```8
b1 + b2 + b3 - (a+b);
b2*c2 + b3*c3 - (1/2 + 1/2*b + b**2 - a*b);
b2*c2**2 + b3*c3**2 - (a*(1/3+b**2) - 4/3*b - b**2 - b**3);
b3*a32*c2 - (a*(1/6 + 1/2*b + b**2) - 2/3*b - b**2 - b**3);
b2*c2**3 + b3*c3**3 - (1/4 + 1/4*b + 5/2*b**2 + 3/2*b**3 + b**4 - a*(b+b**3));
b3*c3*a32*c2 - (1/8 + 3/8*b + 7/4*b**2 + 3/2*b**3 + b**4
- a*(1/2*b + 1/2*b**2 + b**3));
b3*a32*c2**2 - (1/12 + 1/12*b + 7/6*b**2 + 3/2*b**3 + b**4
- a*(2/3*b + b**2 + b**3));
1/24 + 7/24*b + 13/12*b**2 + 3/2*b**3 + b**4 - a*(1/3*b + b**2 + b**3);

TITLE : 8-variable version of Butcher's problem

ROOT COUNTS :

total degree : 4608
5-homogeneous Bezout number : 1361
with partition : {b1 }{b2 b3 a }{b }{c2 c3 }{a32 }
general linear-product Bezout number : 605
based on the set structure :
{ b1 b2 b3 a b }
{ b2 b3 a b }{ b c2 c3 }
{ b2 b3 a b }{ b c2 c3 }{ b c2 c3 }
{ b3 a b }{ b c2 }{ b a32 }
{ b2 b3 a b }{ b c2 c3 }{ b c2 c3 }{ b c2 c3 }
{ b3 a b }{ b c2 }{ b c3 }{ b a32 }
{ b3 a b }{ b c2 }{ b c2 }{ b a32 }
{ a b }{ b }{ b }{ b }
mixed volume : 24

REFERENCES :

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 :

The system has 5 regular solutions.  Two paths converged to highly
singular solutions, which indicates that the system probably has an
positive dimensional solutions component.

THE SOLUTIONS :
7 8
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
b1 : -4.10565491688240E-01   3.76158192263132E-37
b2 : -4.54124145231932E-01  -7.52316384526264E-37
b3 : -2.27062072615965E-01   5.17217514361807E-37
a : -1.00000000000000E+00   5.87747175411144E-38
b : -9.17517095361370E-02   6.39175053259619E-38
c2 : -4.08248290463863E-01  -3.52648305246686E-38
c3 : -8.16496580927726E-01   4.70197740328915E-37
a32 : -8.16496580927727E-01   1.41059322098675E-36
== err :  5.051E-15 = rco :  2.063E-03 = res :  1.422E-16 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
b1 : -3.72792119885564E-01   8.49560898879008E-01
b2 :  2.35528602162567E-01   9.24893027822684E-02
b3 :  3.59196576269339E-01  -1.71748996563447E-01
a :  6.10966529273170E-01   3.85150602548915E-01
b : -3.89033470726828E-01   3.85150602548914E-01
c2 :  1.22905387955472E+00   3.00007066016266E-01
c3 :  1.38903347072683E+00  -3.85150602548913E-01
a32 :  3.87782471854639E-01  -2.22654061728370E-01
== err :  1.629E-14 = rco :  1.346E-03 = res :  1.180E-15 ==
solution 3 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
b1 : -3.72792119885562E-01  -8.49560898879006E-01
b2 :  2.35528602162566E-01  -9.24893027822658E-02
b3 :  3.59196576269336E-01   1.71748996563446E-01
a :  6.10966529273169E-01  -3.85150602548912E-01
b : -3.89033470726828E-01  -3.85150602548915E-01
c2 :  1.22905387955472E+00  -3.00007066016267E-01
c3 :  1.38903347072683E+00   3.85150602548917E-01
a32 :  3.87782471854639E-01   2.22654061728372E-01
== err :  6.667E-15 = rco :  1.346E-03 = res :  1.228E-15 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
b1 : -2.00000000000000E+00   7.85417544673964E-15
b2 : -2.21811329325582E-15  -4.68716554016273E-15
b3 :  7.44660099998350E-16   1.32692288000702E-15
a : -1.00000000000000E+00   4.99325865176077E-16
b : -1.00000000000000E+00   3.99460692140786E-15
c2 :  6.10228849392220E-01   7.80261520828316E-01
c3 : -5.75622432602162E-01  -8.16657458049619E-01
a32 : -4.18737481532424E+00  -1.05540120319375E+00
== err :  6.808E+00 = rco :  7.493E-18 = res :  4.678E-14 ==
solution 5 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
b1 :  1.51135927406468E+00  -5.58329856879419E-46
b2 : -2.44033884709223E-01   2.24426707177021E-46
b3 :  2.88808493551859E-01   3.28429327576129E-46
a :  1.27806694145366E+00   6.36331822178750E-47
b :  2.78066941453657E-01  -6.91070043441438E-47
c2 : -6.24774425776101E-01   1.53267019535527E-46
c3 :  7.21933058546342E-01  -2.66164600889821E-46
a32 : -1.13792449427108E+00   4.26958125848968E-46
== err :  4.383E-15 = rco :  8.828E-03 = res :  7.078E-16 ==
solution 6 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
b1 : -1.83943450831177E+00  -1.24157286016722E-39
b2 : -4.58758547680644E-02   1.03270091626882E-39
b3 : -2.29379273840380E-02  -4.18192303305384E-40
a : -9.99999999999999E-01   7.74974102710197E-41
b : -9.08248290463868E-01  -7.04550447087089E-40
c2 :  4.08248290463805E-01  -2.26037289566993E-39
c3 :  8.16496580927691E-01  -3.95897084218590E-39
a32 :  8.16496580927679E-01   8.36563972814201E-40
== err :  7.525E-13 = rco :  1.468E-04 = res :  1.060E-15 ==
solution 7 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
b1 : -1.99999999999998E+00  -1.77453270253141E-14
b2 : -2.29308802652977E-14   1.60405121178499E-14
b3 :  2.07541705350947E-17   7.73078577002929E-16
a : -1.00000000000000E+00  -1.03526258940137E-16
b : -9.99999999999999E-01  -8.28210071521105E-16
c2 :  3.87642666200723E-02   2.48399890452917E-01
c3 : -5.33762129425664E-01  -4.01820103923349E-01
a32 : -3.94790779697364E+00   2.03513755526150E+01
== err :  2.515E+01 = rco :  2.347E-19 = res :  2.591E-14 ==
```