5
1 + y1 + y2 + y3 + y4 + y5;
y1 + y1*y2 + y2*y3 + y3*y4 + y4*y5 + y5;
y1*y2 + y1*y2*y3 + y2*y3*y4 + y3*y4*y5 + y4*y5 + y5*y1;
y1*y2*y3 + y1*y2*y3*y4 + y2*y3*y4*y5 + y3*y4*y5 + y4*y5*y1 + y5*y1*y2;
y1*y2*y3*y4 + y1*y2*y3*y4*y5 + y2*y3*y4*y5 + y3*y4*y5*y1
+ y4*y5*y1*y2 + y5*y1*y2*y3;
z0**6*y1*y2*y3*y4*y5 - 1;
TITLE : reduced cyclic 6-roots problem
ROOT COUNTS :
total degree : 5! = 120
3-homogeneous Bezout number = 96.
with partition : {y1 y4 }{y2 y5 }{y3 }
generalized Bezout bound : 84
based on
{ y1 y2 y3 y4 y5 }
{ y1 y3 y5 }{ y2 y4 }
{ y1 y4 }{ y2 y5 }{ y3 }
{ y1 }{ y2 }{ y3 }{ y4 }{ y5 }
{ y1 }{ y2 }{ y3 }{ y4 }{ y5 }
mixed volume : 26
REFERENCES :
See Ioannis Z. Emiris:
`Sparse Elimination and Application in Kinematics'
PhD Thesis, Computer Science, University of California at Berkeley, 1994,
page 25.
yi = zi/z0
This reduces the dimension of the problem by 1
and the mixed volume drops with 6.
THE SYMMETRY GROUP :
y1 y2 y3 y4 y5
y5 y4 y3 y2 y1
THE GENERATING SOLUTIONS :
13 5
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : 4.35420544682339E-01 0.00000000000000E+00
y2 : -4.35420544682339E-01 -6.63257745031253E-75
y3 : -1.00000000000000E+00 -8.29072181289067E-75
y4 : -2.29663026288654E+00 -4.55989699708987E-75
y5 : 2.29663026288654E+00 2.10031619259897E-74
== err : 1.757E-15 = rco : 7.119E-02 = res : 1.665E-16 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : 1.00000000000000E+00 -4.86389013022919E-74
y2 : -2.67949192431123E-01 -8.84343660041671E-74
y3 : -3.73205080756888E+00 8.40126477039588E-74
y4 : 1.00000000000000E+00 4.42171830020836E-74
y5 : 1.00000000000000E+00 -5.08497604523961E-74
== err : 2.459E-15 = rco : 1.787E-02 = res : 1.110E-16 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : -3.66025403784439E-01 -9.30604859102100E-01
y2 : -7.32050807568877E-01 6.81250038633213E-01
y3 : 7.32050807568877E-01 -6.81250038633213E-01
y4 : 3.66025403784439E-01 9.30604859102100E-01
y5 : -1.00000000000000E+00 4.70224363277577E-17
== err : 5.957E-16 = rco : 1.413E-01 = res : 6.280E-16 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : -3.73205080756888E+00 -7.82410672410628E-69
y2 : -3.73205080756888E+00 -4.92628941888173E-69
y3 : -3.73205080756888E+00 -3.62227163153069E-69
y4 : -3.73205080756888E+00 -2.39069927681025E-69
y5 : 1.39282032302755E+01 1.97051576755269E-68
== err : 8.118E-15 = rco : 2.117E-03 = res : 4.547E-13 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : -2.67949192431123E-01 4.38555434438390E-79
y2 : -2.67949192431123E-01 1.48434147040686E-78
y3 : -2.67949192431123E-01 -1.34940133673351E-78
y4 : -2.67949192431123E-01 -2.19277717219195E-79
y5 : 7.17967697244908E-02 -3.20482817474208E-79
== err : 3.168E-16 = rco : 6.091E-03 = res : 1.110E-16 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : 1.00000000000000E+00 1.35835186182401E-71
y2 : -3.73205080756888E+00 -5.09381948184003E-72
y3 : -2.67949192431123E-01 -6.22577936669337E-72
y4 : 1.00000000000000E+00 -6.79175930912004E-72
y5 : 1.00000000000000E+00 4.52783953941336E-72
== err : 2.749E-15 = rco : 1.733E-02 = res : 3.331E-16 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : -2.67949192431123E-01 -7.73800702536462E-75
y2 : -3.73205080756888E+00 -3.53737464016668E-74
y3 : 1.00000000000000E+00 1.87923027758855E-74
y4 : 1.00000000000000E+00 1.21597253255730E-74
y5 : 1.00000000000000E+00 1.27124401130990E-74
== err : 2.613E-15 = rco : 1.997E-02 = res : 1.110E-16 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : -1.00000000000000E+00 4.63650768835928E-69
y2 : -4.35420544682339E-01 2.02847211365718E-69
y3 : -1.89591050731465E-01 -2.89781730522455E-70
y4 : 1.89591050731465E-01 -2.31825384417964E-69
y5 : 4.35420544682339E-01 -3.76716249679191E-69
== err : 2.074E-16 = rco : 2.307E-02 = res : 2.220E-16 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : -1.00000000000000E+00 -3.99971315179912E-17
y2 : 3.66025403784439E-01 -9.30604859102100E-01
y3 : 7.32050807568877E-01 6.81250038633213E-01
y4 : -7.32050807568877E-01 -6.81250038633213E-01
y5 : -3.66025403784439E-01 9.30604859102100E-01
== err : 5.225E-16 = rco : 1.207E-01 = res : 3.140E-16 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : 1.00000000000000E+00 -5.48194293047987E-80
y2 : 1.00000000000000E+00 1.68675167091688E-80
y3 : 1.00000000000000E+00 1.68675167091688E-80
y4 : -2.67949192431123E-01 0.00000000000000E+00
y5 : -3.73205080756888E+00 -4.21687917729221E-80
== err : 2.358E-15 = rco : 1.905E-02 = res : 6.661E-16 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : 5.00000000000000E-01 8.66025403784439E-01
y2 : -5.00000000000000E-01 8.66025403784439E-01
y3 : -1.00000000000000E+00 -1.29604050740420E-16
y4 : -5.00000000000000E-01 -8.66025403784439E-01
y5 : 5.00000000000000E-01 -8.66025403784439E-01
== err : 5.178E-16 = rco : 1.754E-01 = res : 4.710E-16 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : 2.29663026288654E+00 1.62277769092575E-68
y2 : 5.27451056440629E+00 3.94103153510539E-68
y3 : -5.27451056440629E+00 -3.59329345847844E-68
y4 : -2.29663026288654E+00 -1.44890865261227E-68
y5 : -1.00000000000000E+00 -1.73869038313473E-69
== err : 1.791E-15 = rco : 3.857E-03 = res : 1.421E-14 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 2
the solution for t :
y1 : -3.66025403784439E-01 -9.30604859102100E-01
y2 : 3.66025403784439E-01 9.30604859102100E-01
y3 : -1.00000000000000E+00 -5.41719466407389E-17
y4 : 3.66025403784439E-01 -9.30604859102100E-01
y5 : -3.66025403784439E-01 9.30604859102100E-01
== err : 7.381E-16 = rco : 1.626E-01 = res : 4.578E-16 ==