7
(x1 + x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6)*x7 - 1;
(x2 + x1*x3 + x2*x4 + x3*x5 + x4*x6)*x7 - 2;
(x3 + x1*x4 + x2*x5 + x3*x6)*x7 - 3;
(x4 + x1*x5 + x2*x6)*x7 - 4;
(x5 + x1*x6)*x7 - 5;
x6*x7 - 6;
x1 + x2 + x3 + x4 + x5 + x6 + 1;
TITLE : 7-dimensional economics problem
ROOT COUNTS :
total degree : 486
3-homogeneous Bezout number : 112
with partition : {x1 x2 x3 x4 x5 }{x6 }{x7 }
generalized Bezout number : 80
based on the set structure :
{x1 x3 x5 }{x2 x4 x6 }{x7 }
{x1 x2 x5 x6 }{x3 x4 }{x7 }
{x1 x2 x3 }{x4 x5 x6 }{x7 }
{x1 x2 x4 }{x5 x6 }{x7 }
{x1 x5 }{x6 }{x7 }
{x6 }{x7 }
{x1 x2 x3 x4 x5 x6 }
mixed volume : 32
REFERENCE :
Alexander Morgan:
`Solving polynomial systems using continuation for engineering
and scientific problems', Prentice-Hall, Englewood Cliffs, New Jersey, 1987,
(p 148).
NOTE :
Transform u = 1/x7 and the total degree equals the number of solutions.
See the reduced economics problem, in the file redeco7.
THE SOLUTIONS :
32 7
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.93369639826478E-03 1.28636083334685E-01
x2 : 5.95495251474475E-01 2.08845973813888E+00
x3 : -6.99468204461881E-01 -4.29895386284063E-01
x4 : -1.32701859744045E+00 -6.35115643463731E-01
x5 : 1.49441734602828E-01 -5.41035850350458E-01
x6 : 2.75616119426760E-01 -6.11048941375316E-01
x7 : 3.68023794115138E+00 8.15919439917735E+00
== err : 4.597E-15 = rco : 1.056E-02 = res : 7.332E-15 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.93369639826586E-03 -3.71479583210633E+00
x2 : -6.29608516467496E+00 2.89485362873523E-01
x3 : 6.85454641806504E-01 6.41622249799803E+00
x4 : 4.94031447364828E+00 -2.36147158475214E-01
x5 : 2.97245117069881E-01 -2.53379026615599E+00
x6 : -6.32862764247967E-01 -2.20974604134012E-01
x7 : -8.45046930263530E+00 2.95062249572451E+00
== err : 1.027E-14 = rco : 1.203E-03 = res : 1.484E-13 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 2.63424603497105E-01 -1.79307987438582E+00
x2 : -1.42230137654078E+00 7.27270787149925E-01
x3 : 6.48059125615115E-01 7.41546225335957E-01
x4 : -1.21625875444223E-01 -4.44285120925777E-01
x5 : -5.08590494417704E-01 4.40081292362660E-01
x6 : 1.41034017290485E-01 3.28466690463057E-01
x7 : 6.62230193448536E+00 -1.54232690910824E+01
== err : 4.346E-14 = rco : 2.736E-03 = res : 8.981E-15 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -2.51557210700575E-01 1.79307987438582E+00
x2 : -5.18994747263476E-01 -1.65067431386248E+00
x3 : 2.54014532482806E+00 1.49593314929199E+00
x4 : -1.73247661935806E+00 1.65911587822018E+00
x5 : -1.53306651761352E+00 -2.14239297415820E+00
x6 : 4.95949770107570E-01 -1.15506161387731E+00
x7 : 1.88319443182410E+00 4.38593932445156E+00
== err : 4.379E-15 = rco : 1.345E-02 = res : 8.894E-15 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 8.27399636935069E-01 -3.71479583210633E+00
x2 : -6.84186578113854E+00 -2.76209288924990E+00
x3 : -4.12071064151692E+00 8.69959556204953E+00
x4 : 8.72511873116428E+00 3.50394294906873E+00
x5 : 1.81846960499532E+00 -5.23487937380794E+00
x6 : -1.40841155043922E+00 -4.91770415954087E-01
