Sun Jan 3 23:45:35 MET 1999
REDUCE 3.7, 15-Jan-99 ...
1: 1:
2: 2: 2: 2: 2: 2: 2: 2: 2:
3: 3: % Tests of the poly package polynomial decomposition and gcds.
% Test for the univariate and multivariate polynomial decomposition.
% Herbert Melenk, ZIB Berlin, 1990.
procedure testdecompose u;
begin scalar r,p,val,nextvar;
write "decomposition of ",u;
r := decompose u;
if length r = 1 then rederr "decomposition failed";
write " leads to ",r;
% test if the result is algebraically correct.
r := reverse r;
nextvar := lhs first r; val := rhs first r;
r := rest r;
while not(r={}) do
<< p := first r; r := rest r;
if 'equal = part(p,0) then
<<val := sub(nextvar=val,rhs p); nextvar := lhs p>>
else
val := sub(nextvar=val,p);
>>;
if val = u then write " O.K. "
else
<<write "**** reconstructed polynomial: ";
write val;
rederr "reconstruction leads to different polynomial";
>>;
end;
testdecompose
% univariate decompositions
testdecompose(x**4+x**2+1);
4 2
decomposition of x + x + 1
2 2
leads to {u + u + 1,u=x }
O.K.
testdecompose(x**6+9x**5+52x**4+177x**3+435x**2+630x+593);
6 5 4 3 2
decomposition of x + 9*x + 52*x + 177*x + 435*x + 630*x + 593
3 2 2
leads to {u + 25*u + 210*u + 593,u=x + 3*x}
O.K.
testdecompose(x**6+6x**4+x**3+9x**2+3x-5);
6 4 3 2
decomposition of x + 6*x + x + 9*x + 3*x - 5
2 3
leads to {u + u - 5,u=x + 3*x}
O.K.
testdecompose(x**8-88*x**7+2924*x**6-43912*x**5+263431*x**4-218900*x**3+
65690*x**2-7700*x+234);
8 7 6 5 4 3
decomposition of x - 88*x + 2924*x - 43912*x + 263431*x - 218900*x
2
+ 65690*x - 7700*x + 234
2
leads to {u + 35*u + 234,
2
u=v + 10*v,
2
v=x - 22*x}
O.K.
% multivariate cases
testdecompose(u**2+v**2+2u*v+1);
2 2
decomposition of u + 2*u*v + v + 1
2
leads to {w + 1,w=u + v}
O.K.
testdecompose(x**4+2x**3*y + 3x**2*y**2 + 2x*y**3 + y**4 + 2x**2*y
+2x*y**2 + 2y**3 + 5 x**2 + 5*x*y + 6*y**2 + 5y + 9);
4 3 2 2 2 2 3 2
decomposition of x + 2*x *y + 3*x *y + 2*x *y + 5*x + 2*x*y + 2*x*y + 5*x*y
4 3 2
+ y + 2*y + 6*y + 5*y + 9
2 2 2
leads to {u + 5*u + 9,u=x + x*y + y + y}
O.K.
testdecompose sub(u=(2 x**2 + 17 x+y + y**3),u**2+2 u + 1);
4 3 2 3 2 2 3
decomposition of 4*x + 68*x + 4*x *y + 4*x *y + 293*x + 34*x*y + 34*x*y
6 4 3 2
+ 34*x + y + 2*y + 2*y + y + 2*y + 1
2 2 3
leads to {u + 2*u + 1,u=2*x + 17*x + y + y}
O.K.
testdecompose sub(u=(2 x**2 *y + 17 x+y + y**3),u**2+2 u + 1);
4 2 3 2 4 2 2 2 2
decomposition of 4*x *y + 68*x *y + 4*x *y + 4*x *y + 4*x *y + 289*x
3 6 4 3 2
+ 34*x*y + 34*x*y + 34*x + y + 2*y + 2*y + y + 2*y + 1
2 2 3
leads to {u + 2*u + 1,u=2*x *y + 17*x + y + y}
O.K.
