6
(x1 + x1*x2 + x2*x3 + x3*x4 + x4*x5)*x6 - 1;
(x2 + x1*x3 + x2*x4 + x3*x5)*x6 - 2;
(x3 + x1*x4 + x2*x5)*x6 - 3;
(x4 + x1*x5)*x6 - 4;
x5*x6 - 5;
x1 + x2 + x3 + x4 + x5 + 1;
TITLE : 6-dimensional economics problem
ROOT COUNTS :
total degree : 162
3-homogeneous Bezout number : 48
with partition : {x1 x2 x3 x4 }{x5 }{x6 }
generalized Bezout number : 36
based on the set structure :
{x1 x3 x5 }{x2 x4 }{x6 }
{x1 x2 x5 }{x3 x4 }{x6 }
{x1 x2 x3 }{x4 x5 }{x6 }
{x1 x4 }{x5 }{x6 }
{x5 }{x6 }
{x1 x2 x3 x4 x5 }
mixed volume : 16
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/x6 and the total degree equals the number of solutions.
See the reduced economics problem, in the file redeco6.
THE SOLUTIONS :
16 6
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.00000000000000E+00 0.00000000000000E+00
x2 : 1.00000000000000E+00 5.93472984109987E-67
x3 : 1.00000000000000E+00 2.96736492054994E-67
x4 : 1.00000000000000E+00 0.00000000000000E+00
x5 : -5.00000000000000E+00 0.00000000000000E+00
x6 : -1.00000000000000E+00 0.00000000000000E+00
== err : 5.346E-51 = rco : 6.489E-02 = res : 2.671E-66 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.00000000000000E-01 -3.31042891395086E+00
x2 : -5.25946979716094E+00 -5.27706843576464E-01
x3 : 5.56851925799105E-01 5.25646392480149E+00
x4 : 3.30816086148002E+00 -4.18343529761888E-01
x5 : -5.54299011818361E-03 -9.99984637512272E-01
x6 : -2.77149505909180E-02 4.99992318756136E+00
== err : 5.621E-15 = rco : 6.151E-03 = res : 4.394E-14 ==
solution 3 :
t : 1.00000000000000E+00 2.16840434497101E-19
m : 3
the solution for t :
x1 : 5.49983204654658E-01 1.21618144467362E+00
x2 : -1.18910100147303E+00 -6.28630424774533E-01
x3 : 4.07202455367483E-01 -2.59518299581422E-01
x4 : -5.83588286261556E-01 1.13892369777438E-01
x5 : -1.84496372287553E-01 -4.41925090095103E-01
x6 : -4.02239098644747E+00 9.63485339599522E+00
== err : 4.533E-15 = rco : 7.742E-03 = res : 2.844E-15 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.00000000000000E-01 -8.78066024603626E-01
x2 : -1.65499971781608E-01 -2.14978127415289E+00
x3 : -1.62144990845010E+00 1.42121425610120E+00
x4 : -3.27974968334248E-01 9.07431740523421E-01
x5 : 7.14924848565961E-01 6.99201302131896E-01
x6 : 3.57462424282980E+00 -3.49600651065948E+00
== err : 4.906E-15 = rco : 1.880E-02 = res : 9.197E-15 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.00000000000000E-01 3.31042891395086E+00
x2 : -5.25946979716094E+00 5.27706843576465E-01
x3 : 5.56851925799103E-01 -5.25646392480149E+00
x4 : 3.30816086148002E+00 4.18343529761887E-01
x5 : -5.54299011818325E-03 9.99984637512272E-01
x6 : -2.77149505909162E-02 -4.99992318756136E+00
== err : 4.327E-15 = rco : 6.151E-03 = res : 3.035E-14 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.14998320465466E+00 -2.09424746927724E+00
x2 : -1.77249865678311E+00 -2.90939448721313E+00
x3 : -3.76795822665312E+00 9.94499923819595E-01
x4 : 1.18599548149207E+00 4.94385765452914E+00
x5 : 2.20447819728949E+00 -9.34715621858359E-01
x6 : 1.92248186143776E+00 8.15147017935889E-01
== err : 5.357E-15 = rco : 1.513E-02 = res : 3.972E-15 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -2.00000000000000E-01 1.89911354915196E-65
x2 : -2.00000000000000E-01 -1.45056101563798E-66
x3 : -2.00000000000000E-01 0.00000000000000E+00
x4 : -2.00000000000000E-01 0.00000000000000E+00
x5 : -2.00000000000000E-01 -2.96736492054994E-67
x6 : -2.50000000000000E+01 1.51929083932157E-64
== err : 4.626E-16 = rco : 3.803E-03 = res : 8.882E-16 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -3.49983204654658E-01 -2.09424746927724E+00
