6
2*x**2+2*y**2+2*z**2+2*t**2+2*u**2+v**2-v;
x*y+y*z+2*z*t+2*t*u+2*u*v-u;
2*x*z+2*y*t+2*z*u+u**2+2*t*v-t;
2*x*t+2*y*u+2*t*u+2*z*v-z;
t**2+2*x*v+2*y*v+2*z*v-y;
2*x+2*y+2*z+2*t+2*u+v-1;
TITLE : a problem of magnetism in physics
ROOT COUNTS :
total degree: 32
mixed volume : 32
REFERENCES :
From the PoSSo test suite.
Shigetoshi Katsura: "Users posing problems to PoSSO",
in the PoSSo Newsletter, no. 2, July 1994,
edited by L. Gonzalez-Vega and T. Recio.
Available at http://janet.dm.unipi.it/
S. Katsura, W. Fukuda, S. Inawashiro, N.M. Fujiki and R. Gebauer,
Cell Biophysics, Vol 11, pages 309--319, 1987.
W. Boege, R. Gebauer, and H. Kredel:
"Some examples for solving systems of algebraic equations by
calculating Groebner bases", J. Symbolic Computation, 2:83-98, 1986.
Shigetoshi Katsura:
"Spin Glass Problem by the Method of Integral Equation
of the Effective Field". In "New Trends in Magnetism",
edited by Mauricio D. Coutinho-Filho and Sergio M. Resende,
pages 110-121, World Scientific, 1990.
NOTE (excerpt from the PoSSo Newsletter) :
The general formulation of the equations is
\sum_{i=-N}^N u(l)*u(m-l) = u(m)
\sum_{i=-N}^N u(l) = 1
with m in {-N+1,-N,..,N-1}, u(l) = u(-l),
and u(l) = 0, for |l| > N,
The number of solutions for a given N is 2^N.
Among them, physically meaningful solutions are restricted
to those for which u(l) is real and 0 <= u(l) <= 1,
since u(l) is a probability.
THE SOLUTIONS :
32 6
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -7.34679413715196E-02 -1.05421979432305E-81
y : 8.96500882133478E-02 7.90664845742289E-82
z : 3.22926736021011E-02 1.44955221719420E-81
t : -1.54099162732096E-01 -1.18599726861343E-81
u : 2.65738935518866E-01 -1.84488464006534E-81
v : 6.79770813538602E-01 3.68976928013068E-81
== err : 4.235E-16 = rco : 6.845E-02 = res : 5.551E-17 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.23989867532776E-01 6.69182486630844E-83
y : 1.15327526760843E-02 -5.45641104483611E-83
z : 8.58389785887035E-02 -6.17706910736164E-84
t : 1.62143145160524E-01 -4.94165528588931E-83
u : 2.25869805737949E-01 2.05902303578721E-83
v : 2.77210370739033E-01 5.76526450020419E-83
== err : 4.821E-16 = rco : 7.475E-02 = res : 8.327E-17 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -2.07358972809625E-01 1.44890865261227E-70
y : 9.35088892161247E-02 -1.01876389636801E-70
z : 2.19257911144064E-01 -7.24454326306137E-71
t : 2.25457387686581E-02 -9.05567907882671E-71
u : 1.51473206244495E-01 1.06121262558422E-70
v : 4.41146454872566E-01 9.05567907882671E-71
== err : 2.406E-16 = rco : 1.067E-01 = res : 5.551E-17 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.57025351946495E+00 5.76457486699560E+00
y : 4.61777915035574E+00 -1.32891381878462E+00
z : -2.85008572379927E+00 -2.96262478568672E+00
t : 3.67679932837071E+00 1.95277814810887E+00
u : -6.44863384847331E-01 -3.92345674167498E+00
v : -5.45875170122980E+00 9.95284662083709E-01
== err : 5.652E-15 = rco : 1.365E-03 = res : 2.010E-14 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.21538696343695E-01 -1.45427136994283E-01
y : 5.41432115293970E-03 1.92234187205295E-01
z : -1.29474720447878E-01 7.44121358558796E-02
t : 4.21087159135082E-01 -4.00121205095676E-02
u : 9.38435315354770E-02 3.49228636012240E-02
v : 4.61336809936150E-01 -2.32259858317095E-01
== err : 4.096E-16 = rco : 3.034E-02 = res : 7.850E-17 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.27009396586336E-02 1.51426720098643E-01
y : 4.31073671633152E-01 9.79652306994441E-03