% some cases which require a special (internal) mapping
testdecompose ( (x + y)**2);
2 2
decomposition of x + 2*x*y + y
2
leads to {u ,u=x + y}
O.K.
testdecompose ((x + y**2)**2);
2 2 4
decomposition of x + 2*x*y + y
2 2
leads to {u ,u=x + y }
O.K.
testdecompose ( (x**2 + y)**2);
4 2 2
decomposition of x + 2*x *y + y
2 2
leads to {u ,u=x + y}
O.K.
testdecompose ( (u + v)**2 +10 );
2 2
decomposition of u + 2*u*v + v + 10
2
leads to {w + 10,w=u + v}
O.K.
% the decomposition is not unique and might generate quite
% different images:
testdecompose ( (u + v + 10)**2 -100 );
2 2
decomposition of u + 2*u*v + 20*u + v + 20*v
leads to {w*(w + 20),w=u + v}
O.K.
% some special (difficult) cases
testdecompose (X**4 + 88*X**3*Y + 2904*X**2*Y**2 - 10*X**2
+ 42592*X*Y**3 - 440*X*Y + 234256*Y**4 - 4840*Y**2);
4 3 2 2 2 3
decomposition of x + 88*x *y + 2904*x *y - 10*x + 42592*x*y - 440*x*y
4 2
+ 234256*y - 4840*y
2
leads to {u*(u - 10),u=v ,v=x + 22*y}
O.K.
% a polynomial with complex coefficients
on complex;
testdecompose(X**4 + (88*I)*X**3*Y - 2904*X**2*Y**2 - 10*X**2 -
(42592*I)*X*Y**3 - (440*I)*X*Y + 234256*Y**4 + 4840*Y**2);
4 3 2 2 2 3
decomposition of x + 88*i*x *y - 2904*x *y - 10*x - 42592*i*x*y - 440*i*x*y
4 2
+ 234256*y + 4840*y
2
leads to {u*(u - 10),u=v ,v=x + 22*i*y}
O.K.
off complex;
% Examples given by J. Gutierrez and J.M. Olazabal.
f1:=x**6-2x**5+x**4-3x**3+3x**2+5$
testdecompose(f1);
6 5 4 3 2
decomposition of x - 2*x + x - 3*x + 3*x + 5
2 3 2
leads to {u - 3*u + 5,u=x - x }
O.K.
f2:=x**32-1$
testdecompose(f2);
32
decomposition of x - 1
2 2 2 2 2
leads to {u - 1,u=v ,v=w ,w=a ,a=x }
O.K.
f3:=x**4-(2/3)*x**3-(26/9)*x**2+x+3$
testdecompose(f3);
4 3 2
9*x - 6*x - 26*x + 9*x + 27
decomposition of --------------------------------
9
2
u - 9*u + 27 2
leads to {---------------,u=3*x - x}
9
O.K.
f4:=sub(x=x**4-x**3-2x+1,x**3-x**2-1)$
testdecompose(f4);
12 11 10 9 8 7 6 5
decomposition of x - 3*x + 3*x - 7*x + 14*x - 10*x + 14*x - 20*x
4 3 2
+ 9*x - 9*x + 8*x - 2*x - 1
3 2 4 3
leads to {u + 2*u + u - 1,u=x - x - 2*x}
O.K.
f5:=sub(x=f4,x**5-5)$
testdecompose(f5);
60 59 58 57 56 55
decomposition of x - 15*x + 105*x - 485*x + 1795*x - 5873*x
54 53 52 51 50
+ 17255*x - 45845*x + 112950*x - 261300*x + 567203*x
49 48 47 46
- 1164475*x + 2280835*x - 4259830*x + 7604415*x
45 44 43 42
- 13053437*x + 21545220*x - 34200855*x + 52436150*x
41 40 39 38
- 77668230*x + 111050794*x - 153746645*x + 206190770*x
37 36 35
- 267484170*x + 336413145*x - 410387890*x
34 33 32
+ 484672110*x - 555048350*x + 616671710*x
31 30 29
- 663135380*x + 690884384*x - 697721320*x
28 27 26
+ 681039235*x - 642661265*x + 586604975*x
25 24 23
- 516016275*x + 437051535*x - 356628245*x
22 21 20
+ 278991765*x - 208571965*x + 149093999*x
19 18 17 16
- 101204325*x + 64656350*x - 38848040*x + 21710870*x
15 14 13 12
- 10971599*x + 4928210*x - 1904450*x + 519730*x
11 10 9 8 7
- 15845*x - 71947*x + 52015*x - 26740*x + 5510*x
6 5 4 3
+ 3380*x - 1972*x - 75*x + 195*x - 10*x - 6
5 4 3 2
leads to {u - 5*u + 10*u - 10*u + 5*u - 6,
3 2
u=v + 2*v + v,
4 3
v=x - x - 2*x}
O.K.
clear f1,f2,f3,f4,f5;
% Tests of gcd code.
