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 ==