x2 : -1.61089899852697E+00 2.31906369483777E-01
x3 : -2.07202455367483E-01 1.40762102476128E+00
x4 : 7.83588286261555E-01 6.17749478619368E-01
x5 : 3.84496372287553E-01 -1.63029403587178E-01
x6 : 1.10223909864475E+01 4.67357810929179E+00
== err : 3.086E-14 = rco : 3.238E-03 = res : 7.324E-15 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.49983204654658E-01 -1.21618144467362E+00
x2 : -1.18910100147303E+00 6.28630424774533E-01
x3 : 4.07202455367483E-01 2.59518299581422E-01
x4 : -5.83588286261556E-01 -1.13892369777438E-01
x5 : -1.84496372287553E-01 4.41925090095103E-01
x6 : -4.02239098644747E+00 -9.63485339599522E+00
== err : 4.533E-15 = rco : 7.742E-03 = res : 2.844E-15 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.14998320465466E+00 2.09424746927724E+00
x2 : -1.77249865678311E+00 2.90939448721313E+00
x3 : -3.76795822665312E+00 -9.94499923819595E-01
x4 : 1.18599548149208E+00 -4.94385765452914E+00
x5 : 2.20447819728949E+00 9.34715621858360E-01
x6 : 1.92248186143776E+00 -8.15147017935890E-01
== err : 6.108E-15 = rco : 1.513E-02 = res : 7.536E-15 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -3.49983204654658E-01 2.09424746927724E+00
x2 : -1.61089899852697E+00 -2.31906369483777E-01
x3 : -2.07202455367483E-01 -1.40762102476128E+00
x4 : 7.83588286261555E-01 -6.17749478619368E-01
x5 : 3.84496372287553E-01 1.63029403587178E-01
x6 : 1.10223909864475E+01 -4.67357810929179E+00
== err : 3.086E-14 = rco : 3.238E-03 = res : 7.324E-15 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -4.99966409309316E-01 -1.63205070630247E-66
x2 : 1.38656942719869E+00 1.18694596821997E-66
x3 : 1.19152225800455E-01 1.55786658328872E-66
x4 : -1.13367501000843E+00 -2.37389193643995E-66
x5 : -8.72080233681396E-01 9.64393599178730E-67
x6 : -5.73341741607079E+00 -4.74778387287990E-66
== err : 3.013E-15 = rco : 2.984E-02 = res : 1.776E-15 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 2.50016795345342E-01 -1.21618144467362E+00
x2 : 1.72498656783105E-01 9.93444005801895E-01
x3 : 2.16795822665312E+00 -1.78564329356081E+00
x4 : -2.78599548149208E+00 8.14100532334901E-02
x5 : -8.04478197289493E-01 1.92697067919904E+00
x6 : -9.22481861437762E-01 -2.20962545047551E+00
== err : 5.006E-15 = rco : 3.564E-02 = res : 5.925E-15 ==
solution 14 :
t : 1.00000000000000E+00 2.16840434497101E-19
m : 14
the solution for t :
x1 : 2.50016795345342E-01 1.21618144467362E+00
x2 : 1.72498656783105E-01 -9.93444005801894E-01
x3 : 2.16795822665312E+00 1.78564329356081E+00
x4 : -2.78599548149208E+00 -8.14100532334898E-02
x5 : -8.04478197289493E-01 -1.92697067919904E+00
x6 : -9.22481861437763E-01 2.20962545047551E+00
== err : 4.743E-15 = rco : 3.564E-02 = res : 7.161E-15 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.29996640930932E+00 3.26410141260493E-66
x2 : -1.36629889313586E-01 1.39466151265847E-65
x3 : -1.58995626049845E+00 -1.30564056504197E-65
x4 : 5.73303223716882E-01 1.78041895232996E-66
x5 : -1.14668348321416E+00 -2.96736492054994E-67
x6 : -4.36040116840698E+00 4.74778387287990E-66
== err : 3.875E-15 = rco : 3.823E-02 = res : 8.882E-16 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.00000000000000E-01 8.78066024603626E-01
x2 : -1.65499971781608E-01 2.14978127415289E+00
x3 : -1.62144990845010E+00 -1.42121425610120E+00
x4 : -3.27974968334248E-01 -9.07431740523421E-01
x5 : 7.14924848565961E-01 -6.99201302131896E-01
x6 : 3.57462424282980E+00 3.49600651065948E+00
== err : 4.906E-15 = rco : 1.880E-02 = res : 9.197E-15 ==