C C. C C C C C A
xk
9 > > > = > > > ; 9 > = > ;
487
mnþ2
n1
It is easy to see that premðG; F ; xk Þ=F 0mnþ1 ¼ C 0 xkn1 þ C 1 xn2 þ þ C n1 ¼ k
n1 X
C i xkn1i .
i¼0
We know that detðBÞ ¼
n1 X ðmnþ2þiÞþ1 ð1Þ C i detðBi Þ; i¼0
where Bi is the submatrix found by deleting the first row and the ith column of A, namely, 1 0 F0 F1 Fn C B .. .. .. C B . . . C B C B B F0 C C B C B 1 xk C B C B C B .. .. Bi ¼ B C. . . C B C B 1 xk C 9 B C B C > B 1 x k C = B C B i . . B .. C .. ; A > @ 1 One can see that detðBi Þ ¼ ðxk Þ This implies that detðBÞ ¼
ni1
F 0mnþ1 for i = 0, , n 1.
n1 X ðmnþ2þiÞþ1 n1i mnþ1 ð1Þ C i ðxk Þ F0 i¼0
¼ ð1Þ
mþ2
F 0mnþ1
n1 X i¼0
C i xkn1i .
488
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
Thus, premðG; F ; xk Þ ¼ ð1Þ The proof is complete.
mþ2
m
detðBÞ ¼ ð1Þ detðAÞ.
h
Example 2.1. Let F = xy2 + 1 and G = 2y3 y2 + x2y. With respect to y, the corresponding prem(G, F, y) can be calculated as follows. Construct matrix A according to Theorem 2.1, 2 3 2 1 x2 0 6x 0 1 0 7 6 7 A¼6 7. 40 x 0 1 5 0
0
1
y
Gaussian elimination on A is used, we obtain that 2 3 2 1 x2 0 60 x 7 0 1 6 7 B¼6 7. 40 0 5 1 y 3 0 0 0 1=2 þ y 1=2yx Then, 3
premðG; F ; yÞ ¼ ð1Þ detðBÞ ¼ x 2xy þ yx4 . The next algorithm newprem follows directly from Theorem 2.1. Algorithm newprem. R prem(G, F, xk). Given two polynomials G, F 2 R[x] and a variable xk (1 6 k 6 n), this algorithm computes the pseudo-remainder R of G with respect to F in xk. P1. Set m deg(G, xk), n deg(F, xk). P2. Construct a m + 1 by m + 1 matrix A according to Theorem 2.1. P3. Gaussian elimination on A is used and return matrix B, set R (1)mdet(B). Moreover, we know that the integer q in form (3) may be determined as small as possible, provided that the division process does not introduce fractions in Q and R. One can take q = 1 instead of 2 in Example 2.1 so that it simplifies to xG ¼ ð2y 1ÞF þ x3 y 2y þ 1. Taking the smallest q is rather crucial for control of the size expansion of the pseudo-remainder in many algorithms (see [14] for the details). Moreover, one
Y.-B. Li / Appl. Math. Comput. 173 (2006) 484–492
489 q
can modify the pseudo-remainder formula (3) by replacing Iq with I 11 I qe e , where I1, . . ., Ie are all the distinct irreducible factors of I, and choosing the smallest q1, . . ., qe so that the corresponding pseudo-remainder formula still holds. Definition 2.1. Let P1, . . ., Ps be polynomials in R[x] which are not all 0. A polynomial G* 2 R[x] is called a greatest common divisor (GCD) of P1, . . ., Ps if G divides P1, . . ., Ps and every common divisor of P1, . . ., Ps divides G*. Now let K be the quotient field of R. A very fundamental problem is to compute GCD(P1, . . ., Ps) in the above definition in K[x]. Now, powerful algorithms have been well developed (see [6] for the details) and implemented in popular computer algebra systems. We can freely use such algorithms and software systems when polynomial factorization over K is necessary. One can modify formula pseudo-remainder (3) as follows: q
I 11 I qe e G ¼ Q F þ R ; q
where R* = R/GCD(Iq, R), I 11 I qe e ¼ I q =GCDðI q ; RÞ and Q* = Q/GCD(Iq, R).