x7 : -3.79717658548780E+00 1.32584762480407E+00
== err : 8.966E-15 = rco : 2.797E-03 = res : 7.105E-14 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.93369639826475E-03 -1.28636083334685E-01
x2 : 5.95495251474475E-01 -2.08845973813888E+00
x3 : -6.99468204461880E-01 4.29895386284063E-01
x4 : -1.32701859744045E+00 6.35115643463731E-01
x5 : 1.49441734602828E-01 5.41035850350458E-01
x6 : 2.75616119426760E-01 6.11048941375316E-01
x7 : 3.68023794115138E+00 -8.15919439917735E+00
== err : 4.866E-15 = rco : 1.056E-02 = res : 3.878E-15 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 3.97308366771297E-01 1.92171595772051E+00
x2 : -2.38300160166170E+00 1.65159813713614E+00
x3 : -1.17635241772214E+00 -2.17748014148655E+00
x4 : 5.01732358554671E-01 -1.22966143182879E+00
x5 : 1.19054568318791E+00 -6.79107470333444E-01
x6 : 4.69767610869959E-01 5.12934948792143E-01
x7 : 5.82616709754244E+00 -6.36153845578694E+00
== err : 4.678E-15 = rco : 5.913E-03 = res : 1.465E-14 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 2.63424603497105E-01 1.79307987438582E+00
x2 : -1.42230137654078E+00 -7.27270787149925E-01
x3 : 6.48059125615115E-01 -7.41546225335957E-01
x4 : -1.21625875444223E-01 4.44285120925777E-01
x5 : -5.08590494417704E-01 -4.40081292362661E-01
x6 : 1.41034017290485E-01 -3.28466690463057E-01
x7 : 6.62230193448536E+00 1.54232690910824E+01
== err : 4.305E-14 = rco : 2.736E-03 = res : 8.882E-15 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.08489054403391E+00 -1.79307987438582E+00
x2 : -1.75656198282473E+00 -7.45683258320039E-01
x3 : -2.96247415748369E-01 2.32634270785216E+00
x4 : 1.04373156113984E+00 -8.97470024637929E-01
x5 : -1.38967844523800E+00 3.78900562083038E-01
x6 : 3.13865738637349E-01 7.30989887408594E-01
x7 : 2.97569862064542E+00 -6.93036968326386E+00
== err : 4.644E-15 = rco : 8.877E-03 = res : 1.421E-14 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.93515873660876E-01 -4.74778387287990E-66
x2 : 7.22177850183674E-01 -1.42433516186397E-65
x3 : -1.65871370380082E+00 -7.59645419660784E-65
x4 : 1.16901800599298E-01 -4.74778387287990E-66
x5 : -1.07196055981947E-01 0.00000000000000E+00
x6 : -7.66685764661078E-01 4.50070940743944E-65
x7 : -7.82589201020625E+00 -4.59406805440176E-64
== err : 2.368E-15 = rco : 2.397E-02 = res : 1.776E-15 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 8.27399636935069E-01 1.28636083334685E-01
x2 : 4.97146350108961E-02 2.19412989932238E+00
x3 : 1.55565100551911E-01 1.21486906126276E+00
x4 : -2.47476413001305E+00 -2.09079838123884E+00
x5 : -1.71288232676718E-01 -8.69709294847598E-02
x6 : 6.13372990191896E-01 -1.35986573319623E+00
x7 : 1.65369671656056E+00 3.66629364825190E+00
== err : 5.475E-15 = rco : 2.262E-02 = res : 1.093E-14 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 3.97308366771298E-01 -1.92171595772051E+00
x2 : -2.38300160166170E+00 -1.65159813713614E+00
x3 : -1.17635241772214E+00 2.17748014148655E+00
x4 : 5.01732358554671E-01 1.22966143182879E+00
x5 : 1.19054568318791E+00 6.79107470333445E-01
