7
 x1 + x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 -   u7;
         x2 + x1*x3 + x2*x4 + x3*x5 + x4*x6 - 2*u7;
                 x3 + x1*x4 + x2*x5 + x3*x6 - 3*u7;
                         x4 + x1*x5 + x2*x6 - 4*u7;
                                 x5 + x1*x6 - 5*u7;
                                         x6 - 6*u7;
                   x1 + x2 + x3 + x4 + x5 + x6 + 1;

TITLE : reduced 7-dimensional economics problem

ROOT COUNTS :

total degree : 32
2-homogeneous Bezout number : 64
  with partition : {x1 x2 x3 x4 x5 }{x6 u7 }
generalized Bezout number : 32
  based on the set structure :
     {x1 x3 x5 u7 }{x2 x4 x6 }
     {x1 x2 x5 x6 u7 }{x3 x4 }
     {x1 x2 x3 u7 }{x4 x5 x6 }
     {x1 x2 x4 u7 }{x5 x6 }
     {x1 x5 u7 }{x6 }
     {x6 u7 }
     {x1 x2 x3 x4 x5 x6 }
mixed volume : 32

REFERENCE :

This is the reduced economics problem (u7 = 1/x7).

Alexander Morgan:
`Solving polynomial systems using continuation for engineering
 and scientific problems', Prentice-Hall, Englewood Cliffs, New Jersey, 1987,
(p 148).

THE SOLUTIONS :

