```

3
6.89081434496698E-05*x2**6+ 6.44001340651120E-05*x2**5+
1.14590985880982E-05*x2**4+ 7.39296683103111E-07*x2**3-7.99985447950654E+00*x1**2*x3+
1.37415740353738E-08*x2**2+ 7.89745634216885E-11*x2+ 1.27997671672105E-13;
-1.81128150008056E-05*x2**5-6.44584163729738E-06*x2**4
-6.23791126190069E-07*x2**3+ 2.24999351824589E+00*x2*x1**2+ 6.74998055473766E+00*x1**2*x3
-1.54595074644379E-08*x2**2+ 7.28997899911668E-07*x2*x1-1.11059680060617E-10*x2
-2.15999377751605E-13;
-1.33333318933335E+00*x1**2+ 2.66666637866670E+00*x1*x3+
4.31999953344005E-07*x2+ 2.39999974080003E-16;

TITLE : problem pb 601 with every equation divided by average coefficient.

ROOT COUNTS :

total degree : 60
3-homogeneous Bezout number : 48
with partition : {x2 }{x1 }{x3 }
generalized Bezout number : 34
based on the set structure :
{x2 x1 }{x2 x1 }{x2 x3 }{x2 }{x2 }{x2 }
{x2 x3 }{x2 x1 }{x2 x1 }{x2 }{x2 }
{x2 x1 }{x1 x3 }
mixed volume : 18

REFERENCES :

L.T. Watson.
`Globally convergent homotopy methods: a tutorial.
Appl. Math. Comput., 31(Spec. Issue):369--396, 1989.

NOTE :

---------------------------------------------------------------------
|     TIMING INFORMATION SUMMARY  on juno, Wed 5 Nov 1997           |
---------------------------------------------------------------------
|   root counts  |  start system  |  continuation  |   total time   |
---------------------------------------------------------------------
|  0h 0m 0s 34ms |  0h 0m 0s787ms |  0h 0m20s178ms |  0h 0m21s325ms |
---------------------------------------------------------------------

THE SOLUTIONS :

18 3
===========================================================================
solution : 1 :         start residual :  7.213E-22
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -7.25805372901499E-01   2.49212288249311E-68
x1 :  1.13267128276082E-03  -8.71609111337071E-71
x3 :  6.70143755976064E-04  -1.58474383879467E-71
== err :  2.006E-15 = rco :  2.094E-03 = res :  7.213E-22 = real regular ==
solution : 2 :         start residual :  3.078E+02
t :  8.18837546585981E-01   0.00000000000000E+00
m : 1
the solution for t :
x2 :  8.50815880957819E+00  -5.24548226861460E+00
x1 : -4.42445711781346E+00   9.20737715301611E-01
x3 : -2.21682655048735E+00   4.88171299800807E-01
== err :  1.553E-10 = rco :  7.740E-08 = res :  3.078E+02 = no solution ==
solution : 3 :         start residual :  1.497E-25
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -2.48030297185225E-05   4.12311836490762E-05
x1 : -1.58621604838585E-05   2.72212652443695E-05
x3 : -8.17846686445923E-06   1.36071825802625E-05
== err :  1.051E-19 = rco :  1.715E-05 = res :  1.497E-25 = complex regular ==
solution : 4 :         start residual :  1.483E-25
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -2.48030297185225E-05  -4.12311836490762E-05
x1 : -1.58621604838585E-05  -2.72212652443695E-05
x3 : -8.17846686445923E-06  -1.36071825802625E-05
== err :  9.233E-20 = rco :  1.715E-05 = res :  1.483E-25 = complex regular ==
solution : 5 :         start residual :  3.211E+02
t :  8.19658881053935E-01   0.00000000000000E+00
m : 1
the solution for t :
x2 : -8.91587080020560E+00  -4.74347729342008E+00
x1 :  3.07046359583193E+00   3.35764905417576E+00
x3 :  1.56137493957089E+00   1.66863445256399E+00
== err :  5.570E-10 = rco :  8.152E-08 = res :  3.211E+02 = no solution ==
solution : 6 :         start residual :  2.602E-29
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -2.79753388607537E-03   3.02700442741163E-04
x1 : -6.11915910154367E-07   1.47229329556450E-06
x3 : -1.37798440237791E-04  -2.49938473158386E-04
== err :  3.392E-17 = rco :  1.173E-08 = res :  2.602E-29 = complex regular ==
solution : 7 :         start residual :  8.416E-25
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -1.69467450435956E-02  -1.06169892227416E-03
x1 : -3.33252787010257E-08  -2.55931624735300E-06
x3 : -8.11742600306795E-05   1.07036125782306E-03
== err :  1.813E-16 = rco :  2.088E-08 = res :  8.416E-25 = complex regular ==
solution : 8 :         start residual :  3.095E+02
