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 ==