% The following examples were introduced in Moses, J. and Yun, D.Y.Y.,
% "The EZ GCD Algorithm", Proc. ACM 73 (1973) 159-166, and considered
% further in Hearn, A.C., "Non-modular Computation of Polynomial GCD's
% Using Trial Division", Proc. EUROSAM 79, 227-239, 72, published as
% Lecture Notes on Comp. Science, # 72, Springer-Verlag, Berlin, 1979.
on gcd;
% The following is the best setting for this file.
on ezgcd;
% In systems that have the heugcd code, the following is also a
% possibility, although not all examples complete in a reasonable time.
% load heugcd; on heugcd;
% The final alternative is to use neither ezgcd nor heugcd. In that case,
% most examples take excessive amounts of computer time.
share n;
operator xx;
% Case 1.
for n := 2:5
do write gcd(((for i:=1:n sum xx(i))-1)*((for i:=1:n sum xx(i)) + 2),
((for i:=1:n sum xx(i))+1)
*(-3xx(2)*xx(1)**2+xx(2)**2-1)**2);
1
1
1
1
% Case 2.
let d = (for i:=1:n sum xx(i)**n) + 1;
for n := 2:7 do write gcd(d*((for i:=1:n sum xx(i)**n) - 2),
d*((for i:=1:n sum xx(i)**n) + 2));
2 2
xx(2) + xx(1) + 1
3 3 3
xx(3) + xx(2) + xx(1) + 1
4 4 4 4
xx(4) + xx(3) + xx(2) + xx(1) + 1
5 5 5 5 5
xx(5) + xx(4) + xx(3) + xx(2) + xx(1) + 1
6 6 6 6 6 6
xx(6) + xx(5) + xx(4) + xx(3) + xx(2) + xx(1) + 1
7 7 7 7 7 7 7
xx(7) + xx(6) + xx(5) + xx(4) + xx(3) + xx(2) + xx(1) + 1
for n := 2:7 do write gcd(d*((for i:=1:n sum xx(i)**n) - 2),
d*((for i:=1:n sum xx(i)**(n-1)) + 2));
2 2
xx(2) + xx(1) + 1
3 3 3
xx(3) + xx(2) + xx(1) + 1
4 4 4 4
xx(4) + xx(3) + xx(2) + xx(1) + 1
5 5 5 5 5
xx(5) + xx(4) + xx(3) + xx(2) + xx(1) + 1
6 6 6 6 6 6
xx(6) + xx(5) + xx(4) + xx(3) + xx(2) + xx(1) + 1
7 7 7 7 7 7 7
xx(7) + xx(6) + xx(5) + xx(4) + xx(3) + xx(2) + xx(1) + 1
% Case 3.
let d = xx(2)**2*xx(1)**2 + (for i := 3:n sum xx(i)**2) + 1;
for n := 2:5
do write gcd(d*(xx(2)*xx(1) + (for i:=3:n sum xx(i)) + 2)**2,
d*(xx(1)**2-xx(2)**2 + (for i:=3:n sum xx(i)**2) - 1));
2 2
xx(2) *xx(1) + 1
2 2 2
xx(3) + xx(2) *xx(1) + 1
2 2 2 2
xx(4) + xx(3) + xx(2) *xx(1) + 1
2 2 2 2 2
xx(5) + xx(4) + xx(3) + xx(2) *xx(1) + 1
% Case 4.