z : 4.97255031475553E-03 -2.20062605132790E-01
t : -1.04214329254954E-01 -1.50525567764471E-02
u : -6.21735758680806E-02 3.61797763381580E-02
v : 4.86085245667523E-01 7.54242848049839E-02
== err : 5.026E-16 = rco : 3.855E-02 = res : 6.592E-17 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -5.83959866995207E-01 8.77442593123539E-01
y : 1.18070251828669E+00 1.16300260901468E-01
z : -3.02272354151035E-01 -7.53820937749179E-01
t : -5.40199676757081E-01 -2.65819846833774E-01
u : 3.27158857148399E-01 5.12015123631312E-01
v : 8.37141044936462E-01 -9.72234386146732E-01
== err : 3.718E-15 = rco : 1.048E-02 = res : 6.661E-16 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 5.94183853341865E-02 1.95697375121334E-01
y : 3.99365516038540E-01 -4.47735495014597E-02
z : 3.56463693311711E-02 -1.98749596752736E-01
t : -2.07078028915640E-01 -4.06794736634173E-02
u : 3.02537558042511E-02 1.02020125595299E-01
v : 3.64788004814983E-01 -2.70297615980414E-02
== err : 4.100E-16 = rco : 2.704E-02 = res : 2.695E-16 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.32387153995653E-02 0.00000000000000E+00
y : -8.73756842370348E-02 -5.52714787526044E-76
z : -6.41927643960892E-02 1.24360827193360E-75
t : 3.27425164118805E-01 -5.52714787526044E-76
u : 4.22144271367303E-02 -8.29072181289067E-76
v : 5.90335145554309E-01 1.65814436257813E-75
== err : 4.302E-16 = rco : 3.254E-02 = res : 3.816E-17 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.81646358312146E-01 -3.05135743680520E-01
y : -2.45661127185011E-01 1.26402934892260E+00
z : 1.28100343359660E+00 1.95756286562680E-02
t : 2.69789837319146E-01 -6.92748789616714E-01
u : -8.81520435955827E-01 -4.69599332750762E-01
v : 5.16069301074466E-01 3.67757776938245E-01
== err : 5.229E-15 = rco : 2.422E-02 = res : 5.620E-16 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.21538696343695E-01 1.45427136994283E-01
y : 5.41432115293969E-03 -1.92234187205295E-01
z : -1.29474720447878E-01 -7.44121358558796E-02
t : 4.21087159135082E-01 4.00121205095676E-02
u : 9.38435315354771E-02 -3.49228636012240E-02
v : 4.61336809936150E-01 2.32259858317095E-01
== err : 4.336E-16 = rco : 3.034E-02 = res : 6.206E-17 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.27009396586336E-02 -1.51426720098643E-01
y : 4.31073671633152E-01 -9.79652306994440E-03
z : 4.97255031475552E-03 2.20062605132790E-01
t : -1.04214329254954E-01 1.50525567764471E-02
u : -6.21735758680806E-02 -3.61797763381580E-02
v : 4.86085245667523E-01 -7.54242848049839E-02
== err : 5.075E-16 = rco : 3.855E-02 = res : 8.100E-17 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 4.22327061275954E-01 -3.86856051726076E-01
y : -5.39101147413182E-01 -3.67652911266504E-02
z : -7.13331690209261E-02 1.11472099190827E-01
t : 1.25961423554398E-01 2.27162877399557E-01
u : 3.28700380323012E-01 3.47451620579309E-01
v : 4.66890902561487E-01 -5.24930508633935E-01
== err : 5.281E-16 = rco : 3.983E-02 = res : 1.943E-16 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.57025351946497E+00 -5.76457486699559E+00
y : 4.61777915035575E+00 1.32891381878461E+00
z : -2.85008572379926E+00 2.96262478568673E+00
t : 3.67679932837070E+00 -1.95277814810888E+00
u : -6.44863384847322E-01 3.92345674167498E+00
v : -5.45875170122980E+00 -9.95284662083694E-01
== err : 1.842E-14 = rco : 1.365E-03 = res : 1.446E-14 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -1.81646358312146E-01 3.05135743680519E-01
y : -2.45661127185011E-01 -1.26402934892261E+00
z : 1.28100343359660E+00 -1.95756286562676E-02