x6 : 4.69767610869959E-01 -5.12934948792143E-01
x7 : 5.82616709754244E+00 6.36153845578694E+00
== err : 4.728E-15 = rco : 5.913E-03 = res : 1.375E-14 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.66666666666667E-01 -2.49628917422963E-52
x2 : -1.66666666666667E-01 -8.35238971903811E-53
x3 : -1.66666666666667E-01 5.01143383142287E-52
x4 : -1.66666666666667E-01 4.17619485951906E-52
x5 : -1.66666666666667E-01 5.84667280332668E-52
x6 : -1.66666666666667E-01 5.01143383142287E-52
x7 : -3.60000000000000E+01 6.41463530422127E-50
== err : 6.281E-15 = rco : 2.220E-03 = res : 1.776E-15 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 8.27399636935068E-01 3.71479583210633E+00
x2 : -6.84186578113855E+00 2.76209288924990E+00
x3 : -4.12071064151691E+00 -8.69959556204954E+00
x4 : 8.72511873116429E+00 -3.50394294906871E+00
x5 : 1.81846960499532E+00 5.23487937380794E+00
x6 : -1.40841155043922E+00 4.91770415954085E-01
x7 : -3.79717658548780E+00 -1.32584762480407E+00
== err : 2.537E-14 = rco : 2.797E-03 = res : 3.553E-14 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -4.24157573765506E-01 -1.92171595772051E+00
x2 : -1.48391565543637E+00 -7.29739304826774E-02
x3 : -5.60052371456211E-01 1.17922334308334E+00
x4 : 5.48670634671490E-01 9.30133142907219E-01
x5 : 7.08367213492227E-01 1.15818184498564E-01
x6 : 2.11087752494370E-01 -2.30484782285939E-01
x7 : 1.29659090383023E+01 1.41573572436994E+01
== err : 3.463E-14 = rco : 1.367E-03 = res : 1.489E-14 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.36024966562036E-01 1.92171595772051E+00
x2 : -8.16560321165552E-01 1.72600044478268E+00
x3 : -3.45095823467105E+00 8.67158668830608E-01
x4 : -5.62843410349120E-01 -4.13034050901962E+00
x5 : 2.42330915003328E+00 -1.44479097161200E+00
x6 : 9.71027849590408E-01 1.06025640929782E+00
x7 : 2.81860566521976E+00 -3.07760969275286E+00
== err : 6.365E-15 = rco : 9.009E-03 = res : 2.512E-14 ==
solution 17 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.51498181419768E+00 1.18504685467082E-62
x2 : 7.41222573661469E-01 1.21543267145725E-63
x3 : -1.07823548534975E+00 1.27620430503012E-62
x4 : -1.63478816493804E+00 -7.29259602874353E-63
x5 : 1.16304871296224E+00 1.09388940431153E-62
x6 : -1.70622945053361E+00 -1.52792069897868E-62
x7 : -3.51652586826676E+00 3.14903289293149E-62
== err : 5.155E-15 = rco : 3.603E-02 = res : 3.553E-15 ==
solution 18 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -6.81648480864347E-01 -3.03858167864314E-64
x2 : 9.58129145190158E-01 7.21663148677745E-64
x3 : 6.31940496052593E-01 7.59645419660784E-65
x4 : -4.34421392403970E-01 -4.55787251796470E-64
x5 : -8.87912123263308E-01 -1.80415787169436E-64
x6 : -5.86087644711127E-01 1.89911354915196E-64
x7 : -1.02373767032016E+01 8.07127252475485E-64
== err : 2.856E-14 = rco : 1.259E-02 = res : 3.553E-15 ==
solution 19 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.25749090709884E+00 -1.92171595772051E+00
x2 : -1.00903560786738E+00 -3.30462465143614E+00
x3 : -3.99271237385452E+00 -1.22680685072913E+00
x4 : -3.03453290093010E+00 3.65908289972281E+00
