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 ==