5
 x1 + x2 + x3 + x4 + x5;
 x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1;
 x1*x2*x3 + x2*x3*x4 + x3*x4*x5 + x4*x5*x1 + x5*x1*x2;
 x1*x2*x3*x4 + x2*x3*x4*x5 + x3*x4*x5*x1 + x4*x5*x1*x2 + x5*x1*x2*x3;
 x1*x2*x3*x4*x5 - 1;

TITLE : cyclic 5-roots problem

ROOT COUNTS :

total degree : 120
5-homogeneous Bezout number : 120
  with partition : {x1 }{x2 }{x3 }{x4 }{x5 }
general linear-product Bezout number : 106
  based on the set structure :
     {x1 x2 x3 x4 x5 }
     {x1 x3 x5 }{x2 x4 x5 }
     {x1 x4 }{x2 x4 x5 }{x3 x5 }
     {x1 x5 }{x2 x5 }{x3 x5 }{x4 x5 }
     {x1 }{x2 }{x3 }{x4 }{x5 }
mixed volume : 70

SYMMETRY GROUP :

  b c d e a
  e d c b a

SYMMETRIC SET STRUCTURE :

 { a b c d e }
 { a } { b } { c } { d } { e }
 { a } { b } { c } { d } { e }
 { a } { b } { c } { d } { e }
 { a } { b } { c } { d } { e }

with generalized Bezout bound : 120, leading to 12 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 SOLUTIONS : (generating)

7 5
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 10
the solution for t :
 x1 :  3.09016994374947E-01  -9.51056516295154E-01
 x2 :  3.09016994374947E-01  -9.51056516295154E-01
 x3 : -8.09016994374948E-01   2.48989828488278E+00
 x4 : -1.18033988749895E-01   3.63271264002680E-01
 x5 :  3.09016994374948E-01  -9.51056516295154E-01
== err :  8.556E-16 = rco :  6.220E-02 = res :  7.022E-16 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 10
the solution for t :
 x1 :  1.00000000000000E+00  -3.31628872515627E-75
 x2 :  1.00000000000000E+00  -8.84343660041671E-75
 x3 : -2.61803398874990E+00   3.31628872515627E-75
 x4 : -3.81966011250105E-01   1.65814436257813E-75
 x5 :  1.00000000000000E+00   6.90893484407556E-75
== err :  4.713E-15 = rco :  6.850E-02 = res :  4.441E-16 ==
solution 3 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 10
the solution for t :
 x1 :  3.09016994374948E-01   9.51056516295154E-01
 x2 :  3.09016994374947E-01   9.51056516295154E-01
 x3 : -8.09016994374948E-01  -2.48989828488278E+00
 x4 : -1.18033988749895E-01  -3.63271264002680E-01
 x5 :  3.09016994374947E-01   9.51056516295154E-01
== err :  6.582E-16 = rco :  6.220E-02 = res :  4.965E-16 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 10
the solution for t :
 x1 : -8.09016994374947E-01   5.87785252292473E-01
 x2 : -8.09016994374947E-01   5.87785252292473E-01
 x3 :  2.11803398874990E+00  -1.53884176858763E+00
 x4 :  3.09016994374947E-01  -2.24513988289793E-01
 x5 : -8.09016994374948E-01   5.87785252292473E-01
== err :  5.945E-15 = rco :  6.765E-02 = res :  4.003E-16 ==
solution 5 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 10
the solution for t :
 x1 : -8.09016994374947E-01  -5.87785252292473E-01
 x2 : -8.09016994374947E-01  -5.87785252292473E-01
 x3 :  2.11803398874990E+00   1.53884176858763E+00
 x4 :  3.09016994374947E-01   2.24513988289793E-01
 x5 : -8.09016994374948E-01  -5.87785252292473E-01
== err :  5.945E-15 = rco :  6.765E-02 = res :  4.003E-16 ==
solution 6 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 10
the solution for t :
 x1 :  1.00000000000000E+00  -7.24393703353565E-18
 x2 : -8.09016994374947E-01  -5.87785252292473E-01
 x3 :  3.09016994374947E-01   9.51056516295154E-01
 x4 :  3.09016994374947E-01  -9.51056516295154E-01
 x5 : -8.09016994374948E-01   5.87785252292473E-01
== err :  7.269E-16 = rco :  2.571E-01 = res :  4.442E-16 ==
solution 7 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 10
the solution for t :
 x1 : -8.09016994374947E-01   5.87785252292473E-01
 x2 : -8.09016994374947E-01  -5.87785252292473E-01
 x3 :  3.09016994374947E-01  -9.51056516295153E-01
 x4 :  1.00000000000000E+00   2.82553319327192E-17
 x5 :  3.09016994374947E-01   9.51056516295153E-01
== err :  6.769E-16 = rco :  2.221E-01 = res :  7.022E-16 ==