let u = xx(1) - xx(2)*xx(3) + 1,
v = xx(1) - xx(2) + 3xx(3);
gcd(u*v**2,v*u**2);
2 2
3*xx(3) *xx(2) - xx(3)*xx(2) + xx(3)*xx(2)*xx(1) - 3*xx(3)*xx(1) - 3*xx(3)
2
+ xx(2)*xx(1) + xx(2) - xx(1) - xx(1)
gcd(u*v**3,v*u**3);
2 2
3*xx(3) *xx(2) - xx(3)*xx(2) + xx(3)*xx(2)*xx(1) - 3*xx(3)*xx(1) - 3*xx(3)
2
+ xx(2)*xx(1) + xx(2) - xx(1) - xx(1)
gcd(u*v**4,v*u**4);
2 2
3*xx(3) *xx(2) - xx(3)*xx(2) + xx(3)*xx(2)*xx(1) - 3*xx(3)*xx(1) - 3*xx(3)
2
+ xx(2)*xx(1) + xx(2) - xx(1) - xx(1)
gcd(u**2*v**4,v**2*u**4);
4 2 3 3 3 2
9*xx(3) *xx(2) - 6*xx(3) *xx(2) + 6*xx(3) *xx(2) *xx(1)
3 3 2 4
- 18*xx(3) *xx(2)*xx(1) - 18*xx(3) *xx(2) + xx(3) *xx(2)
2 3 2 2 2 2 2
- 2*xx(3) *xx(2) *xx(1) + xx(3) *xx(2) *xx(1) + 12*xx(3) *xx(2) *xx(1)
2 2 2 2 2
+ 12*xx(3) *xx(2) - 12*xx(3) *xx(2)*xx(1) - 12*xx(3) *xx(2)*xx(1)
2 2 2 2 3
+ 9*xx(3) *xx(1) + 18*xx(3) *xx(1) + 9*xx(3) - 2*xx(3)*xx(2) *xx(1)
3 2 2 2
- 2*xx(3)*xx(2) + 4*xx(3)*xx(2) *xx(1) + 4*xx(3)*xx(2) *xx(1)
3 2
- 2*xx(3)*xx(2)*xx(1) - 8*xx(3)*xx(2)*xx(1) - 12*xx(3)*xx(2)*xx(1)
3 2
- 6*xx(3)*xx(2) + 6*xx(3)*xx(1) + 12*xx(3)*xx(1) + 6*xx(3)*xx(1)
2 2 2 2 3 2
+ xx(2) *xx(1) + 2*xx(2) *xx(1) + xx(2) - 2*xx(2)*xx(1) - 4*xx(2)*xx(1)
4 3 2
- 2*xx(2)*xx(1) + xx(1) + 2*xx(1) + xx(1)
% Case 5.
let d = (for i := 1:n product (xx(i)+1)) - 3;
for n := 2:5 do write gcd(d*for i := 1:n product (xx(i) - 2),
d*for i := 1:n product (xx(i) + 2));
xx(2)*xx(1) + xx(2) + xx(1) - 2
xx(3)*xx(2)*xx(1) + xx(3)*xx(2) + xx(3)*xx(1) + xx(3) + xx(2)*xx(1) + xx(2)
+ xx(1) - 2
xx(4)*xx(3)*xx(2)*xx(1) + xx(4)*xx(3)*xx(2) + xx(4)*xx(3)*xx(1) + xx(4)*xx(3)
+ xx(4)*xx(2)*xx(1) + xx(4)*xx(2) + xx(4)*xx(1) + xx(4) + xx(3)*xx(2)*xx(1)
+ xx(3)*xx(2) + xx(3)*xx(1) + xx(3) + xx(2)*xx(1) + xx(2) + xx(1) - 2
xx(5)*xx(4)*xx(3)*xx(2)*xx(1) + xx(5)*xx(4)*xx(3)*xx(2)
+ xx(5)*xx(4)*xx(3)*xx(1) + xx(5)*xx(4)*xx(3) + xx(5)*xx(4)*xx(2)*xx(1)
+ xx(5)*xx(4)*xx(2) + xx(5)*xx(4)*xx(1) + xx(5)*xx(4) + xx(5)*xx(3)*xx(2)*xx(1)
+ xx(5)*xx(3)*xx(2) + xx(5)*xx(3)*xx(1) + xx(5)*xx(3) + xx(5)*xx(2)*xx(1)
+ xx(5)*xx(2) + xx(5)*xx(1) + xx(5) + xx(4)*xx(3)*xx(2)*xx(1)
+ xx(4)*xx(3)*xx(2) + xx(4)*xx(3)*xx(1) + xx(4)*xx(3) + xx(4)*xx(2)*xx(1)
+ xx(4)*xx(2) + xx(4)*xx(1) + xx(4) + xx(3)*xx(2)*xx(1) + xx(3)*xx(2)
+ xx(3)*xx(1) + xx(3) + xx(2)*xx(1) + xx(2) + xx(1) - 2
clear d,u,v;
% The following examples were discussed in Char, B.W., Geddes, K.O.,
% Gonnet, G.H., "GCDHEU: Heuristic Polynomial GCD Algorithm Based
% on Integer GCD Computation", Proc. EUROSAM 84, 285-296, published as
% Lecture Notes on Comp. Science, # 174, Springer-Verlag, Berlin, 1984.
