6
 z0 + z1 + z2 + z3 + z4 + z5;
 z0*z1 + z1*z2 + z2*z3 + z3*z4 + z4*z5 + z5*z0;
 z0*z1*z2 + z1*z2*z3 + z2*z3*z4 + z3*z4*z5 + z4*z5*z0 + z5*z0*z1;
 z0*z1*z2*z3 + z1*z2*z3*z4 + z2*z3*z4*z5 + z3*z4*z5*z0 + z4*z5*z0*z1 
 + z5*z0*z1*z2;
 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;
 z0*z1*z2*z3*z4*z5 - 1;

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 ==