3
x^3+y^2+z^2-1;
x^2+y^3+z^2-1;
x^2+y^2+z^3-1;
TITLE : problem with 15 regular roots and 3 roots of multiplicity 4
ROOT COUNTS :
total degree : 27
1-homogeneous Bezout number : 27
with partition : {x y z }
general linear-product Bezout number : 27
based on the set structure :
{x y z }{x y z }{x }
{x y z }{x y z }{y }
{x y z }{x y z }{z }
mixed volume : 27
DESCRIPTION :
There are 27 solutions, counted with multiplicities.
Besides 9 real regular roots and 6 conjugated complex ones,
the three solutions (1,0,0), (0,1,0) and (0,0,1) each have
multiplicity four, which all adds up to 27.
You can count the multiplicities simply by counting the paths
that end up close to these roots.
One deflation suffices to deal with the multiple roots.
This problem comes from mth191, a course taught by Bernd Sturmfels.
THE SOLUTIONS :
27 3
===========================================================================
solution 1 : start residual : 1.111E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 3.93703666064922E-07 -2.07091288705183E-07
y : 1.00000000000000E+00 -7.09749727919967E-19
z : 3.97074282279030E-07 -2.09849313278767E-07
== err : 1.208E-06 = rco : 5.068E-08 = res : 1.111E-12 = complex regular ==
solution 2 : start residual : 3.220E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.07832554546136E+00 1.42029731261303E+00
y : -1.07832554546136E+00 1.42029731261303E+00
z : -2.60767292300353E-01 -1.86692741261055E+00
== err : 6.960E-16 = rco : 1.898E-01 = res : 3.220E-15 = complex regular ==
solution 3 : start residual : 3.543E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.14254920187881E-07 -2.28578488849644E-07
y : -1.20013773176305E-07 -2.18423329854419E-07
z : 1.00000000000000E+00 1.58943696131505E-20
== err : 6.813E-07 = rco : 2.243E-08 = res : 3.543E-13 = complex regular ==
solution 4 : start residual : 1.400E-39
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.00000000000000E+00 3.67341984631965E-40
y : -1.00000000000000E+00 3.67341984631965E-40
z : -1.00000000000000E+00 1.14794370197489E-40
== err : 2.519E-24 = rco : 1.429E-01 = res : 1.400E-39 = real regular ==
solution 5 : start residual : 2.220E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -2.60767292300353E-01 1.86692741261055E+00
y : -1.07832554546136E+00 -1.42029731261303E+00
z : -1.07832554546136E+00 -1.42029731261303E+00
== err : 4.760E-16 = rco : 1.587E-01 = res : 2.220E-15 = complex regular ==
solution 6 : start residual : 4.126E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -3.86909521106789E-08 -9.59585245159877E-08
y : 1.65466398470997E-07 -3.37891752463884E-07
z : 1.00000000000000E+00 4.51069551162661E-20
== err : 6.380E-07 = rco : 1.310E-08 = res : 4.126E-13 = complex regular ==
solution 7 : start residual : 5.329E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.07832554546136E+00 1.42029731261303E+00
y : -2.60767292300353E-01 -1.86692741261055E+00
z : -1.07832554546136E+00 1.42029731261303E+00
== err : 4.151E-16 = rco : 2.107E-01 = res : 5.329E-15 = complex regular ==
solution 8 : start residual : 4.048E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 3.08613754201721E-07 -4.45578839436618E-08
y : 1.00000000000000E+00 -1.15817684998345E-19
z : 2.01215480726998E-07 -2.02845624912646E-07
== err : 7.572E-07 = rco : 2.351E-08 = res : 4.048E-13 = complex regular ==
solution 9 : start residual : 1.943E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 6.18033988749895E-01 -3.47946327843397E-36
y : 6.18033988749895E-01 3.19734463423662E-36
z : 6.18033988749894E-01 7.05296610493373E-37
== err : 8.106E-16 = rco : 2.484E-02 = res : 1.943E-16 = real regular ==
solution 10 : start residual : 2.220E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -2.60767292300353E-01 -1.86692741261055E+00
y : -1.07832554546136E+00 1.42029731261303E+00
z : -1.07832554546136E+00 1.42029731261303E+00
== err : 2.324E-16 = rco : 1.587E-01 = res : 2.220E-15 = complex regular ==
solution 11 : start residual : 3.109E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.07832554546136E+00 -1.42029731261303E+00
y : -2.60767292300353E-01 1.86692741261055E+00
z : -1.07832554546136E+00 -1.42029731261303E+00
== err : 5.608E-16 = rco : 2.107E-01 = res : 3.109E-15 = complex regular ==
solution 12 : start residual : 1.203E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -2.76006882523289E-07 1.14700342969912E-07
y : -3.82225622816641E-07 -5.63036539524554E-07
z : 1.00000000000000E+00 -1.74326453148164E-19
== err : 1.336E-06 = rco : 4.224E-08 = res : 1.203E-12 = complex regular ==
solution 13 : start residual : 4.441E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.07832554546136E+00 -1.42029731261303E+00
y : -1.07832554546136E+00 -1.42029731261303E+00
z : -2.60767292300353E-01 1.86692741261055E+00
== err : 3.110E-16 = rco : 1.898E-01 = res : 4.441E-15 = complex regular ==
solution 14 : start residual : 8.397E-50
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.61803398874989E+00 5.34552942018439E-51
y : -1.61803398874989E+00 0.00000000000000E+00
z : -1.61803398874989E+00 5.34552942018439E-51
