var z0;
var z1;
var z2;
var z3;
var z4;
var z5;

subject to

cons1:  z0 + z1 + z2 + z3 + z4 + z5 = 0;
cons2:  z0*z1 + z1*z2 + z2*z3 + z3*z4 + z4*z5 + z5*z0 = 0;
cons3:  z0*z1*z2 + z1*z2*z3 + z2*z3*z4 + z3*z4*z5 + z4*z5*z0 + z5*z0*z1 = 0;
cons4:  z0*z1*z2*z3 + z1*z2*z3*z4 + z2*z3*z4*z5 + z3*z4*z5*z0 + z4*z5*z0*z1 
 + z5*z0*z1*z2 = 0;
cons5:  z0*z1*z2*z3*z4 + z1*z2*z3*z4*z5 + z2*z3*z4*z5*z0 + z3*z4*z5*z0*z1 
 + z4*z5*z0*z1*z2 + z5*z0*z1*z2*z3 = 0;
cons6:  z0*z1*z2*z3*z4*z5 - 1 = 0;

solve;
display z0, z1, z2, z3, z4, z5;

# TITLE : cyclic 6-roots problem

# ROOT COUNTS :

# total degree : 6! = 720

# bound based on set structure analysis : 504
#   with set structure
#     {z0 z1 z2 z3 z4 z5 }
#     {z0 z2 z4 }{z1 z3 z5 }
#     {z0 z3 }{z1 z4 }{z2 z5 }
#     {z0 }{z1 }{z2 }{z3 }{z4 }{z5 }
#     {z0 }{z1 }{z2 }{z3 }{z4 }{z5 }
#     {z0 }{z1 }{z2 }{z3 }{z4 }{z5 }

# mixed volume : 156 = 13*12

# SYMMETRY GROUP :

# z1 z2 z3 z4 z5 z0
# z5 z4 z3 z2 z1 z0

# SYMMETRIC SET STRUCTURE :

#     {z0 z1 z2 z3 z4 z5 }
#     {z0 }{z1 }{z2 }{z3 }{z4 }{z5 }
#     {z0 }{z1 }{z2 }{z3 }{z4 }{z5 }
#     {z0 }{z1 }{z2 }{z3 }{z4 }{z5 }
#     {z0 }{z1 }{z2 }{z3 }{z4 }{z5 }
#     {z0 }{z1 }{z2 }{z3 }{z4 }{z5 }

# with generalized Bezout bound : 720 and 60 generating solutions

# REFERENCES :

# See G\"oran Bj\"orck and Ralf Fr\"oberg:
# `A faster way to count the solutions of inhomogeneous systems
#  of algebraic equations, with applications to cyclic n-roots',
# in J. Symbolic Computation (1991) 12, pp 329--336.

