5
-1 + 2*x**2 - 2*y**2 + 2*z**2 - 2*t**2 + 2*u**2;
-1 + 2*x**3 - 2*y**3 + 2*z**3 - 2*t**3 + 2*u**3;
-1 + 2*x**4 - 2*y**4 + 2*z**4 - 2*t**4 + 2*u**4;
-1 + 2*x**5 - 2*y**5 + 2*z**5 - 2*t**5 + 2*u**5;
-1 + 2*x**6 - 2*y**6 + 2*z**6 - 2*t**6 + 2*u**6;
TITLE : The 5-dimensional system of Reimer.
ROOT COUNTS :
total degree : 720
mixed volume : 720
REFERENCES :
See the PoSSo test suite.
For general dimension n, the system looks like
-1/2 + \sum_{i=1}^{n}(-1)^(i+1)x_{i}^k = 0, k=2..n+1
GENERATORS OF SYMMETRY GROUP :
z y x t u
u y z t x
x t z y u
NOTE :
The system has 12 roots with all components positive,
but any fine mixed subdivision has only one mixed cell.
THE GENERATING SOLUTIONS :
12 5
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : -1.25705460701026E-01 4.80449609667978E-01
y : -3.96732207927890E-01 -2.01295133009924E-01
z : 3.39026130046387E-01 4.97108436261838E-01
t : -9.17817062157943E-02 -4.55073462282180E-01
u : 8.75016837867993E-01 1.54169279604962E-02
== err : 5.257E-16 = rco : 2.886E-02 = res : 4.381E-16 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : 1.65179856275237E-01 4.01306224781909E-53
y : 3.71809860927029E-01 6.21535250576860E-53
z : 6.00580975840425E-01 6.78631664671847E-53
t : 8.06590703105641E-01 4.86135182865890E-53
u : 9.49130782115133E-01 1.47635013588467E-53
== err : 4.081E-15 = rco : 1.931E-04 = res : 3.400E-16 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : 3.28353058900760E-01 3.83293571654000E-01
y : -3.65037597430609E-01 -1.77941422105434E-01
z : 8.83954216708990E-01 7.47522218167706E-19
t : -3.65037597430609E-01 1.77941422105434E-01
u : 3.28353058900760E-01 -3.83293571654000E-01
== err : 4.158E-16 = rco : 1.470E-02 = res : 3.643E-16 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : -8.20501169693851E-01 4.81804127164467E-02
y : 1.92649800357694E-02 -8.15105711153907E-01
z : 2.83453963159985E-02 8.39642248705884E-01
t : 6.63305484227383E-01 -6.19401872141988E-01
u : -6.12614994436549E-01 6.70606441286687E-01
== err : 3.890E-16 = rco : 1.109E-01 = res : 7.022E-16 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : -2.89250316112716E-01 1.93720295400755E-02
y : 1.25951647109466E-01 -3.11152873990998E-01
z : 9.17357766838520E-01 -3.21236642103152E-03
t : 6.77654385257208E-01 -1.36742906837876E-02
u : -1.51240072064018E-01 2.63861320151929E-01
== err : 4.402E-16 = rco : 6.008E-03 = res : 3.334E-16 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : 2.92426652768470E-02 1.98441472348304E-01
y : -1.15079341958084E-01 -9.12609179052688E-02
z : 4.88543274566396E-01 1.87476161310206E-02
t : 7.55781391744456E-01 8.67380109394292E-03
u : 9.36082697242564E-01 2.23886684350489E-03
== err : 5.112E-16 = rco : 1.351E-03 = res : 2.624E-16 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : -8.23715781042161E-01 -2.47741481660290E-19
y : 5.91859519233860E-01 -5.00136991500666E-01
z : -5.37294659901288E-01 5.27047166093312E-01
t : 5.91859519233860E-01 5.00136991500666E-01
u : -5.37294659901288E-01 -5.27047166093312E-01
== err : 4.104E-16 = rco : 1.415E-01 = res : 7.706E-16 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : 8.75016837867993E-01 -1.54169279604962E-02
y : -9.17817062157942E-02 4.55073462282180E-01
z : 3.39026130046387E-01 -4.97108436261838E-01
t : -3.96732207927890E-01 2.01295133009924E-01
u : -1.25705460701026E-01 -4.80449609667978E-01
== err : 5.296E-16 = rco : 2.849E-02 = res : 6.661E-16 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : -1.51240072064018E-01 -2.63861320151929E-01
y : 1.25951647109466E-01 3.11152873990998E-01
z : 9.17357766838520E-01 3.21236642103152E-03
t : 6.77654385257208E-01 1.36742906837876E-02
u : -2.89250316112716E-01 -1.93720295400755E-02
== err : 4.648E-16 = rco : 4.638E-03 = res : 3.268E-16 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : -6.12614994436549E-01 -6.70606441286687E-01
y : 6.63305484227383E-01 6.19401872141988E-01
z : 2.83453963159985E-02 -8.39642248705884E-01
t : 1.92649800357694E-02 8.15105711153907E-01
u : -8.20501169693851E-01 -4.81804127164468E-02
== err : 4.093E-16 = rco : 9.277E-02 = res : 1.423E-15 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : 2.92426652768470E-02 -1.98441472348305E-01
y : 7.55781391744457E-01 -8.67380109394305E-03
z : 9.36082697242564E-01 -2.23886684350493E-03
t : -1.15079341958084E-01 9.12609179052692E-02
u : 4.88543274566397E-01 -1.87476161310209E-02
== err : 1.018E-15 = rco : 1.359E-03 = res : 3.501E-16 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 12
the solution for t :
x : -8.90691982819378E-02 8.72390401374093E-48
y : 3.04390884130488E-01 -2.35203294488113E-48
z : 5.59254489061997E-01 2.56585412168851E-49
t : 7.86835443685312E-01 4.49024471295489E-49
u : 9.43962351030283E-01 1.71056941445901E-49
== err : 2.854E-15 = rco : 5.084E-04 = res : 4.003E-16 ==