10
5*x1*x2+ 5*x1+ 3*x2+ 55;
7*x2*x3+ 9*x2+ 9*x3+ 19;
3*x3*x4+ 6*x3+ 5*x4-4;
6*x4*x5+ 6*x4+ 7*x5+ 118;
x5*x6+ 3*x5+ 9*x6+ 27;
6*x6*x7+ 7*x6+x7+ 72;
9*x7*x8+ 7*x7+x8+ 35;
4*x8*x9+ 4*x8+ 6*x9+ 16;
8*x9*x10+ 4*x9+ 3*x10-51;
3*x1*x10-6*x1+x10+ 5;
TITLE : 10-dimensional system of Ku
ROOT COUNTS :
total degree : 1024
10-homogeneous Bezout number : 2
with partition : {x1 }{x2 }{x3 }{x4 }{x5 }{x6 }{x7 }{x8 }{x9 }{x10 }
generalized Bezout number : 2
based on the set structure :
{x1 }{x2 }
{x2 }{x3 }
{x3 }{x4 }
{x4 }{x5 }
{x5 }{x6 }
{x6 }{x7 }
{x7 }{x8 }
{x8 }{x9 }
{x9 }{x10 }
{x1 }{x10 }
mixed volume : 2
REFERENCES :
M.C. Steenkamp :
`Die numeriese oplos van stelsels polinoomvergelykings'.
Technical report, Nasionale Navorsingsinstituut vir Wiskundige Wetenskappe,
Pretoria, 1982.
NOTE :
This system was known as an easy system for elimination, but difficult
for homotopy continuation. With multi-homogenization, this is also an
easy system for homotopy continuation methods.
THE SOLUTIONS :
2 10
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 2.00000000000000E+00 -2.67276471009220E-51
x2 : -5.00000000000000E+00 -5.34552942018439E-51
x3 : -1.00000000000000E+00 -4.17619485951906E-52
x4 : 5.00000000000000E+00 4.00914706513829E-51
x5 : -4.00000000000000E+00 2.00457353256915E-51
x6 : -3.00000000000000E+00 4.17619485951906E-53
x7 : 3.00000000000000E+00 8.35238971903811E-53
x8 : -2.00000000000000E+00 2.08809742975953E-53
x9 : 4.00000000000000E+00 3.34095588761525E-52
x10 : 1.00000000000000E+00 -1.67047794380762E-52
== err : 3.762E-35 = rco : 6.097E-03 = res : 1.604E-50 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 7.73143904674611E+00 -6.81046792009123E-57
x2 : -2.24828375286041E+00 -9.75768795568627E-58
x3 : -1.83222958057395E-01 -1.11516433779272E-57
x4 : 1.14583333333333E+00 2.34981056892037E-57
x5 : -9.00000000000000E+00 7.16891360009604E-57
x6 : -3.32551987064446E+00 -1.54330778890956E-58
x7 : 2.57062493671946E+00 -1.24460305557223E-58
x8 : -2.19569105036795E+00 -1.19481893334934E-58
x9 : 2.59354917484962E+00 -1.99136488891557E-59
x10 : 1.71067585997878E+00 -9.95682444457783E-59
== err : 4.749E-15 = rco : 3.440E-03 = res : 1.421E-14 ==