```
5
a + b + c + d + x;
a*b + b*c + c*d + d*x + x*a;
a*b*c + b*c*d + c*d*x + d*x*a + x*a*b;
a*b*c*d + b*c*d*x + c*d*x*a + d*x*a*b + x*a*b*c;
a*b*c*d*x - 1;

TITLE : cyclic 5-roots problem

ROOT COUNTS :

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

SYMMETRY GROUP :

b c d x a
x d c b a

SYMMETRIC SET STRUCTURE :

{ a b c d x }
{ a } { b } { c } { d } { x }
{ a } { b } { c } { d } { x }
{ a } { b } { c } { d } { x }
{ a } { b } { c } { d } { x }

with generalized Bezout bound : 120, leading to 12 generating solutions.

REFERENCES :

Sx 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
x :  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
x :  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
x :  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
x : -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
x : -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
x : -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
x :  3.09016994374947E-01   9.51056516295153E-01
== err :  6.769E-16 = rco :  2.221E-01 = res :  7.022E-16 ==
```