% Maple Problem 1.
gcd(34*x**80-91*x**99+70*x**31-25*x**52+20*x**76-86*x**44-17*x**33
-6*x**89-56*x**54-17,
91*x**49+64*x**10-21*x**52-88*x**74-38*x**76-46*x**84-16*x**95
-81*x**72+96*x**25-20);
1
% Maple Problem 2.
g := 34*x**19-91*x+70*x**7-25*x**16+20*x**3-86;
19 16 7 3
g := 34*x - 25*x + 70*x + 20*x - 91*x - 86
gcd(g * (64*x**34-21*x**47-126*x**8-46*x**5-16*x**60-81),
g * (72*x**60-25*x**25-19*x**23-22*x**39-83*x**52+54*x**10+81) );
19 16 7 3
34*x - 25*x + 70*x + 20*x - 91*x - 86
% Maple Problem 3.
gcd(3427088418+8032938293*x-9181159474*x**2-9955210536*x**3
+7049846077*x**4-3120124818*x**5-2517523455*x**6+5255435973*x**7
+2020369281*x**8-7604863368*x**9-8685841867*x**10+4432745169*x**11
-1746773680*x**12-3351440965*x**13-580100705*x**14+8923168914*x**15
-5660404998*x**16 +5441358149*x**17-1741572352*x**18
+9148191435*x**19-4940173788*x**20+6420433154*x**21+980100567*x**22
-2128455689*x**23+5266911072*x**24-8800333073*x**25-7425750422*x**26
-3801290114*x**27-7680051202*x**28-4652194273*x**29-8472655390*x**30
-1656540766*x**31+9577718075*x**32-8137446394*x**33+7232922578*x**34
+9601468396*x**35-2497427781*x**36-2047603127*x**37-1893414455*x**38
-2508354375*x**39-2231932228*x**40,
2503247071-8324774912*x+6797341645*x**2+5418887080*x**3
-6779305784*x**4+8113537696*x**5+2229288956*x**6+2732713505*x**7
+9659962054*x**8-1514449131*x**9+7981583323*x**10+3729868918*x**11
-2849544385*x**12-5246360984*x**13+2570821160*x**14-5533328063*x**15
-274185102*x**16+8312755945*x**17-2941669352*x**18-4320254985*x**19
+9331460166*x**20-2906491973*x**21-7780292310*x**22-4971715970*x**23
-6474871482*x**24-6832431522*x**25-5016229128*x**26-6422216875*x**27
-471583252*x**28+3073673916*x**29+2297139923*x**30+9034797416*x**31
+6247010865*x**32+5965858387*x**33-4612062748*x**34+5837579849*x**35
-2820832810*x**36-7450648226*x**37+2849150856*x**38+2109912954*x**39
+2914906138*x**40);
1
% Maple Problem 4.
