```3
x1**3  - 3*x1**2*x2 + 3*x1*x2**2  - x2**3  - x3**2;
x3**3  - 3*x3**2*x1 + 3*x3*x1**2  - x1**3  - x2**2;
x2**3  - 3*x2**2*x3 + 3*x2*x3**2  - x3**3  - x1**2;

TITLE : system with 6 regular roots and (0,0,0) of multiplicity 8

One deflation suffices to find the eightfold root at (0,0,0)
correctly.

REFERENCES :

Bernd Sturmfels: "Solving Systems of Polynomial Equations."
CBMS Regional Conference Series in Mathematics, Number 97,
AMS 2002, page 13.

THE SOLUTIONS :
14 3
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  5.37277086211826E-09   2.18551897649950E-10
x2 :  5.28162126749316E-09  -3.69855914141030E-09
x3 :  7.39506616258108E-09   1.88871673942021E-09
== err :  2.386E-08 = rco :  6.294E-01 = res :  1.635E-16 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -5.23545990215904E-09   1.07673420770961E-08
x2 :  1.09480653581837E-08   1.22315306516400E-08
x3 : -4.65687341671419E-09  -1.25449523691313E-09
== err :  4.509E-08 = rco :  2.937E-01 = res :  5.306E-16 ==
solution 3 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  3.63243594028686E-09  -8.69030893384238E-09
x2 :  2.70366348870743E-09  -1.89352374969738E-09
x3 : -1.04669667684142E-08   7.06940817391873E-09
== err :  3.446E-08 = rco :  3.343E-01 = res :  3.470E-16 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.42331834475307E-01   3.58782271022646E-01
x2 :  1.42331834475307E-01   3.58782271022647E-01
x3 : -6.15292468146106E-19  -1.51879117096055E-01
== err :  1.709E-15 = rco :  4.421E-02 = res :  1.527E-16 ==
solution 5 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.35727462454695E-17   1.51879117096055E-01
x2 : -1.42331834475307E-01  -3.58782271022646E-01
x3 :  1.42331834475307E-01  -3.58782271022647E-01
== err :  1.747E-15 = rco :  4.581E-02 = res :  1.776E-16 ==
solution 6 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  2.88660590905561E-09   7.66240042036625E-10
x2 : -6.87183237259284E-09  -7.53509140870451E-09
x3 :  3.25031345997487E-09  -6.24640101900724E-09
== err :  2.756E-08 = rco :  3.506E-01 = res :  1.943E-16 ==
solution 7 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.15389975396580E-17  -1.51879117096055E-01
x2 : -1.42331834475307E-01   3.58782271022646E-01
x3 :  1.42331834475307E-01   3.58782271022647E-01
== err :  1.747E-15 = rco :  4.581E-02 = res :  1.770E-16 ==
solution 8 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.07010212905962E-10   8.00722640240981E-11
x2 :  3.83842668264870E-11   1.43499544213946E-10
x3 :  2.90305874905303E-11  -1.38906488476950E-13
== err :  3.981E-10 = rco :  2.154E-01 = res :  5.316E-20 ==
solution 9 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.42331834475307E-01  -3.58782271022646E-01
x2 :  1.42331834475307E-01  -3.58782271022646E-01
x3 : -4.57001783878961E-18   1.51879117096055E-01
== err :  1.610E-15 = rco :  4.421E-02 = res :  1.214E-16 ==
solution 10 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  5.14577405651810E-08  -3.74452876770802E-08
x2 : -1.37080915817790E-08   9.60478197316042E-09
x3 : -2.00434987875522E-08   4.18880330533515E-08
== err :  1.741E-07 = rco :  2.527E-01 = res :  8.490E-15 ==
solution 11 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -3.18617999286573E-10  -1.36780215380865E-11
x2 : -4.23072795508151E-10  -1.57594008874193E-09
x3 :  1.20969388659049E-09  -8.34025083844505E-10
== err :  4.375E-09 = rco :  2.039E-01 = res :  6.534E-18 ==
solution 12 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  1.42331834475307E-01  -3.58782271022647E-01
x2 :  1.40345504410974E-17   1.51879117096055E-01
x3 : -1.42331834475307E-01  -3.58782271022647E-01
== err :  1.781E-15 = rco :  4.193E-02 = res :  1.527E-16 ==
solution 13 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  1.42331834475307E-01   3.58782271022646E-01
x2 :  1.30502637400813E-17  -1.51879117096055E-01
x3 : -1.42331834475307E-01   3.58782271022646E-01
== err :  1.692E-15 = rco :  4.193E-02 = res :  1.700E-16 ==
solution 14 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.01150302946056E-08  -2.61617893541228E-09
x2 : -7.27442303883107E-09   5.09406845182344E-09
x3 : -7.38574185120420E-09   4.64437788320316E-11
== err :  3.253E-08 = rco :  7.158E-01 = res :  3.047E-16 ==
```