== err : 1.444E-34 = rco : 9.646E-02 = res : 8.397E-50 = real regular ==
solution 15 : start residual : 8.839E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.72950810988886E-07 -3.63451294533619E-07
y : -1.55487607733046E-07 -3.56766215707846E-07
z : 1.00000000000000E+00 1.39716992749556E-19
== err : 1.049E-06 = rco : 3.400E-08 = res : 8.839E-13 = complex regular ==
solution 16 : start residual : 1.076E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 3.64170367781606E-07 -3.61967837245771E-07
y : 1.00000000000000E+00 -6.44958111189010E-19
z : 4.29235995647959E-07 -1.20105350239433E-07
== err : 1.275E-06 = rco : 4.758E-08 = res : 1.076E-12 = complex regular ==
solution 17 : start residual : 9.696E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 4.03783992949629E-07 -1.18607519679620E-07
y : 1.00000000000000E+00 -4.45217683266378E-19
z : 3.99455607188134E-07 -1.18263866793045E-07
== err : 1.040E-06 = rco : 4.363E-08 = res : 9.696E-13 = complex regular ==
solution 18 : start residual : 1.735E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -3.30349280373268E-01 -4.10536659470161E-47
y : 7.19739963708203E-01 4.81696347111656E-46
z : 7.19739963708202E-01 -3.06534039071054E-46
== err : 5.215E-16 = rco : 2.711E-02 = res : 1.735E-16 = real regular ==
solution 19 : start residual : 3.331E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 8.51883864973974E-01 -6.65616770554288E-45
y : 4.36911127214512E-01 7.18165462966469E-45
z : 4.36911127214512E-01 7.18165462966469E-45
== err : 1.166E-15 = rco : 7.982E-03 = res : 3.331E-16 = real regular ==
solution 20 : start residual : 5.009E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 1.00000000000000E+00 5.94228538510084E-21
y : -2.31090561190662E-07 -2.96334342196165E-07
z : 2.54220611170702E-07 1.29396164571412E-07
== err : 9.109E-07 = rco : 1.987E-08 = res : 5.009E-13 = complex regular ==
solution 21 : start residual : 1.804E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 4.36911127214511E-01 -8.39433832069138E-41
y : 8.51883864973975E-01 5.73971850987445E-41
z : 4.36911127214512E-01 -7.60512702558365E-41
== err : 7.624E-16 = rco : 7.982E-03 = res : 1.804E-16 = real regular ==
solution 22 : start residual : 1.238E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 1.00000000000000E+00 -9.75322556383625E-19
y : -1.59519024817330E-07 -4.60378547867266E-07
z : 5.60529678802230E-07 -3.85952472095994E-07
== err : 1.566E-06 = rco : 3.605E-08 = res : 1.238E-12 = complex regular ==
solution 23 : start residual : 3.653E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 1.00000000000000E+00 1.21926062145209E-19
y : 1.81362204329137E-07 2.25707251761889E-07
z : 2.37056317227989E-07 1.30642422952020E-07
== err : 7.747E-07 = rco : 1.702E-08 = res : 3.653E-13 = complex regular ==
solution 24 : start residual : 3.331E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 7.19739963708203E-01 1.23160997841048E-47
y : -3.30349280373269E-01 1.36845553156720E-48
z : 7.19739963708202E-01 -1.36845553156720E-47
== err : 8.054E-17 = rco : 2.711E-02 = res : 3.331E-16 = real regular ==
solution 25 : start residual : 1.098E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 1.00000000000000E+00 -1.22166867653063E-18
y : 4.32555727278952E-08 3.28722635637267E-07
z : 5.35563523922606E-07 -3.79185805724897E-07
== err : 1.287E-06 = rco : 3.034E-08 = res : 1.098E-12 = complex regular ==
solution 26 : start residual : 3.886E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 7.19739963708203E-01 4.37905770101505E-47
y : 7.19739963708202E-01 1.31371731030452E-46
z : -3.30349280373269E-01 -1.47793197409258E-46
== err : 1.033E-15 = rco : 2.711E-02 = res : 3.886E-16 = real regular ==
solution 27 : start residual : 1.943E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 4.36911127214511E-01 -1.40623103491924E-40
y : 4.36911127214511E-01 -1.72191555296234E-40
z : 8.51883864973975E-01 8.32259183931795E-41
== err : 1.637E-15 = rco : 7.982E-03 = res : 1.943E-16 = real regular ==
===========================================================================
A list of 27 solutions has been refined :
Number of regular solutions : 27.
Number of singular solutions : 0.
Number of real solutions : 9.
Number of complex solutions : 18.
Number of clustered solutions : 0.
Number of failures : 0.
===========================================================================
TIMING INFORMATION for Solving the polynomial system
The elapsed time in seconds was 0.480000000 = 0h 0m 0s480ms
User time in seconds was 0.480000000 = 0h 0m 0s480ms
System CPU time in seconds was 0.000000000 = 0h 0m 0s 0ms
Non-I/O page faults was 10
I/O page faults was 80
Signals delivered was 0
Swaps was 0
Total context switches was 0
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| TIMING INFORMATION SUMMARY |
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| root counts | start system | continuation | total time |
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| 0h 0m 0s 10ms | 0h 0m 0s 0ms | 0h 0m 0s450ms | 0h 0m 0s480ms |
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