t :  8.18800817136328E-01   0.00000000000000E+00
m : 1
the solution for t :
x2 :  2.69742739549773E-01   1.00222317934163E+01
x1 :  1.39018208048578E+00  -4.31043715863374E+00
x3 :  6.73139597211546E-01  -2.17300894722606E+00
== err :  3.312E-10 = rco :  9.523E-08 = res :  3.095E+02 = no solution ==
solution : 9 :         start residual :  1.654E-24
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -9.12134388423492E-02  -3.29126160686788E-02
x1 : -1.56461412462451E-05   3.58863386679535E-06
x3 : -8.30790906178327E-04  -5.27740340761897E-04
== err :  2.532E-16 = rco :  7.540E-07 = res :  1.654E-24 = complex regular ==
solution : 10 :         start residual :  2.524E-29
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -4.92207628089560E-03  -4.56516765330819E-45
x1 : -1.47535767566305E-06  -5.29207412598255E-49
x3 : -5.41200700556774E-04  -1.25897908904183E-45
== err :  1.034E-16 = rco :  6.382E-09 = res :  2.524E-29 = real singular ==
solution : 11 :         start residual :  6.899E-22
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -7.24468371562951E-01   1.73768889747404E-63
x1 : -1.12504448363903E-03   5.74926953356550E-66
x3 : -6.66841562353025E-04   2.46662209020714E-66
== err :  2.081E-15 = rco :  2.076E-03 = res :  6.899E-22 = real regular ==
solution : 12 :         start residual :  8.336E-25
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -1.69467450435956E-02   1.06169892227416E-03
x1 : -3.33252787010254E-08   2.55931624735300E-06
x3 : -8.11742600306793E-05  -1.07036125782306E-03
== err :  1.755E-16 = rco :  2.088E-08 = res :  8.336E-25 = complex regular ==
solution : 13 :         start residual :  1.654E-24
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -9.28516277142570E-02  -3.39881530129233E-02
x1 :  1.60131916619861E-05  -4.09845171837017E-06
x3 :  8.07010140554413E-04   5.46296064967122E-04
== err :  2.193E-16 = rco :  8.445E-07 = res :  1.654E-24 = complex regular ==
solution : 14 :         start residual :  4.136E-25
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -6.34684428050861E-03  -7.43028589405630E-49
x1 :  1.12082450749626E-06   4.69821921695894E-53
x3 :  9.17910699087857E-04  -2.24512235647744E-49
== err :  1.097E-16 = rco :  5.530E-09 = res :  4.136E-25 = real singular ==
solution : 15 :         start residual :  6.617E-24
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -9.12134388423492E-02   3.29126160686787E-02
x1 : -1.56461412462451E-05  -3.58863386679535E-06
x3 : -8.30790906178328E-04   5.27740340761897E-04
== err :  2.260E-16 = rco :  7.540E-07 = res :  6.617E-24 = complex regular ==
solution : 16 :         start residual :  2.132E-25
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -2.79753388607537E-03  -3.02700442741164E-04
x1 : -6.11915910154367E-07  -1.47229329556450E-06
x3 : -1.37798440237790E-04   2.49938473158385E-04
== err :  3.252E-17 = rco :  1.173E-08 = res :  2.132E-25 = complex regular ==
solution : 17 :         start residual :  6.821E-24
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 : -9.28516277142570E-02   3.39881530129233E-02
x1 :  1.60131916619861E-05   4.09845171837017E-06
x3 :  8.07010140554413E-04  -5.46296064967122E-04
== err :  2.610E-16 = rco :  8.445E-07 = res :  6.821E-24 = complex regular ==
solution : 18 :         start residual :  4.201E-26
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x2 :  4.66889844311746E-05  -3.50324616081204E-46
x1 :  3.22097501038582E-05  -2.73691106313441E-48
x3 :  1.58700484713607E-05  -2.05268329735081E-48
== err :  7.222E-20 = rco :  1.957E-05 = res :  4.201E-26 = real regular ==
===========================================================================
A list of 18 solutions has been refined :
Number of regular solutions   : 13.
Number of singular solutions  : 2.
Number of real solutions      : 5.
Number of complex solutions   : 10.
Number of clustered solutions : 0.
Number of failures            : 3.
===========================================================================
```