var a;
var a32;
var b;
var b1;
var b2;
var b3;
var c2;
var c3;

subject to

cons1:  b1 + b2 + b3 - (a+b) = 0;
cons2:  b2*c2 + b3*c3 - (1/2 + 1/2*b + b^2 - a*b) = 0;
cons3:  b2*c2^2 + b3*c3^2 - (a*(1/3+b^2) - 4/3*b - b^2 - b^3) = 0;
cons4:  b3*a32*c2 - (a*(1/6 + 1/2*b + b^2) - 2/3*b - b^2 - b^3) = 0;
cons5:  b2*c2*83 + b3*c3^3 - (1/4 + 1/4*b + 5/2*b^2 + 3/2*b^3 + b^4 - a*(b+b^3)) = 0;
cons6:  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)) = 0;
cons7:  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)) = 0;
cons8:  1/24 + 7/24*b + 13/12*b^2 + 3/2*b^3 + b^4 - a*(1/3*b + b^2 + b^3) = 0;

solve;
display a, a32, b, b1, b2, b3, c2, c3;

# TITLE : 8-variable version of Butcher's problem

# ROOT COUNTS :

# total degree : 4608
# 6-homogeneous Bezout number : 1461
#   with partition : {b1 }{b2 b3 a }{b }{c2 }{c3 }{a32 }
# generalized Bezout number : 587
#   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 c3 }{b 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 : 26

# 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 16 regular solutions.  Four paths converged to highly
# singular solutions, which indicates that the system probably has an
# infinite component of solutions.

