5
 a + b + c + d + e;
 a*b + b*c + c*d + d*e + e*a;
 a*b*c + b*c*d + c*d*e + d*e*a + e*a*b;
 a*b*c*d + b*c*d*e + c*d*e*a + d*e*a*b + e*a*b*c;
 a*b*c*d*e - 1;

TITLE : cyclic 5-roots problem

ROOT COUNTS :

total degree : 120
5-homogeneous Bezout number : 120
  with partition : {a }{b }{c }{d }{e }
generalized Bezout number : 106
  based on the set structure :
     {a b c d e }
     {a c e }{b d e }
     {a d }{b d e }{c e }
     {a e }{b e }{c e }{d e }
     {a }{b }{c }{d }{e }
mixed volume : 70 = 14*5

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 GENERATING SOLUTIONS :

7 5
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 10
the solution for t :
 a :  3.09016994374947E-01  -9.51056516295154E-01
 b :  3.09016994374947E-01  -9.51056516295154E-01
 c : -8.09016994374948E-01   2.48989828488278E+00
 d : -1.18033988749895E-01   3.63271264002680E-01
 e :  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 :
 a :  1.00000000000000E+00  -3.31628872515627E-75
 b :  1.00000000000000E+00  -8.84343660041671E-75
 c : -2.61803398874990E+00   3.31628872515627E-75
 d : -3.81966011250105E-01   1.65814436257813E-75
 e :  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 :
 a :  3.09016994374948E-01   9.51056516295154E-01
 b :  3.09016994374947E-01   9.51056516295154E-01
 c : -8.09016994374948E-01  -2.48989828488278E+00
 d : -1.18033988749895E-01  -3.63271264002680E-01
 e :  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 :
 a : -8.09016994374947E-01   5.87785252292473E-01
 b : -8.09016994374947E-01   5.87785252292473E-01
 c :  2.11803398874990E+00  -1.53884176858763E+00
 d :  3.09016994374947E-01  -2.24513988289793E-01
 e : -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 :
 a : -8.09016994374947E-01  -5.87785252292473E-01
 b : -8.09016994374947E-01  -5.87785252292473E-01
 c :  2.11803398874990E+00   1.53884176858763E+00
 d :  3.09016994374947E-01   2.24513988289793E-01
 e : -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 :
 a :  1.00000000000000E+00  -7.24393703353565E-18
 b : -8.09016994374947E-01  -5.87785252292473E-01
 c :  3.09016994374947E-01   9.51056516295154E-01
 d :  3.09016994374947E-01  -9.51056516295154E-01
 e : -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 :
 a : -8.09016994374947E-01   5.87785252292473E-01
 b : -8.09016994374947E-01  -5.87785252292473E-01
 c :  3.09016994374947E-01  -9.51056516295153E-01
 d :  1.00000000000000E+00   2.82553319327192E-17
 e :  3.09016994374947E-01   9.51056516295153E-01
== err :  6.769E-16 = rco :  2.221E-01 = res :  7.022E-16 ==