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

TITLE : extended cyclic 6-roots problem, to exploit the symmetry

ROOT COUNTS :

total degree : 6! = 720
mixed volume : 156

REFERENCES :

This is the Arnborg's system or Davenport's problem,
extended with the constant term to exploit symmetry.

For the original problem :

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',
J. Symbolic Computation (1991) 12, pp 329--336.

THE SOLUTIONS :

13 6
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 : -1.09145911498337E-01  -2.82051661977533E-01
z1 : -1.19329529302722E+00  -3.08367868303911E+00
z2 :  6.84847142039119E-01   1.11902672009036E+00
z3 :  6.68044546894126E-01   8.74453630967107E-01
z4 :  5.51668586068374E-01   7.22120404126932E-01
z5 :  3.97880929523935E-01   6.50129589832241E-01
== err :  3.150E-15 = rco :  2.299E-02 = res :  1.404E-15 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 : -3.53129181444054E-01  -4.55887812930893E-01
z1 : -1.06193432801704E+00  -1.37095132239199E+00
z2 : -2.50000000000000E-01  -9.68245836551854E-01
z3 :  1.34883344892200E-01   1.74133649482747E-01
z4 :  2.78018016456890E+00   3.58919715894384E+00
z5 : -2.50000000000000E-01  -9.68245836551854E-01
== err :  5.300E-15 = rco :  2.085E-02 = res :  1.256E-15 ==
solution 3 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 :  3.21007829191040E-01  -2.54057382351715E+00
z1 :  3.21007829191040E-01   2.54057382351715E+00
z2 :  1.30039852922610E-01   9.91508767813914E-01
z3 :  4.89523178863507E-02   3.87426617400472E-01
z4 :  4.89523178863507E-02  -3.87426617400472E-01
z5 :  1.30039852922610E-01  -9.91508767813914E-01
== err :  4.347E-16 = rco :  6.306E-02 = res :  9.155E-16 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 :  2.66839454551322E-01  -9.63740995026544E-01
z1 :  7.89153930342621E-01   6.14195469068922E-01
z2 : -4.44985611119495E-01   8.95537718857564E-01
z3 :  7.44236290740844E-01  -6.67916419579807E-01
z4 : -4.96471667324206E-01  -8.68052926695327E-01
z5 :  1.41227602808914E-01   9.89977153375192E-01
== err :  8.917E-16 = rco :  1.092E-01 = res :  7.109E-16 ==
solution 5 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 :  7.44236290740845E-01   6.67916419579807E-01
z1 : -4.96471667324207E-01   8.68052926695327E-01
z2 :  1.41227602808914E-01  -9.89977153375192E-01
z3 :  2.66839454551322E-01   9.63740995026544E-01
z4 :  7.89153930342621E-01  -6.14195469068922E-01
z5 : -4.44985611119495E-01  -8.95537718857564E-01
== err :  6.069E-16 = rco :  1.043E-01 = res :  8.951E-16 ==
solution 6 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 : -1.30493744993987E+00   1.10325388562335E+00
z1 :  2.47735189059043E+00  -1.20477380171330E+00
z2 :  8.86709471424394E-01  -4.62327063112546E-01
z3 :  3.26450487727244E-01  -1.58757823894198E-01
z4 : -4.46891660260174E-01   3.77822676985343E-01
z5 : -9.38682739542032E-01   3.44782126111355E-01
== err :  6.011E-15 = rco :  6.878E-02 = res :  7.216E-16 ==
solution 7 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 : -3.81966011250105E-01   6.04048609987806E-92
z1 :  6.85410196624969E+00   8.43750439348047E-91
z2 :  1.45898033750315E-01  -4.45845402610047E-92
z3 : -2.61803398874990E+00  -5.44602556306467E-91
z4 : -3.81966011250105E-01   6.61577049034264E-92
z5 : -2.61803398874990E+00  -3.83522926976385E-91
== err :  4.788E-15 = rco :  1.766E-03 = res :  8.882E-16 ==
solution 8 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 :  5.00000000000000E-01  -8.66025403784439E-01
z1 : -9.57427107756338E-01  -2.88675134594813E-01
z2 : -9.57427107756338E-01   2.88675134594813E-01
z3 :  5.00000000000000E-01   8.66025403784439E-01
z4 :  9.57427107756338E-01   2.88675134594813E-01
z5 :  9.57427107756338E-01  -2.88675134594813E-01
== err :  4.711E-16 = rco :  1.213E-01 = res :  7.109E-16 ==
solution 9 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 :  3.26450487727244E-01   1.58757823894198E-01
z1 :  8.86709471424394E-01   4.62327063112546E-01
z2 :  2.47735189059043E+00   1.20477380171330E+00
z3 : -1.30493744993987E+00  -1.10325388562335E+00
z4 : -9.38682739542032E-01  -3.44782126111355E-01
z5 : -4.46891660260174E-01  -3.77822676985343E-01
== err :  5.690E-15 = rco :  5.844E-02 = res :  6.713E-16 ==
solution 10 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 :  1.00000000000000E+00   3.41829996175611E-86
z1 :  1.00000000000000E+00  -3.01614702507892E-86
z2 :  1.00000000000000E+00   3.61937643009471E-86
z3 :  1.00000000000000E+00  -2.61399408840173E-86
z4 : -3.81966011250105E-01  -3.21722349341752E-86
z5 : -2.61803398874990E+00   4.12206760094120E-86
== err :  4.720E-15 = rco :  3.715E-02 = res :  2.220E-16 ==
solution 11 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 :  5.51668586068374E-01  -7.22120404126932E-01
z1 :  3.97880929523935E-01  -6.50129589832240E-01
z2 : -1.09145911498336E-01   2.82051661977533E-01
z3 : -1.19329529302722E+00   3.08367868303911E+00
z4 :  6.84847142039118E-01  -1.11902672009036E+00
z5 :  6.68044546894127E-01  -8.74453630967107E-01
== err :  3.182E-15 = rco :  2.489E-02 = res :  1.201E-15 ==
solution 12 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 : -6.04864265798915E-01  -7.96328587933109E-01
z1 :  1.15709953428924E-01   9.93283044593774E-01
z2 :  9.89154312369991E-01  -1.46880040576826E-01
z3 : -6.04864265798915E-01   7.96328587933109E-01
z4 :  1.15709953428924E-01  -9.93283044593774E-01
z5 :  9.89154312369991E-01   1.46880040576826E-01
== err :  6.660E-16 = rco :  1.458E-01 = res :  8.006E-16 ==
solution 13 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 12
the solution for t :
z0 :  1.34883344892200E-01  -1.74133649482747E-01
z1 : -2.50000000000000E-01   9.68245836551854E-01
z2 : -1.06193432801704E+00   1.37095132239199E+00
z3 : -3.53129181444054E-01   4.55887812930893E-01
z4 : -2.50000000000000E-01   9.68245836551854E-01
z5 :  2.78018016456890E+00  -3.58919715894384E+00
== err :  5.011E-15 = rco :  1.055E-02 = res :  9.155E-16 ==
```