g := 34271+80330*x-91812*x**2-99553*x**3+70499*x**4-31201*x**5
-25175*x**6+52555*x**7+20204*x**8-76049*x**9-86859*x**10;
10 9 8 7 6 5
g := - 86859*x - 76049*x + 20204*x + 52555*x - 25175*x - 31201*x
4 3 2
+ 70499*x - 99553*x - 91812*x + 80330*x + 34271
gcd(g * (44328-17468*x-33515*x**2-5801*x**3+89232*x**4-56604*x**5
+54414*x**6-17416*x**7+91482*x**8-49402*x**9+64205*x**10
+9801*x**11-21285*x**12+52669*x**13-88004*x**14-74258*x**15
-38013*x**16-76801*x**17-46522*x**18-84727*x**19-16565*x**20
+95778*x**21-81375*x**22+72330*x**23+96015*x**24-24974*x**25
-20476*x**26-18934*x**27-25084*x**28-22319*x**29+25033*x**30),
g * (-83248+67974*x+54189*x**2-67793*x**3+81136*x**4+22293*x**5
+27327*x**6+96600*x**7-15145*x**8+79816*x**9+37299*x**10
-28496*x**11-52464*x**12+25708*x**13-55334*x**14-2742*x**15
+83128*x**16-29417*x**17-43203*x**18+93315*x**19-29065*x**20
-77803*x**21-49717*x**22-64749*x**23-68325*x**24-50163*x**25
-64222*x**26-4716*x**27+30737*x**28+22972*x**29+90348*x**30));
10 9 8 7 6 5 4
86859*x + 76049*x - 20204*x - 52555*x + 25175*x + 31201*x - 70499*x
3 2
+ 99553*x + 91812*x - 80330*x - 34271
% Maple Problem 5.
gcd(-8472*x**4*y**10-8137*x**9*y**10-2497*x**4*y**4-2508*x**4*y**6
-8324*x**9*y**8-6779*x**9*y**6+2733*x**10*y**4+7981*x**7*y**3
-5246*x**6*y**2-274*x**10*y**3-4320,
15168*x**3*y-4971*x*y-2283*x*y**5+3074*x**6*y**10+6247*x**8*y**2
+2849*x**6*y**7-2039*x**7-2626*x**2*y**7+9229*x**6*y**5+2404*y**5
+1387*x**4*y**8+5602*x**5*y**2-6212*x**3*y**7-8561);
1
% Maple Problem 6.
g := -19*x**4*y**4+25*y**9+54*x*y**9+22*x**7*y**10-15*x**9*y**7-28;
9 7 7 10 4 4 9 9
g := - 15*x *y + 22*x *y - 19*x *y + 54*x*y + 25*y - 28
gcd(g*(91*x**2*y**9+10*x**4*y**8-88*x*y**3-76*x**2-16*x**10*y
+72*x**10*y**4-20),
g*(34*x**9-99*x**9*y**3-25*x**8*y**6-76*y**7-17*x**3*y**5
+89*x**2*y**8-17));
9 7 7 10 4 4 9 9
15*x *y - 22*x *y + 19*x *y - 54*x*y - 25*y + 28
% Maple Problem 7.
gcd(6713544209*x**9+8524923038*x**3*y**3*z**7+6010184640*x*z**7
+4126613160*x**3*y**4*z**9+2169797500*x**7*y**4*z**9
+2529913106*x**8*y**5*z**3+7633455535*y*z**3+1159974399*x**2*z**4
+9788859037*y**8*z**9+3751286109*x**3*y**4*z**3,
3884033886*x**6*z**8+7709443539*x*y**9*z**6
+6366356752*x**9*y**4*z**8+6864934459*x**3*y**2*z**6
+2233335968*x**4*y**9*z**3+2839872507*x**9*y**3*z
+2514142015*x*y*z**2+1788891562*x**4*y**6*z**6
+9517398707*x**8*y**7*z**2+7918789924*x**3*y*z**6
+6054956477*x**6*y**3*z**6);
1
% Maple Problem 8.
g := u**3*(x**2-y)*z**2+(u-3*u**2*x)*y*z-u**4*x*y+3;
4 3 2 2 3 2 2
g := - u *x*y + u *x *z - u *y*z - 3*u *x*y*z + u*y*z + 3
gcd(g * ((y**2+x)*z**2+u**5*(x*y+x**2)*z-y+5),
g * ((y**2-x)*z**2+u**5*(x*y-x**2)*z+y+9) );
4 3 2 2 3 2 2
u *x*y - u *x *z + u *y*z + 3*u *x*y*z - u*y*z - 3
% Maple Problem 9.