3. Implementation The algorithm newprem has been implemented in Maple system. The efficiency of algorithm newprem depends heavily on the one of Gaussian elimination. Our experiments indicate that it has not any advantage when computing the pseudo-remainder of univariate polynomials, but it is rather efficient in some cases of multivariate polynomials. Here the set of eight examples is taken to show the performance of our draft implementation. The following timings in CPU seconds are obtained in Maple 9.5 on Pentium III PC. Timings
Ex. 1
Ex. 2
Ex. 3
Ex. 4
Ex. 5
Ex. 6
Ex. 7
Ex. 8
prem 41.820 24.134 48.610 234.296 P500 92.372 28.391 55.911 newprem 23.644 8.832 3.013 9.774 55.810 57.972 10.995 10.525
Example 3.1 G1 ¼ 25y 7 x27 uz49 53y 27 x18 uz31 þ 57x20 u11 z25 þ 5y 17 x16 u37 z7 þ 81y 52 x16 u4 z3 37y 30 x7 u16 z; F 1 ¼ 57yu7 z5 2y 2 x4 uz5 31y 5 x4 z3 34x8 uz2 þ 24y 4 x5 u2 z þ 89yx5 u2 .
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Example 3.2 G2 ¼ 65y 8 x30 u2 z39 þ 34y 2 x41 u4 z26 þ 59y 24 x20 u23 z16 32y 19 x2 z11 þ 13y 47 u13 z9 32y 4 x19 u53 z2 ; F 2 ¼ 34u2 z3 þ 77xu4 z3 49y 3 x2 u3 z þ 83y 6 u5 z þ 32y 3 xu 55y 2 x7 u3 . Example 3.3 G3 ¼78y 4 x47 z77 þ 68y 65 x43 z68 þ 94y 47 x85 z21 66y 14 x24 z16 þ 39y 94 x29 z14 32y 47 x9 z5 ; F 3 ¼ 88y 2 z6 59y 3 x7 z6 þ 25y 4 x5 z6 73x2 y 10 z2 þ 4y 3 x10 z2 43y 4 x6 z2 . Example 3.4 G4 ¼ 75y 7 x22 z119 þ 12y 25 x19 z78 71xy 59 z68 49y 46 x7 z59 þ 23y 63 x72 z7 46y 62 x53 z4 ; F 4 ¼ 79y 5 z7 53y 6 z7 80y 2 x11 z þ 26y 2 x5 z 85y 3 þ 32y 13 x. Example 3.5 G5 ¼ 45y 5 x6 z86 þ 8y 24 x48 z31 þ 22y 5 x44 z30 99y 62 x55 z14 þ 60y 72 x47 z14 þ 11y 17 x38 z8 ; F 5 ¼ 50y 2 z6 63y 3 x5 z6 þ 98y 8 x2 z6 21yx9 z6 þ x9 z3 24x15 z. Example 3.6 G6 ¼ 38yx5 u35 w7 z40 50y 14 x13 u36 v5 w54 z17 þ 50y 21 x4 u55 v16 w8 z17 99y 19 x13 u5 v7 w55 z11 þ 77y 20 x8 u12 v52 w35 z9 þ 11y 38 x58 uv13 w16 z5 ; F 6 ¼ 81uv4 wz þ 81x7 uv2 z 35y 3 xuv2 þ 53y 5 x4 u þ 23y 3 u3 w9 23y 2 x7 v7 .
Example 3.7 G7 ¼ 53y 31 x10 u5 vw21 z50 þ 72y 28 x9 u5 v42 w21 z33 15y 48 x28 u5 w29 z24 þ 85y 59 x11 u3 v43 w5 z8 64y 20 x25 u19 v8 w27 z6 þ 31y 39 x6 u5 v31 w14 z3 ; F 7 ¼ 89y 4 x2 u3 vwz5 64x4 v3 w4 z5 62x4 u2 v7 z2 þ 28y 2 x6 u2 z2 þ 80y 4 u6 v3 z 66y 8 v3 wz.
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Example 3.8 G8 ¼ 31y 2 xu27 v31 w23 z52 þ 6y 48 x6 u13 v11 w10 z49 þ 50y 10 x5 u8 v82 w6 z29 þ 90y 9 x40 u6 v5 w34 z10 þ 82y 16 x2 u6 v14 w21 z3 þ 80y 10 x18 u34 v63 z2 ; F 8 ¼F 7 .
Acknowledgement The author thanks specially his friend Prof. Mingsheng Wang who mentioned this interest topic on a walk in Chengdu, January, 2005.
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