# THE SOLUTIONS :

# 20 8
# ===========================================================
# solution 1 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -1.54868742915591E+00   1.10984586753349E-01
#  b2 :  2.45505205023205E-03  -1.98840610261098E-04
#  b3 :  1.90480330453772E-01   7.11458772979691E-02
#  a : -9.02280049972086E-01   1.32262749440421E-01
#  b : -4.53471996679816E-01   4.96688740006357E-02
#  c2 :  3.67885421140448E-01   2.23562655640229E-01
#  c3 :  4.80574364999927E-01   2.62040941134371E-01
#  a32 :  6.09280158170986E-01  -4.30372908822721E-01
# == err :  3.846E-15 = rco :  2.015E-04 = res :  2.225E-16 ==
# solution 2 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -1.71888103839571E+00  -1.93162394224810E-57
#  b2 : -4.21182839890999E-02   3.09856376715262E-56
#  b3 :  1.72262192175605E-02  -3.16228744359792E-56
#  a : -1.11411504415350E+00   5.97409466674670E-58
#  b : -6.29658059013750E-01  -3.14635652448659E-57
#  c2 :  7.55553790471265E+00  -1.12918354661068E-54
#  c3 :  1.15147156906762E+01   1.38662719944969E-54
#  a32 : -2.84251025589465E-02  -2.00729580802689E-56
# == err :  5.907E-14 = rco :  1.869E-06 = res :  7.105E-15 ==
# solution 3 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -8.91439313414223E-01   2.84072246785365E+00
#  b2 :  2.75139464570149E-02  -1.84200109581071E-02
#  b3 :  7.50735107390070E-04  -9.51801936712548E-04
#  a : -1.88754000466580E-01   1.32962334709711E+00
#  b : -6.74420631383238E-01   1.49172730786171E+00
#  c2 :  7.27675649003600E+00   2.77127125872537E+00
#  c3 :  1.81312600660805E+01  -1.71806056022904E+01
#  a32 :  8.95628415149649E+00   4.01389814545182E-01
# == err :  5.070E-13 = rco :  6.346E-08 = res :  2.220E-15 ==
# solution 4 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 :  3.26056134161109E+00  -1.79278659840263E+00
#  b2 : -1.14798576732338E-02  -2.35765602643108E-02
#  b3 : -3.16020516034117E-04   4.68070394354158E-04
#  a :  1.92851446497257E+00  -8.78141159303192E-01
#  b :  1.32025099844925E+00  -9.37753928969399E-01
#  c2 : -5.18539450338994E+00   9.50585171067483E+00
#  c3 : -3.13463433262412E+00  -3.53140384372959E+01
#  a32 : -1.07841734207531E+01   1.50126695694059E+01
# == err :  7.313E-13 = rco :  1.391E-08 = res :  6.033E-15 ==
# solution 5 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -3.55858011781411E+00   3.42390300587920E-57
#  b2 : -2.81832179394210E-02  -5.28227039015323E-60
#  b3 :  1.00704763099191E-03   8.53233735362993E-61
#  a : -1.63035345948598E+00   1.58430190205247E-57
#  b : -1.95540282863656E+00   1.83516720544125E-57
#  c2 : -6.53590540536284E+00   5.89817388035679E-57
#  c3 : -2.61177367346545E+01  -1.19880166312717E-56
#  a32 : -6.85719071588913E+00  -1.41386907113005E-57
# == err :  1.956E-12 = rco :  5.160E-08 = res :  1.721E-15 ==
# solution 6 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -8.34275476922321E-01  -9.23993308456822E-57
#  b2 :  1.54297081464225E-01   7.64684117343577E-57
#  b3 : -7.22435800199501E-02  -8.28407793788875E-57
#  a : -6.11410030984530E-01  -1.08330249957007E-56
#  b : -1.40811944493516E-01   8.76200551122849E-58
#  c2 :  7.50008412839803E+00   1.63132611699963E-55
#  c3 :  1.09893902739762E+01  -1.93719976393706E-55
#  a32 : -1.08167128370528E-02   6.96977711120448E-58
# == err :  6.169E-15 = rco :  4.006E-06 = res :  1.421E-14 ==
# solution 7 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -2.00000000000000E+00  -4.95496457394533E-16
#  b2 :  9.81697592740941E-18   1.44253943690846E-18
#  b3 :  3.48443312844541E-16   4.02010615093877E-17
#  a : -1.00000000000000E+00  -5.04280951614286E-17
#  b : -1.00000000000000E+00  -4.03424761286809E-16
#  c2 :  5.06282151098845E-01   4.53679915659373E-02
#  c3 :  5.96260455108259E-01   9.16597652437176E-02
#  a32 :  3.72557168652312E-01   1.02268973958993E-02
# == err :  9.973E-313 = rco :  1.138E-18 = res :  0.000E+00 ==
# solution 8 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -2.00000000000001E+00  -9.23785533592253E-17
#  b2 :  2.04391768504228E-15   1.24148748189622E-15
#  b3 : -1.63658692022580E-15  -1.25746237041559E-15
#  a : -1.00000000000000E+00  -1.20392713198447E-17
#  b : -1.00000000000001E+00  -9.63141705587465E-17
#  c2 :  8.86985542366175E+00   1.27740351671357E-01
#  c3 :  8.79337014820927E+00  -7.75638432783920E-02
#  a32 : -3.85504280503917E-03  -1.57263018605654E-03
# == err :  9.973E-313 = rco :  3.574E-20 = res :  0.000E+00 ==
# solution 9 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -8.91439313414197E-01  -2.84072246785366E+00
#  b2 :  2.75139464570150E-02   1.84200109581070E-02
#  b3 :  7.50735107390112E-04   9.51801936712518E-04
#  a : -1.88754000466566E-01  -1.32962334709712E+00
#  b : -6.74420631383226E-01  -1.49172730786172E+00
#  c2 :  7.27675649003601E+00  -2.77127125872534E+00
#  c3 :  1.81312600660803E+01   1.71806056022907E+01
#  a32 :  8.95628415149655E+00  -4.01389814544759E-01
# == err :  1.335E-12 = rco :  6.346E-08 = res :  7.161E-15 ==
# solution 10 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -2.00000000000000E+00  -2.92392686882682E-16
#  b2 :  7.85137967194930E-18  -1.55451732201970E-18
#  b3 : -2.11129408981433E-17   8.85844138615643E-17
#  a : -1.00000000000000E+00  -2.28180878006245E-17
#  b : -1.00000000000000E+00  -1.82544702542522E-16
#  c2 :  1.76172086709783E+00   3.51234877536242E-01
#  c3 :  1.76866072663750E+00  -1.74512275061876E-01
#  a32 :  7.83679152587805E-01  -4.63247687258756E-01
# == err :  9.973E-313 = rco :  2.354E-19 = res :  0.000E+00 ==
# solution 11 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -7.09869839778797E-01  -5.79229632491348E-64
#  b2 : -1.39629197970438E-01  -9.49556774575980E-65
#  b3 :  5.87691127059789E-02   1.16320704885558E-64
#  a : -6.59713826937515E-01  -4.98517306652389E-64
#  b : -1.31016098105741E-01  -5.93472984109987E-65
#  c2 : -7.38286882089761E+00  -3.79822709830392E-64
#  c3 : -1.13263580832817E+01  -3.03858167864314E-63
#  a32 :  1.29806460468640E-02  -2.55193383167295E-65
# == err :  3.133E-14 = rco :  3.389E-06 = res :  1.421E-14 ==
# solution 12 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -1.89270285756505E+00   4.60244253126165E-55
#  b2 :  2.14134038122082E-05  -2.55765927920093E-57
#  b3 : -4.34643527786413E-02  -1.86949335771393E-55
#  a : -9.97315422530658E-01   1.20378007534946E-56
#  b : -9.38830374409218E-01   2.58877435559023E-55
#  c2 :  2.72352838082211E-01  -1.82759504045115E-54
#  c3 :  5.59735904846106E-01  -5.06204954762337E-56
#  a32 :  4.33249618635895E-01  -1.24796447950472E-53
# == err :  1.365E-13 = rco :  1.194E-04 = res :  2.001E-16 ==
# solution 13 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -4.71553639145464E-01  -6.01085225033391E-01
#  b2 : -5.07627521635982E-05  -7.28211053839031E-04
#  b3 :  5.11602857020603E-01  -1.71702170405392E-01
#  a :  3.45354921336824E-01  -4.21485360244454E-01
#  b : -3.05356466213849E-01  -3.52030246248168E-01
#  c2 :  1.11004417986242E+00  -1.03964923338075E-01
#  c3 :  9.83164051899763E-01   3.93783706265552E-01
#  a32 :  1.78910560306613E-01   2.52079968211080E-01
# == err :  2.764E-15 = rco :  5.202E-04 = res :  7.837E-17 ==
# solution 14 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 :  8.06827148509337E-01  -3.58891745739919E-54
#  b2 : -1.42448574529814E-03  -8.66641999656054E-56
#  b3 :  7.54607190707658E-01  -4.07831529249908E-55
#  a :  1.35012723779024E+00  -2.93638701059934E-54
#  b :  2.09882615681460E-01  -9.38012517274788E-55
#  c2 : -3.98538306917890E-01   5.87277402119867E-54
#  c3 :  4.83769670358321E-01   3.26265223399926E-54
#  a32 : -7.74627768583179E-01  -2.61012178719941E-54
# == err :  2.812E-15 = rco :  1.222E-03 = res :  1.665E-16 ==
# solution 15 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -1.54868742915591E+00  -1.10984586753349E-01
#  b2 :  2.45505205023205E-03   1.98840610261100E-04
#  b3 :  1.90480330453771E-01  -7.11458772979697E-02
#  a : -9.02280049972086E-01  -1.32262749440421E-01
#  b : -4.53471996679816E-01  -4.96688740006359E-02
#  c2 :  3.67885421140447E-01  -2.23562655640230E-01
#  c3 :  4.80574364999928E-01  -2.62040941134371E-01
#  a32 :  6.09280158170985E-01   4.30372908822724E-01
# == err :  4.639E-15 = rco :  2.015E-04 = res :  2.238E-16 ==
# solution 16 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -4.71553639145464E-01   6.01085225033390E-01
#  b2 : -5.07627521635952E-05   7.28211053839031E-04
#  b3 :  5.11602857020603E-01   1.71702170405392E-01
#  a :  3.45354921336824E-01   4.21485360244453E-01
#  b : -3.05356466213849E-01   3.52030246248168E-01
#  c2 :  1.11004417986242E+00   1.03964923338074E-01
#  c3 :  9.83164051899762E-01  -3.93783706265552E-01
#  a32 :  1.78910560306613E-01  -2.52079968211080E-01
# == err :  2.599E-15 = rco :  5.202E-04 = res :  1.608E-16 ==
# solution 17 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -6.20681684032930E-01  -3.70920615068742E-67
#  b2 :  8.95850880561880E-04   1.73869038313473E-69
#  b3 : -5.38246012441459E-01   2.04006338287808E-67
#  a : -1.07134874113107E+00  -7.41841230137484E-68
#  b : -8.66831044627572E-02  -8.34571383904670E-68
#  c2 : -3.29372192907320E-01   5.19288861096239E-67
#  c3 : -6.90389957944890E-01   0.00000000000000E+00
#  a32 : -5.03421273025244E-01  -7.41841230137484E-67
# == err :  6.017E-16 = rco :  1.903E-03 = res :  9.660E-17 ==
# solution 18 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -1.73994795522882E+00   3.32344871101593E-65
#  b2 :  4.99670061024409E-02   4.53413359860030E-63
#  b3 : -2.37928809546032E-02  -4.57686365345622E-63
#  a : -1.11228981517644E+00  -1.38635289088093E-63
#  b : -6.01484014904543E-01   1.37567037716695E-63
#  c2 : -7.31987332040536E+00  -1.09996656766882E-61
#  c3 : -1.08338550646840E+01   8.14339889876360E-62
#  a32 :  2.03036070257183E-02  -3.60831574338872E-64
# == err :  2.974E-14 = rco :  2.286E-06 = res :  4.441E-16 ==
# solution 19 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 : -2.00000000000000E+00   3.13212001649522E-17
#  b2 : -2.87669384196252E-15  -7.47661995621575E-16
#  b3 :  2.80406118127485E-15   7.91594866046253E-16
#  a : -1.00000000000000E+00   8.36156339884810E-18
#  b : -1.00000000000000E+00   6.68925071907823E-17
#  c2 : -8.64056895134541E+00  -4.60767532370001E-01
#  c3 : -9.21942041985285E+00   2.48894218752857E-01
#  a32 : -2.88994579001811E-03   4.80205697111369E-03
# == err :  9.973E-313 = rco :  1.548E-19 = res :  0.000E+00 ==
# solution 20 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 1
# the solution for t :
#  b1 :  3.26056134161102E+00   1.79278659840270E+00
#  b2 : -1.14798576732334E-02   2.35765602643111E-02
#  b3 : -3.16020516034195E-04  -4.68070394354121E-04
#  a :  1.92851446497254E+00   8.78141159303226E-01
#  b :  1.32025099844921E+00   9.37753928969435E-01
#  c2 : -5.18539450338971E+00  -9.50585171067495E+00
#  c3 : -3.13463433262608E+00   3.53140384372956E+01
#  a32 : -1.07841734207505E+01  -1.50126695694069E+01
# == err :  1.006E-12 = rco :  1.391E-08 = res :  1.123E-14 ==