32 7
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 : -6.81648480864347E-01  -3.40321148008031E-62
 x2 :  9.58129145190158E-01   5.83407682299482E-62
 x3 :  6.31940496052593E-01  -3.88938454866321E-62
 x4 : -4.34421392403970E-01   4.86173068582902E-62
 x5 : -8.87912123263308E-01  -3.88938454866321E-62
 x6 : -5.86087644711127E-01  -9.72346137165803E-63
 u7 : -9.76812741185211E-02  -3.03858167864314E-63
== err :  6.304E-16 = rco :  2.962E-02 = res :  2.220E-16 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  5.93369639826469E-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
 u7 :  4.59360199044601E-02   1.01841490229219E-01
== err :  2.366E-15 = rco :  2.512E-02 = res :  2.118E-16 ==
solution 3 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  6.93515873660876E-01   1.60919707978251E-54
 x2 :  7.22177850183674E-01  -7.83036536159823E-54
 x3 : -1.65871370380082E+00   2.61012178719941E-54
 x4 :  1.16901800599298E-01   0.00000000000000E+00
 x5 : -1.07196055981947E-01  -3.26265223399926E-55
 x6 : -7.66685764661078E-01   2.61012178719941E-54
 u7 : -1.27780960776846E-01   6.52530446799852E-55
== err :  2.177E-15 = rco :  2.812E-02 = res :  2.220E-16 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 : -1.66666666666667E-01   3.98272977783113E-59
 x2 : -1.66666666666667E-01   0.00000000000000E+00
 x3 : -1.66666666666667E-01  -9.95682444457783E-60
 x4 : -1.66666666666667E-01   9.95682444457783E-60
 x5 : -1.66666666666667E-01  -4.48057100006002E-59
 x6 : -1.66666666666667E-01   0.00000000000000E+00
 u7 : -2.77777777777778E-02  -3.38667498114907E-61
== err :  3.132E-16 = rco :  3.765E-02 = res :  2.220E-16 ==
solution 5 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  5.93369639826462E-03   3.71479583210633E+00
 x2 : -6.29608516467496E+00  -2.89485362873526E-01
 x3 :  6.85454641806507E-01  -6.41622249799802E+00
 x4 :  4.94031447364827E+00   2.36147158475215E-01
 x5 :  2.97245117069881E-01   2.53379026615599E+00
 x6 : -6.32862764247967E-01   2.20974604134012E-01
 u7 : -1.05477127374661E-01   3.68291006890019E-02
== err :  5.334E-15 = rco :  1.183E-02 = res :  1.067E-14 ==
solution 6 :
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
 u7 :  8.26582950179284E-02   1.92510268979552E-01
== err :  3.738E-15 = rco :  2.485E-02 = res :  9.930E-16 ==
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   6.79107470333445E-01
 x6 :  4.69767610869959E-01  -5.12934948792143E-01
 u7 :  7.82946018116599E-02  -8.54891581320238E-02
== err :  4.453E-15 = rco :  1.532E-02 = res :  8.882E-16 ==
solution 8 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  5.93369639826422E-03  -3.71479583210633E+00
 x2 : -6.29608516467496E+00   2.89485362873526E-01
 x3 :  6.85454641806506E-01   6.41622249799802E+00
 x4 :  4.94031447364827E+00  -2.36147158475213E-01
 x5 :  2.97245117069882E-01  -2.53379026615599E+00
 x6 : -6.32862764247966E-01  -2.20974604134012E-01
 u7 : -1.05477127374661E-01  -3.68291006890020E-02
== err :  4.397E-15 = rco :  1.183E-02 = res :  3.878E-15 ==
solution 9 :
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   8.47407879790141E-01
 x4 :  2.99663982501566E-01  -4.45879059994048E+00
 x5 : -4.31844602385446E+00   2.65619747762992E+00
 x6 :  1.10371698908089E+00   2.57054484851373E+00
 u7 :  1.83952831513482E-01   4.28424141418954E-01
== err :  4.440E-15 = rco :  2.857E-02 = res :  1.831E-15 ==
solution 10 :
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 : -5.62843410349119E-01   4.13034050901962E+00
 x5 :  2.42330915003328E+00   1.44479097161200E+00
 x6 :  9.71027849590407E-01  -1.06025640929782E+00
 u7 :  1.61837974931735E-01  -1.76709401549637E-01
== err :  5.610E-15 = rco :  1.976E-02 = res :  2.402E-15 ==
solution 11 :
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 : -5.60052371456210E-01   1.17922334308334E+00
 x4 :  5.48670634671490E-01   9.30133142907219E-01
 x5 :  7.08367213492226E-01   1.15818184498564E-01
 x6 :  2.11087752494370E-01  -2.30484782285939E-01
 u7 :  3.51812920823950E-02  -3.84141303809898E-02
== err :  2.990E-15 = rco :  1.721E-02 = res :  8.006E-16 ==
solution 12 :
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 :  3.61780513583610E+00  -5.15362410077954E+00
 x6 :  2.16098483971706E+00   2.35955954061657E+00
 u7 :  3.60164139952843E-01   3.93259923436095E-01
== err :  7.855E-15 = rco :  1.195E-02 = res :  5.403E-15 ==
solution 13 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  8.27399636935069E-01   1.28636083334685E-01
 x2 :  4.97146350108965E-02   2.19412989932238E+00
 x3 :  1.55565100551911E-01   1.21486906126276E+00
 x4 : -2.47476413001306E+00  -2.09079838123884E+00
 x5 : -1.71288232676718E-01  -8.69709294847598E-02
 x6 :  6.13372990191896E-01  -1.35986573319623E+00
 u7 :  1.02228831698649E-01  -2.26644288866038E-01
== err :  5.124E-15 = rco :  2.241E-02 = res :  1.057E-15 ==
solution 14 :
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  -6.79107470333445E-01
 x6 :  4.69767610869959E-01   5.12934948792143E-01
 u7 :  7.82946018116599E-02   8.54891581320238E-02
== err :  4.453E-15 = rco :  1.532E-02 = res :  8.882E-16 ==
solution 15 :
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 :  3.61780513583610E+00   5.15362410077954E+00
 x6 :  2.16098483971706E+00  -2.35955954061657E+00
 u7 :  3.60164139952843E-01  -3.93259923436095E-01
== err :  1.282E-14 = rco :  1.195E-02 = res :  7.324E-15 ==
solution 16 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  1.00000000000000E+00   0.00000000000000E+00
 x2 :  1.00000000000000E+00   0.00000000000000E+00
 x3 :  1.00000000000000E+00   5.93472984109987E-67