x5 : 3.61780513583611E+00 5.15362410077954E+00
x6 : 2.16098483971706E+00 -2.35955954061657E+00
x7 : 1.26652651496622E+00 1.38290869371563E+00
== err : 9.623E-15 = rco : 1.153E-02 = res : 1.066E-14 ==
solution 20 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.39817459672457E-01 -3.20475411419393E-65
x2 : -1.52476811214796E-01 -4.46083501852424E-65
x3 : 2.16348572029908E+00 1.85163571042316E-64
x4 : -9.41947644616723E-01 -6.64689742203186E-65
x5 : -9.04563389105643E-01 -6.88428661567585E-65
x6 : -1.30431533503437E+00 5.45995145381188E-65
x7 : -4.60011458796647E+00 9.49556774575980E-65
== err : 4.665E-15 = rco : 6.219E-02 = res : 2.887E-15 ==
solution 21 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.69908729836229E-01 -1.79307987438582E+00
x2 : -1.27629537390668E+00 1.77720268392520E-01
x3 : 2.62145169634245E+00 8.47407879790140E-01
x4 : 2.99663982501565E-01 -4.45879059994048E+00
x5 : -4.31844602385446E+00 2.65619747762992E+00
x6 : 1.10371698908089E+00 2.57054484851373E+00
x7 : 8.46204103742907E-01 -1.97080014277834E+00
== err : 4.554E-15 = rco : 2.347E-02 = res : 6.169E-15 ==
solution 22 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.00000000000000E+00 -6.68191177523049E-52
x2 : 1.00000000000000E+00 -1.78184314006146E-51
x3 : 1.00000000000000E+00 5.34552942018439E-51
x4 : 1.00000000000000E+00 5.34552942018439E-51
x5 : 1.00000000000000E+00 5.34552942018439E-51
x6 : -6.00000000000000E+00 -1.06910588403688E-50
x7 : -1.00000000000000E+00 6.68191177523049E-52
== err : 4.815E-35 = rco : 4.762E-02 = res : 2.717E-50 ==
solution 23 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.25749090709884E+00 1.92171595772051E+00
x2 : -1.00903560786738E+00 3.30462465143614E+00
x3 : -3.99271237385452E+00 1.22680685072913E+00
x4 : -3.03453290093010E+00 -3.65908289972281E+00
x5 : 3.61780513583611E+00 -5.15362410077954E+00
x6 : 2.16098483971706E+00 2.35955954061657E+00
x7 : 1.26652651496622E+00 -1.38290869371563E+00
== err : 8.689E-15 = rco : 1.153E-02 = res : 1.422E-14 ==
solution 24 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.08489054403391E+00 1.79307987438582E+00
x2 : -1.75656198282473E+00 7.45683258320039E-01
x3 : -2.96247415748368E-01 -2.32634270785216E+00
x4 : 1.04373156113984E+00 8.97470024637929E-01
x5 : -1.38967844523800E+00 -3.78900562083039E-01
x6 : 3.13865738637349E-01 -7.30989887408594E-01
x7 : 2.97569862064542E+00 6.93036968326386E+00
== err : 4.402E-15 = rco : 8.877E-03 = res : 1.589E-14 ==
solution 25 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.78534059463196E-01 -1.03311777073867E-62
x2 : 1.40399529688145E+00 1.94469227433161E-62
x3 : -4.24145311642535E-01 1.21543267145725E-62
x4 : 2.30307117019040E+00 -9.72346137165803E-63
x5 : -1.76538489062995E+00 5.34790375441192E-62
x6 : -2.69607032426256E+00 -6.32024989157772E-62
x7 : -2.22546123741826E+00 -3.88938454866321E-62
== err : 4.397E-15 = rco : 7.427E-02 = res : 1.221E-15 ==
solution 26 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.93369639826430E-03 3.71479583210633E+00