# THE GENERATING SOLUTIONS :

# 13 6
# ===========================================================
# solution 1 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 :  8.86365602936828E-18   1.00000000000000E+00
#  z1 : -9.30604859102100E-01  -3.66025403784439E-01
#  z2 :  9.30604859102100E-01   3.66025403784439E-01
#  z3 : -4.26190968730363E-17  -1.00000000000000E+00
#  z4 : -9.30604859102100E-01   3.66025403784439E-01
#  z5 :  9.30604859102100E-01  -3.66025403784439E-01
# == err :  6.592E-16 = rco :  2.143E-01 = res :  4.530E-16 ==
# solution 2 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 : -8.66025403784439E-01   5.00000000000000E-01
#  z1 : -1.48315131443269E-01  -9.88940150759876E-01
#  z2 :  1.48315131443269E-01   9.88940150759876E-01
#  z3 :  8.66025403784439E-01  -5.00000000000000E-01
#  z4 : -7.82289727658830E-01  -6.22914746975437E-01
#  z5 :  7.82289727658831E-01   6.22914746975437E-01
# == err :  6.084E-16 = rco :  2.286E-01 = res :  4.965E-16 ==
# solution 3 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 : -8.66025403784439E-01  -5.00000000000000E-01
#  z1 :  7.82289727658830E-01  -6.22914746975437E-01
#  z2 : -7.82289727658831E-01   6.22914746975437E-01
#  z3 :  8.66025403784439E-01   5.00000000000000E-01
#  z4 :  1.48315131443269E-01  -9.88940150759876E-01
#  z5 : -1.48315131443269E-01   9.88940150759876E-01
# == err :  4.762E-16 = rco :  2.399E-01 = res :  2.483E-16 ==
# solution 4 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 : -1.86602540378444E+00  -3.23205080756888E+00
#  z1 : -1.33974596215561E-01  -2.32050807568877E-01
#  z2 :  5.00000000000000E-01   8.66025403784438E-01
#  z3 :  5.00000000000000E-01   8.66025403784439E-01
#  z4 :  5.00000000000000E-01   8.66025403784439E-01
#  z5 :  5.00000000000000E-01   8.66025403784439E-01
# == err :  3.111E-15 = rco :  3.021E-02 = res :  8.006E-16 ==
# solution 5 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 : -1.86602540378444E+00   3.23205080756888E+00
#  z1 : -1.33974596215561E-01   2.32050807568877E-01
#  z2 :  5.00000000000000E-01  -8.66025403784438E-01
#  z3 :  5.00000000000000E-01  -8.66025403784439E-01
#  z4 :  5.00000000000000E-01  -8.66025403784439E-01
#  z5 :  5.00000000000000E-01  -8.66025403784439E-01
# == err :  3.111E-15 = rco :  3.021E-02 = res :  8.006E-16 ==
# solution 6 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 :  3.73205080756888E+00   1.18072616964182E-79
#  z1 :  2.67949192431123E-01   9.44580935713455E-79
#  z2 : -1.00000000000000E+00   2.15904213877361E-78
#  z3 : -1.00000000000000E+00  -5.39760534693403E-79
#  z4 : -1.00000000000000E+00  -5.39760534693403E-79
#  z5 : -1.00000000000000E+00  -2.15904213877361E-78
# == err :  2.484E-15 = rco :  3.113E-02 = res :  3.331E-16 ==
# solution 7 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 :  1.86602540378444E+00   3.23205080756888E+00
#  z1 :  1.33974596215561E-01   2.32050807568877E-01
#  z2 : -5.00000000000000E-01  -8.66025403784438E-01
#  z3 : -5.00000000000000E-01  -8.66025403784439E-01
#  z4 : -5.00000000000000E-01  -8.66025403784439E-01
#  z5 : -5.00000000000000E-01  -8.66025403784439E-01
# == err :  3.111E-15 = rco :  3.021E-02 = res :  8.006E-16 ==
# solution 8 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 :  1.86602540378444E+00  -3.23205080756888E+00
#  z1 :  1.33974596215561E-01  -2.32050807568877E-01
#  z2 : -5.00000000000000E-01   8.66025403784438E-01
#  z3 : -5.00000000000000E-01   8.66025403784439E-01
#  z4 : -5.00000000000000E-01   8.66025403784439E-01
#  z5 : -5.00000000000000E-01   8.66025403784439E-01
# == err :  3.111E-15 = rco :  3.021E-02 = res :  8.006E-16 ==
# solution 9 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 : -3.73205080756888E+00  -1.18072616964182E-79
#  z1 : -2.67949192431123E-01  -9.44580935713455E-79
#  z2 :  1.00000000000000E+00  -2.15904213877361E-78
#  z3 :  1.00000000000000E+00   5.39760534693403E-79
#  z4 :  1.00000000000000E+00   5.39760534693403E-79
#  z5 :  1.00000000000000E+00   2.15904213877361E-78
# == err :  2.484E-15 = rco :  3.113E-02 = res :  3.331E-16 ==
# solution 10 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 : -8.66025403784439E-01  -5.00000000000000E-01
#  z1 : -8.66025403784439E-01   5.00000000000000E-01
#  z2 : -5.17587871677573E-17   1.00000000000000E+00
#  z3 :  8.66025403784439E-01   5.00000000000000E-01
#  z4 :  8.66025403784439E-01  -5.00000000000000E-01
#  z5 :  7.00201700631743E-17  -1.00000000000000E+00
# == err :  5.113E-16 = rco :  2.821E-01 = res :  4.996E-16 ==
# solution 11 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 :  1.51132949714153E-77   2.29663026288654E+00
#  z1 : -8.09640802040104E-79   1.00000000000000E+00
#  z2 : -3.10362307448707E-78   4.35420544682339E-01
#  z3 : -5.39760534693403E-78  -4.35420544682339E-01
#  z4 :  1.21446120306016E-78  -1.00000000000000E+00
#  z5 : -6.47712641632083E-78  -2.29663026288654E+00
# == err :  1.915E-15 = rco :  8.556E-02 = res :  6.661E-16 ==
# solution 12 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 : -1.98894015075988E+00   1.14831513144327E+00
#  z1 : -8.66025403784439E-01   5.00000000000000E-01
#  z2 : -3.77085253024563E-01   2.17710272341170E-01
#  z3 :  3.77085253024563E-01  -2.17710272341169E-01
#  z4 :  8.66025403784439E-01  -5.00000000000000E-01
#  z5 :  1.98894015075988E+00  -1.14831513144327E+00
# == err :  4.422E-15 = rco :  8.405E-02 = res :  1.193E-15 ==
# solution 13 :
# t :  1.00000000000000E+00   0.00000000000000E+00
# m : 12
# the solution for t :
#  z0 : -1.98894015075988E+00  -1.14831513144327E+00
#  z1 : -8.66025403784439E-01  -5.00000000000000E-01
#  z2 : -3.77085253024563E-01  -2.17710272341170E-01
#  z3 :  3.77085253024563E-01   2.17710272341169E-01
#  z4 :  8.66025403784439E-01   5.00000000000000E-01
#  z5 :  1.98894015075988E+00   1.14831513144327E+00
# == err :  4.422E-15 = rco :  8.405E-02 = res :  1.193E-15 ==