 x4 :  1.00000000000000E+00  -9.08755506918418E-67
 x5 :  1.00000000000000E+00  -5.93472984109987E-67
 x6 : -6.00000000000000E+00  -4.74778387287990E-66
 u7 : -1.00000000000000E+00  -1.18694596821997E-66
== err :  2.940E-50 = rco :  4.762E-02 = res :  5.657E-66 ==
solution 17 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 : -4.24157573765507E-01   1.92171595772051E+00
 x2 : -1.48391565543637E+00   7.29739304826769E-02
 x3 : -5.60052371456210E-01  -1.17922334308334E+00
 x4 :  5.48670634671490E-01  -9.30133142907219E-01
 x5 :  7.08367213492226E-01  -1.15818184498564E-01
 x6 :  2.11087752494370E-01   2.30484782285939E-01
 u7 :  3.51812920823950E-02   3.84141303809898E-02
== err :  2.809E-15 = rco :  1.721E-02 = res :  4.475E-16 ==
solution 18 :
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
 u7 :  2.35056695484141E-02   5.47444484105095E-02
== err :  2.618E-15 = rco :  2.056E-02 = res :  3.554E-16 ==
solution 19 :
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
 u7 :  2.35056695484141E-02  -5.47444484105095E-02
== err :  2.558E-15 = rco :  2.056E-02 = res :  2.483E-16 ==
solution 20 :
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
 u7 : -2.34735258406536E-01   8.19617359923479E-02
== err :  1.048E-14 = rco :  3.851E-03 = res :  1.421E-14 ==
solution 21 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  6.54799273870137E-01   2.91703841149741E-62
 x2 : -8.54232602712374E-01  -2.91703841149741E-62
 x3 :  1.57319825015525E-01   1.97800470374835E-62
 x4 : -5.20756185121201E-01  -1.59209508504746E-62
 x5 : -6.62201048157115E-02   1.21543267145725E-63
 x6 : -3.70910206236376E-01  -1.09388940431153E-62
 u7 : -6.18183677060627E-02  -1.79360058965448E-63
== err :  4.779E-16 = rco :  2.506E-02 = res :  1.110E-16 ==
solution 22 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  8.27399636935069E-01  -1.28636083334685E-01
 x2 :  4.97146350108963E-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
 u7 :  1.02228831698649E-01   2.26644288866038E-01
== err :  4.845E-15 = rco :  2.241E-02 = res :  1.116E-15 ==
solution 23 :
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
 u7 :  8.26582950179284E-02  -1.92510268979552E-01
== err :  3.232E-15 = rco :  2.485E-02 = res :  1.351E-15 ==
solution 24 :
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   8.67158668830609E-01
 x4 : -5.62843410349120E-01  -4.13034050901962E+00
 x5 :  2.42330915003328E+00  -1.44479097161200E+00
 x6 :  9.71027849590408E-01   1.06025640929782E+00
 u7 :  1.61837974931735E-01   1.76709401549637E-01
== err :  6.794E-15 = rco :  1.976E-02 = res :  3.202E-15 ==
solution 25 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  5.93369639826470E-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
 u7 :  4.59360199044601E-02  -1.01841490229219E-01
== err :  2.370E-15 = rco :  2.512E-02 = res :  2.178E-16 ==
solution 26 :
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 : -4.12071064151691E+00   8.69959556204954E+00
 x4 :  8.72511873116428E+00   3.50394294906872E+00
 x5 :  1.81846960499532E+00  -5.23487937380794E+00
 x6 : -1.40841155043922E+00  -4.91770415954087E-01
 u7 : -2.34735258406536E-01  -8.19617359923478E-02
== err :  1.532E-14 = rco :  3.851E-03 = res :  2.383E-14 ==
solution 27 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  1.08489054403391E+00   1.79307987438582E+00
 x2 : -1.75656198282473E+00   7.45683258320038E-01
 x3 : -2.96247415748368E-01  -2.32634270785216E+00
 x4 :  1.04373156113984E+00   8.97470024637929E-01
 x5 : -1.38967844523799E+00  -3.78900562083039E-01
 x6 :  3.13865738637350E-01  -7.30989887408594E-01
 u7 :  5.23109564395583E-02  -1.21831647901432E-01
== err :  4.986E-15 = rco :  1.450E-02 = res :  1.299E-15 ==
solution 28 :
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 :  3.13865738637350E-01   7.30989887408594E-01
 u7 :  5.23109564395583E-02   1.21831647901432E-01
== err :  4.894E-15 = rco :  1.450E-02 = res :  1.986E-15 ==
solution 29 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  1.39817459672457E-01  -2.62619866977784E-63
 x2 : -1.52476811214796E-01  -4.70980160189686E-63
 x3 :  2.16348572029908E+00  -3.03858167864314E-63
 x4 : -9.41947644616723E-01   5.01365976976117E-63
 x5 : -9.04563389105642E-01   9.11574503592941E-64
 x6 : -1.30431533503437E+00   4.86173068582902E-63
 u7 : -2.17385889172396E-01   8.35609961626862E-64
== err :  4.735E-15 = rco :  4.114E-02 = res :  1.943E-16 ==
solution 30 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  1.51498181419768E+00   2.22552369041245E-67
 x2 :  7.41222573661469E-01   5.93472984109987E-67
 x3 : -1.07823548534975E+00   7.78933291644359E-67
 x4 : -1.63478816493804E+00   0.00000000000000E+00
 x5 :  1.16304871296224E+00  -3.70920615068742E-67
 x6 : -1.70622945053360E+00  -1.11276184520623E-66
 u7 : -2.84371575088934E-01  -1.85460307534371E-67
== err :  5.014E-15 = rco :  2.480E-02 = res :  4.441E-16 ==
solution 31 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 :  1.78534059463196E-01   9.11574503592941E-64
 x2 :  1.40399529688145E+00  -4.55787251796470E-63
 x3 : -4.24145311642535E-01  -1.01602574879630E-63
 x4 :  2.30307117019040E+00  -7.29259602874353E-63
 x5 : -1.76538489062995E+00   4.25401435010039E-63
 x6 : -2.69607032426256E+00   5.92523427335412E-63
 u7 : -4.49345054043761E-01   1.06350358752510E-63
== err :  2.862E-15 = rco :  3.735E-02 = res :  7.772E-16 ==
solution 32 :
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
 u7 :  1.83952831513482E-01  -4.28424141418954E-01
== err :  5.346E-15 = rco :  2.857E-02 = res :  2.220E-15 ==