7
z*u+y*v+t*w-w**2-1/2*w-1/2;
z*u**2+y*v**2-t*w**2+w**3+w**2-1/3*t+4/3*w;
x*z*v-t*w**2+w**3-1/2*t*w+w**2-1/6*t+2/3*w;
z*u**3+y*v**3+t*w**3-w**4-3/2*w**3+t*w-5/2*w**2-1/4*w-1/4;
x*z*u*v+t*w**3-w**4+1/2*t*w**2-3/2*w**3+1/2*t*w-7/4*w**2-3/8*w-1/8;
x*z*v**2+t*w**3-w**4+t*w**2-3/2*w**3+2/3*t*w-7/6*w**2-1/12*w-1/12;
-t*w**3+w**4-t*w**2+3/2*w**3-1/3*t*w+13/12*w**2+7/24*w+1/24;
TITLE : Butcher's problem
ROOT COUNTS :
total degree : 4608
4-homogeneous Bezout number : 1361
with partition : {{z y t }{u v }{w }{x }}
multi-homogeneous Bezout number 1209,
with the following degree structure :
The partition for equation 1 : {{z y t }{u v }{w }}
The partition for equation 2 : {{z y t }{u v }{w }}
The partition for equation 3 : {{z t }{v }{w }{x }}
The partition for equation 4 : {{z y t }{u v }{w }}
The partition for equation 5 : {{z t }{u }{v }{w }{x }}
The partition for equation 6 : {{z t }{v }{w }{x }}
The partition for equation 7 : {{t }{w }}
generalized Bezout number : 605
based on the set structure :
{z y t w }{u v w }
{z y t w }{u v w }{u v w }
{z t w }{v w }{w x }
{z y t w }{u v w }{u v w }{u v w }
{z t w }{u w }{v w }{w x }
{z t w }{v w }{v w }{w x }
{t w }{w }{w }{w }
mixed volume: 24
REFERENCES :
The example has been retrieved from the POSSO test suite,
available by anonymous ftp from the site gauss.dm.unipi.it,
from the directory pub/posso.
See also
W. Boege, R. Gebauer, and H. Kredel:
"Some examples for solving systems of algebraic equations by
calculating Groebner bases", J. Symbolic Computation, 2:83-98, 1986.
C. Butcher: "An application of the Runge-Kutta space".
BIT, 24, pages 425--440, 1984.
NOTE:
There are 5 regular solutions and two singular solutions
The two singular solutions belong to a manifold of solutions:
t=-1=w, z=0=y, with u and v arbitrary complex numbers.
There are 3 regular real solutions.
THE SOLUTIONS :
7 7
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : -2.29379273840329E-02 7.44087924070508E-36
u : 8.16496580927762E-01 -1.29774576330781E-35
y : -4.58758547680744E-02 -6.84137712178571E-36
v : 4.08248290463845E-01 -1.10731567847459E-35
t : -1.00000000000000E+00 3.37954625861408E-37
w : -9.08248290463859E-01 -3.28550671054829E-36
x : 8.16496580927830E-01 -6.65800000305744E-35
== err : 4.965E-14 = rco : 1.736E-04 = res : 3.331E-16 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : 2.88808493551858E-01 -1.92592994438724E-34
u : 7.21933058546343E-01 9.62964972193618E-34
y : -2.44033884709223E-01 -9.14816723583937E-34
v : -6.24774425776102E-01 1.73333694994851E-33
t : 1.27806694145366E+00 -1.17963209093718E-33
w : 2.78066941453658E-01 -5.05556610401649E-34
x : -1.13792449427108E+00 -5.87408633038107E-33
== err : 2.785E-15 = rco : 1.273E-02 = res : 8.327E-17 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : -2.27062072615966E-01 -3.85185988877447E-34
u : -8.16496580927726E-01 1.92592994438724E-34
y : -4.54124145231932E-01 3.61111864572607E-34
v : -4.08248290463863E-01 1.38426214752833E-34
t : -1.00000000000000E+00 -2.46383615932351E-35
w : -9.17517095361370E-02 1.55165254308542E-36
x : -8.16496580927726E-01 0.00000000000000E+00
== err : 8.898E-16 = rco : 2.386E-03 = res : 2.776E-17 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : 7.32450810630072E-16 1.37145635748911E-15
u : 4.17630643926077E-01 -7.58875284416709E-01
y : 1.67709576191771E-15 -4.57212901658091E-15
v : 4.94521940247853E-01 -3.05129784662510E-02
t : -1.00000000000000E+00 4.68824600419586E-16
w : -1.00000000000000E+00 3.75059625315153E-15
x : -1.50630009976240E+00 -2.13582124544133E+00
== err : 0.000E+00 = rco : 2.854E-17 = res : 0.000E+00 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : 1.27097621050230E-16 3.03259676112193E-17
u : 2.34958226863350E+00 -1.87549206427436E+00
y : -1.91328685223823E-15 -3.04962129204523E-16
v : 1.73977586284498E+00 4.84530770879628E-01
t : -1.00000000000000E+00 -3.24271157948272E-15
w : -9.99999999999994E-01 -2.68584216774777E-15
x : 5.62376352154760E+00 -2.71542574259875E+00
== err : 0.000E+00 = rco : 8.236E-19 = res : 0.000E+00 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : 3.59196576269340E-01 -1.71748996563448E-01
u : 1.38903347072683E+00 -3.85150602548912E-01
y : 2.35528602162567E-01 9.24893027822689E-02
v : 1.22905387955472E+00 3.00007066016267E-01
t : 6.10966529273171E-01 3.85150602548917E-01
w : -3.89033470726829E-01 3.85150602548913E-01
x : 3.87782471854638E-01 -2.22654061728370E-01
== err : 4.135E-15 = rco : 1.198E-03 = res : 2.776E-16 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : 3.59196576269339E-01 1.71748996563450E-01
u : 1.38903347072683E+00 3.85150602548910E-01
y : 2.35528602162569E-01 -9.24893027822688E-02
v : 1.22905387955472E+00 -3.00007066016269E-01
t : 6.10966529273174E-01 -3.85150602548915E-01
w : -3.89033470726829E-01 -3.85150602548913E-01
x : 3.87782471854640E-01 2.22654061728369E-01
== err : 5.151E-15 = rco : 1.198E-03 = res : 4.965E-16 ==