x2 : -6.29608516467496E+00 -2.89485362873527E-01
x3 : 6.85454641806506E-01 -6.41622249799802E+00
x4 : 4.94031447364827E+00 2.36147158475214E-01
x5 : 2.97245117069882E-01 2.53379026615599E+00
x6 : -6.32862764247966E-01 2.20974604134012E-01
x7 : -8.45046930263531E+00 -2.95062249572452E+00
== err : 9.137E-15 = rco : 1.203E-03 = res : 8.527E-14 ==
solution 27 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -2.51557210700575E-01 -1.79307987438582E+00
x2 : -5.18994747263476E-01 1.65067431386248E+00
x3 : 2.54014532482806E+00 -1.49593314929199E+00
x4 : -1.73247661935806E+00 -1.65911587822018E+00
x5 : -1.53306651761352E+00 2.14239297415820E+00
x6 : 4.95949770107570E-01 1.15506161387731E+00
x7 : 1.88319443182410E+00 -4.38593932445156E+00
== err : 4.379E-15 = rco : 1.345E-02 = res : 8.894E-15 ==
solution 28 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.36024966562036E-01 -1.92171595772051E+00
x2 : -8.16560321165553E-01 -1.72600044478268E+00
x3 : -3.45095823467105E+00 -8.67158668830608E-01
x4 : -5.62843410349120E-01 4.13034050901962E+00
x5 : 2.42330915003328E+00 1.44479097161200E+00
x6 : 9.71027849590408E-01 -1.06025640929782E+00
x7 : 2.81860566521976E+00 3.07760969275286E+00
== err : 5.695E-15 = rco : 9.009E-03 = res : 1.962E-14 ==
solution 29 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 8.27399636935069E-01 -1.28636083334685E-01
x2 : 4.97146350108961E-02 -2.19412989932238E+00
x3 : 1.55565100551911E-01 -1.21486906126276E+00
x4 : -2.47476413001305E+00 2.09079838123884E+00
x5 : -1.71288232676718E-01 8.69709294847598E-02
x6 : 6.13372990191896E-01 1.35986573319623E+00
x7 : 1.65369671656056E+00 -3.66629364825190E+00
== err : 5.425E-15 = rco : 2.262E-02 = res : 1.084E-14 ==
solution 30 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.69908729836229E-01 1.79307987438582E+00
x2 : -1.27629537390668E+00 -1.77720268392520E-01
x3 : 2.62145169634245E+00 -8.47407879790140E-01
x4 : 2.99663982501565E-01 4.45879059994048E+00
x5 : -4.31844602385446E+00 -2.65619747762992E+00
x6 : 1.10371698908089E+00 -2.57054484851373E+00
x7 : 8.46204103742907E-01 1.97080014277834E+00
== err : 4.554E-15 = rco : 2.347E-02 = res : 6.169E-15 ==
solution 31 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -4.24157573765507E-01 1.92171595772051E+00
x2 : -1.48391565543637E+00 7.29739304826770E-02
x3 : -5.60052371456211E-01 -1.17922334308334E+00
x4 : 5.48670634671490E-01 -9.30133142907219E-01
x5 : 7.08367213492227E-01 -1.15818184498564E-01
x6 : 2.11087752494370E-01 2.30484782285939E-01
x7 : 1.29659090383023E+01 -1.41573572436994E+01
== err : 4.016E-14 = rco : 1.367E-03 = res : 1.676E-14 ==
solution 32 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.54799273870137E-01 -1.51929083932157E-63
x2 : -8.54232602712374E-01 6.07716335728627E-64
x3 : 1.57319825015525E-01 0.00000000000000E+00
x4 : -5.20756185121201E-01 7.59645419660784E-64
x5 : -6.62201048157115E-02 3.03858167864314E-64
x6 : -3.70910206236376E-01 1.13946812949118E-64
x7 : -1.61764219455754E+01 -1.52687825261831E-62
== err : 2.186E-14 = rco : 6.932E-03 = res : 1.776E-15 ==