t : 2.69789837319146E-01 6.92748789616714E-01
u : -8.81520435955827E-01 4.69599332750762E-01
v : 5.16069301074466E-01 -3.67757776938245E-01
== err : 5.120E-15 = rco : 2.422E-02 = res : 1.054E-15 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -4.53090455165408E-01 -2.65159314313777E-02
y : 4.72369413963952E-01 -1.72189347661563E-01
z : 7.61392026176293E-02 -2.59841065181448E-02
t : 3.90830532173818E-02 -3.10677427065331E-02
u : 1.66041794189728E-02 -1.21033708197555E-01
v : 6.97789211894943E-01 7.53581673030348E-01
== err : 5.479E-16 = rco : 2.561E-02 = res : 9.021E-17 ==
solution 17 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -4.53090455165408E-01 2.65159314313777E-02
y : 4.72369413963952E-01 1.72189347661563E-01
z : 7.61392026176293E-02 2.59841065181447E-02
t : 3.90830532173818E-02 3.10677427065331E-02
u : 1.66041794189728E-02 1.21033708197555E-01
v : 6.97789211894943E-01 -7.53581673030348E-01
== err : 6.846E-16 = rco : 2.561E-02 = res : 1.804E-16 ==
solution 18 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 2.19202174282559E-01 -8.29072181289067E-76
y : 1.86196233915891E-01 3.21265470249513E-75
z : -2.33164321950951E-02 -1.10542957505209E-75
t : -6.22139522220197E-02 -3.05720366850343E-75
u : 6.08345024330872E-02 5.52714787526044E-76
v : 2.38594947571154E-01 2.59085056652833E-75
== err : 4.761E-16 = rco : 3.716E-02 = res : 5.551E-17 ==
solution 19 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 1.77720242269329E-01 -2.69880267346701E-79
y : 2.52054104063177E-01 -2.15904213877361E-78
z : -1.26609638729233E-01 4.93380435074275E-79
t : 6.57264116780968E-02 1.07952106938681E-78
u : -7.31811773405752E-02 -2.69880267346701E-79
v : 4.08580116118410E-01 1.61928160408021E-78
== err : 5.089E-16 = rco : 4.527E-02 = res : 5.551E-17 ==
solution 20 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 9.16958134062345E-02 -3.86900351268231E-74
y : -8.43879405799624E-02 1.10542957505209E-75
z : -1.02027573161162E-01 1.76868732008334E-74
t : 5.52705192655442E-02 1.98977323509376E-74
u : 3.08655664592089E-01 8.29072181289067E-75
v : 4.61587032954515E-01 -1.54760140507292E-74
== err : 5.099E-16 = rco : 6.751E-02 = res : 2.776E-17 ==
solution 21 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -3.71434890128193E+00 -2.26721676217087E+00
y : -1.30211184710235E+00 4.55637566992507E+00
z : 4.36802284527241E+00 -3.22491009143098E+00
t : -2.53969151312992E+00 -2.00811793381777E+00
u : 2.20426421471743E+00 2.90935925832586E+00
v : 2.96773040304869E+00 6.90197183373700E-02
== err : 9.879E-15 = rco : 2.934E-03 = res : 1.465E-14 ==
solution 22 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 1.40862400739638E-01 -1.38178696881511E-76
y : 1.92885306946496E-01 -2.76357393763022E-76
z : -5.91323323306972E-02 5.52714787526044E-76
t : 1.80509696917380E-01 -4.14536090644533E-76
u : -1.01057866510769E-01 5.52714787526044E-76
v : 2.91865588475904E-01 -5.52714787526044E-76
== err : 2.834E-16 = rco : 5.475E-02 = res : 2.220E-16 ==
solution 23 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 5.94183853341865E-02 -1.95697375121334E-01
y : 3.99365516038540E-01 4.47735495014597E-02
z : 3.56463693311711E-02 1.98749596752736E-01
t : -2.07078028915640E-01 4.06794736634173E-02
u : 3.02537558042511E-02 -1.02020125595299E-01
v : 3.64788004814983E-01 2.70297615980414E-02
== err : 4.389E-16 = rco : 2.704E-02 = res : 8.777E-17 ==
solution 24 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -3.17327431527865E-01 1.79678795707145E-02
