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 ==