``` 3
x1**3-x2*x3;
x2**3-x1*x3;
x3**3-x1*x2;

TITLE : example from CBMS lectures, origin has multiplicy 11

This system has 16 regular solutions and (0,0,0) occurs with
multiplicity 11.  One deflation with tolerance = 1.0E-4
(instead of the default 1.0E-6) suffices to find the origin correctly.

REFERENCES :

Bernd Sturmfels: "Solving Systems of Polynomial Equations"
CBMS Regional Conference Series in Mathematics, Number 97,
AMS 2002, page 134.

THE SOLUTIONS :
27 3
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  0.00000000000000E+00   1.00000000000000E+00
x2 :  0.00000000000000E+00   1.00000000000000E+00
x3 : -1.00000000000000E+00   0.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  1.00000000000000E+00   0.00000000000000E+00
x2 :  0.00000000000000E+00   1.00000000000000E+00
x3 :  0.00000000000000E+00  -1.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 3 :
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   0.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -2.73765171436056E-11  -4.85744930238757E-12
x2 : -1.24454719919523E-05  -1.06446388283176E-06
x3 :  2.18825088701436E-11  -1.71529701297422E-11
== err :  6.755E-06 = rco :  3.436E-05 = res :  3.285E-15 ==
solution 5 :
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   6.78874747444356E-152
== err :  7.127E-51 = rco :  2.000E-01 = res :  3.394E-151 ==
solution 6 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  5.78076263737166E-06   6.64470804759034E-06
x2 :  1.13062313881419E-11  -7.63945627158148E-12
x3 : -1.12332131107518E-11   7.74642249009998E-12
== err :  6.213E-06 = rco :  2.114E-05 = res :  1.239E-15 ==
solution 7 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  1.00000000000000E+00   0.00000000000000E+00
x2 :  0.00000000000000E+00  -1.00000000000000E+00
x3 :  0.00000000000000E+00   1.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 8 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  0.00000000000000E+00   1.00000000000000E+00
x2 :  1.00000000000000E+00   0.00000000000000E+00
x3 :  0.00000000000000E+00  -1.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 9 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -2.74352356727701E-11  -4.50977098352231E-12
x2 : -2.18271337147256E-11   1.72222708001136E-11
x3 :  1.24425541434375E-05   1.10449778023456E-06
== err :  6.774E-06 = rco :  3.215E-05 = res :  3.296E-15 ==
solution 10 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.00000000000000E+00   0.00000000000000E+00
x2 :  0.00000000000000E+00   1.00000000000000E+00
x3 :  0.00000000000000E+00   1.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 11 :
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   0.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 12 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  8.56611743187778E-11  -1.01658570259253E-10
x2 :  5.82368592585610E-06   1.23832320148551E-05
x3 :  1.85802291649511E-11   1.31632636487452E-10
== err :  9.104E-06 = rco :  3.105E-05 = res :  7.866E-15 ==
solution 13 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.29646151622107E-62   1.00000000000000E+00
x2 : -1.00000000000000E+00  -1.94469227433161E-62
x3 : -1.94469227433161E-62   1.00000000000000E+00
== err :  1.752E-46 = rco :  2.000E-01 = res :  5.186E-62 ==
solution 14 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  8.43674661920762E-11  -1.02734290520474E-10
x2 : -1.89969769281707E-11  -1.31573215629036E-10
x3 : -5.78497070501530E-06  -1.24014956986517E-05
== err :  9.094E-06 = rco :  4.226E-05 = res :  7.870E-15 ==
solution 15 :
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   0.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 16 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.28719830471098E-05   2.52797852311137E-06
x2 : -8.45340486736735E-11  -4.12605172252170E-11
x3 :  8.49228721699610E-11   4.04564381039593E-11
== err :  7.700E-06 = rco :  4.371E-05 = res :  6.138E-15 ==
solution 17 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.00000000000000E+00   7.87880842051049E-126
x2 :  0.00000000000000E+00  -1.00000000000000E+00
x3 : -2.36364252615315E-125  -1.00000000000000E+00
== err :  1.955E-109 = rco :  2.000E-01 = res :  9.455E-125 ==
solution 18 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  0.00000000000000E+00   1.00000000000000E+00
x2 :  0.00000000000000E+00  -1.00000000000000E+00
x3 :  1.00000000000000E+00   0.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 19 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  0.00000000000000E+00  -1.00000000000000E+00
x2 :  1.00000000000000E+00   0.00000000000000E+00
x3 :  0.00000000000000E+00   1.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 20 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -2.10805935709363E-09  -3.14085347951474E-10
x2 : -2.53984400896948E-10  -3.83259588534239E-09
x3 :  3.94734707374156E-10   3.84737631815935E-09
== err :  1.075E-08 = rco :  2.760E-01 = res :  3.457E-17 ==
solution 21 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  0.00000000000000E+00  -1.00000000000000E+00
x2 :  0.00000000000000E+00   1.00000000000000E+00
x3 :  1.00000000000000E+00   0.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 22 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  9.19089056347244E-12   2.45430821845003E-11
x2 :  2.43229858604610E-11   9.75847298028484E-12
x3 : -6.92469040053085E-06   9.95470179110771E-06
== err :  8.440E-06 = rco :  2.233E-05 = res :  2.999E-15 ==
solution 23 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  0.00000000000000E+00  -1.00000000000000E+00
x2 : -1.00000000000000E+00   0.00000000000000E+00
x3 :  0.00000000000000E+00  -1.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 24 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.01416115658826E-08   8.78966775138949E-09
x2 :  2.69219399961400E-09   2.67725274022063E-09
x3 : -2.51393368434959E-09  -2.62401788925816E-09
== err :  2.944E-08 = rco :  9.658E-02 = res :  1.215E-16 ==
solution 25 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  4.63733305530515E-06  -1.35240451127704E-05
x2 :  7.71400925571445E-12   1.09240696195101E-10
x3 : -8.75022751637894E-12  -1.09162621975080E-10
== err :  9.081E-06 = rco :  4.738E-05 = res :  7.866E-15 ==
solution 26 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  0.00000000000000E+00  -1.00000000000000E+00
x2 :  0.00000000000000E+00  -1.00000000000000E+00
x3 : -1.00000000000000E+00   0.00000000000000E+00
== err :  0.000E+00 = rco :  2.000E-01 = res :  0.000E+00 ==
solution 27 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  8.87905192612294E-12   2.46566429790388E-11
x2 :  6.89344988759544E-06  -9.97730266233753E-06
x3 : -2.42910644755182E-11  -9.83523469851215E-12
== err :  8.436E-06 = rco :  2.600E-05 = res :  2.989E-15 ==
```