y : 2.51074725113845E-01 1.35840540037664E+00
z : 1.60334509980202E+00 -4.17460309000618E-01
t : -4.52423320845073E-01 -1.14940404120786E+00
u : -7.68030460085420E-01 2.53099446870810E-01
v : 3.66722775084977E-01 -1.25216753219381E-01
== err : 4.568E-15 = rco : 2.054E-02 = res : 1.153E-15 ==
solution 25 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 2.10588109558726E-01 5.52714787526044E-76
y : 9.63978681241064E-02 -9.39615138794276E-75
z : 4.03916112981482E-02 8.84343660041671E-75
t : 4.17053473397812E-02 7.73800702536462E-75
u : 4.27934029063517E-02 -3.59264611891929E-75
v : 1.36247321545774E-01 -6.63257745031253E-75
== err : 5.003E-16 = rco : 4.736E-02 = res : 5.551E-17 ==
solution 26 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 4.22327061275954E-01 3.86856051726075E-01
y : -5.39101147413182E-01 3.67652911266506E-02
z : -7.13331690209261E-02 -1.11472099190827E-01
t : 1.25961423554398E-01 -2.27162877399557E-01
u : 3.28700380323012E-01 -3.47451620579309E-01
v : 4.66890902561488E-01 5.24930508633935E-01
== err : 5.897E-16 = rco : 3.983E-02 = res : 3.664E-16 ==
solution 27 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 0.00000000000000E+00 0.00000000000000E+00
y : 0.00000000000000E+00 0.00000000000000E+00
z : 1.36566297568363E-158 0.00000000000000E+00
t : 2.42427351708672E-191 0.00000000000000E+00
u : 0.00000000000000E+00 0.00000000000000E+00
v : 1.00000000000000E+00 0.00000000000000E+00
== err : 1.215E-63 = rco : 6.598E-02 = res : 2.731E-158 ==
solution 28 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : 1.39125672609318E-01 -5.43582081447824E-82
y : -1.45563258086646E-01 2.71791040723912E-82
z : -1.14364175483778E-01 2.80027132867061E-82
t : 1.90920510948048E-01 6.09470818593015E-82
u : 5.32023462057030E-02 -2.96499317153359E-82
v : 7.53357807614709E-01 -5.92998634306717E-82
== err : 4.485E-16 = rco : 4.737E-02 = res : 5.551E-17 ==
solution 29 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -5.83959866995208E-01 -8.77442593123539E-01
y : 1.18070251828669E+00 -1.16300260901468E-01
z : -3.02272354151035E-01 7.53820937749179E-01
t : -5.40199676757081E-01 2.65819846833775E-01
u : 3.27158857148399E-01 -5.12015123631312E-01
v : 8.37141044936462E-01 9.72234386146732E-01
== err : 4.110E-15 = rco : 1.048E-02 = res : 4.441E-16 ==
solution 30 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -3.71434890128193E+00 2.26721676217088E+00
y : -1.30211184710235E+00 -4.55637566992507E+00
z : 4.36802284527241E+00 3.22491009143098E+00
t : -2.53969151312992E+00 2.00811793381777E+00
u : 2.20426421471743E+00 -2.90935925832586E+00
v : 2.96773040304869E+00 -6.90197183373696E-02
== err : 6.334E-15 = rco : 2.934E-03 = res : 9.441E-15 ==
solution 31 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -3.17327431527865E-01 -1.79678795707145E-02
y : 2.51074725113845E-01 -1.35840540037664E+00
z : 1.60334509980202E+00 4.17460309000618E-01
t : -4.52423320845073E-01 1.14940404120786E+00
u : -7.68030460085420E-01 -2.53099446870810E-01
v : 3.66722775084977E-01 1.25216753219381E-01
== err : 4.622E-15 = rco : 2.054E-02 = res : 1.096E-15 ==
solution 32 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x : -2.07926968038444E-01 1.34940133673351E-78
y : 1.09542324450522E-01 -1.55181153724353E-78
z : 1.63551292586952E-01 1.34940133673351E-79
t : 1.21988131355882E-01 1.24819623647849E-78
u : -5.03053401613212E-02 -7.42170735203429E-79
v : 7.26301119612819E-01 -1.21446120306016E-78
== err : 5.627E-16 = rco : 4.755E-02 = res : 5.551E-17 ==