g := 34*u**2*y**2*z-25*u**2*v*z**2-18*v*x**2*z**2-18*u**2*x**2*y*z+53
+x**3;
2 2 2 2 2 2 2 2 3
g := - 25*u *v*z - 18*u *x *y*z + 34*u *y *z - 18*v*x *z + x + 53
gcd( g * (-85*u*v**2*y**2*z**2-25*u*v*x*y*z-84*u**2*v**2*y**2*z
+27*u**2*v*x**2*y**2*z-53*u*x*y**2*z+34*x**3),
g * (48*x**3-99*u*x**2*y**2*z-69*x*y*z-75*u*v*x*y*z**2
-43*u**2*v+91*u**2*v**2*y**2*z) );
2 2 2 2 2 2 2 2 3
25*u *v*z + 18*u *x *y*z - 34*u *y *z + 18*v*x *z - x - 53
% Maple Problem 10.
gcd(-9955*v**9*x**3*y**4*z**8+2020*v*y**7*z**4
-3351*v**5*x**10*y**2*z**8-1741*v**10*x**2*y**9*z**6
-2128*v**8*y*z**3-7680*v**2*y**4*z**10-8137*v**9*x**10*y**4*z**4
-1893*v**4*x**4*y**6+6797*v**8*x*y**9*z**6
+2733*v**10*x**4*y**9*z**7-2849*v**2*x**6*y**2*z**5
+8312*v**3*x**3*y**10*z**3-7780*v**2*x*y*z**2
-6422*v**5*x**7*y**6*z**10+6247*v**8*x**2*y**8*z**3
-7450*v**7*x**6*y**7*z**4+3625*x**4*y**2*z**7+9229*v**6*x**5*y**6
-112*v**6*x**4*y**8*z**7-7867*v**5*x**8*y**5*z**2
-6212*v**3*x**7*z**5+8699*v**8*x**2*y**2*z**5
+4442*v**10*x**5*y**4*z+1965*v**10*y**3*z**3-8906*v**6*x*y**4*z**5
+5552*x**10*y**4+3055*v**5*x**3*y**6*z**2+6658*v**7*x**10*z**6
+3721*v**8*x**9*y**4*z**8+9511*v*x**6*y+5437*v**3*x**9*y**9*z**7
-1957*v**6*x**4*y*z**3+9214*v**3*x**9*y**3*z**7
+7273*v**2*x**8*y**4*z**10+1701*x**10*y**7*z**2
+4944*v**5*x**5*y**8*z**8-1935*v**3*x**6*y**10*z**7
+4029*x**6*y**10*z**3+9462*v**6*x**5*y**4*z**8-3633*v**4*x*y**7*z**5
-1876,
-5830*v**7*x**8*y*z**2-1217*v**8*x*y**2*z**5
-1510*v**9*x**3*y**10*z**10+7036*v**6*x**8*y**3*z**3
+1022*v**9*y**3*z**8+3791*v**8*x**3*y**7+6906*v**6*x*y*z**10
+117*v**7*x**2*y**4*z**4+6654*v**6*x**5*y**2*z**3
-7302*v**10*x**8*y**3-5343*v**8*x**5*y**9*z
-2244*v**9*x**3*y**8*z**9-3719*v**5*x**10*y**6*z**8
+2629*x**3*y**2*z**10+8517*x**9*y**6*z**7-9551*v**5*x**6*y**6*z**2
-7750*x**10*y**7*z**4-5035*v**5*x**2*y**5*z-5967*v**9*x**5*y**9*z**5
-8517*v**3*x**2*y**7*z**6-2668*v**10*y**9*z**4+1630*v**5*x**5*y*z**8
+9099*v**7*x**9*y**4*z**3-5358*v**9*x**5*y**6*z**2
+5766*v**5*y**3*z**4-3624*v*x**4*y**10*z**10
+8839*v**6*x**9*y**10*z**4+3378*x**7*y**2*z**5+7582*v**7*x*y**8*z**7
-85*v*x**2*y**9*z**6-9495*v**9*x**10*y**6*z**3+1983*v**9*x**3*y
-4613*v**10*x**4*y**7*z**6+5529*v**10*x*y**6
+5030*v**4*x**5*y**4*z**9-9202*x**6*y**3*z**9
-4988*v**2*x**2*y**10*z**4-8572*v**9*x**7*y**10*z**10
+4080*v**4*x**8*z**8-382*v**9*x**9*y**2*z**2-7326);
1
end;
4: 4: 4: 4: 4: 4: 4: 4: 4:
Time for test: 1980 ms, plus GC time: 60 ms
5: 5:
Quitting
Sun Jan 3 23:45:44 MET 1999