5 x1 + x2 + x3 + x4 +(-6.92520438442030E-01 - 7.21398255015950E-01*i)*zz1; x1*x2 + x1*x4 + x2*x3 + x3*x4 +(-9.99481186966167E-01 - 3.22080254083045E-02*i)*zz1; x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 +(-6.95281866926750E-01 + 7.18737174162331E-01*i)*zz1; x1*x2*x3*x4 - 1 +(-3.91346845929466E-01 - 9.20243253809045E-01*i)*zz1; +(-7.73739801949262E-01 + 6.33503527124764E-01*i)*x1 +( 4.78231258247626E-01 + 8.78233945845236E-01*i)*x2 +( 9.52550263480145E-01 + 3.04381332449785E-01*i)*x3 +(-8.63344595438512E-01 - 5.04614813027830E-01*i)*x4 +( 2.12360886105879E-01 + 9.77191308829712E-01*i)*zz1 +(-5.74631821960278E-01 + 8.18412041205780E-01*i); TITLE : embedded cyclic 4-roots system DESCRIPTION : There is curve of degree four which factors in two quadrics: 2 2 : 1 4 2 : 2 3 Solutions 1 and 4 lie on the first quadric, while solutions 2 and 3 are witness points for the second irreducible quadric. THE SOLUTIONS with zz = 0 : 4 5 =========================================================== solution 1 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 7.00257227754609E-01 6.59718482088328E-01 x2 : 7.56553377908548E-01 -7.12755579393360E-01 x3 : -7.00257227754609E-01 -6.59718482088329E-01 x4 : -7.56553377908548E-01 7.12755579393361E-01 zz1 : 4.69347030751973E-17 -3.12160612016110E-17 == err : 5.584E-16 = rco : 1.000E+00 = res : 1.228E-15 == solution 2 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -7.34981313260901E-01 1.13714827345112E+00 x2 : 4.00905372402036E-01 6.20272711453857E-01 x3 : 7.34981313260901E-01 -1.13714827345112E+00 x4 : -4.00905372402036E-01 -6.20272711453857E-01 zz1 : 1.66790175346363E-17 1.84355822937642E-16 == err : 8.021E-16 = rco : 1.000E+00 = res : 9.249E-16 == solution 3 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 3.26569430380819E-01 -7.40925970097483E-01 x2 : -4.98108638024790E-01 -1.13011688023611E+00 x3 : -3.26569430380819E-01 7.40925970097483E-01 x4 : 4.98108638024790E-01 1.13011688023611E+00 zz1 : -5.67369757529988E-17 -7.11493127463934E-17 == err : 6.866E-16 = rco : 1.000E+00 = res : 1.334E-15 == solution 4 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -1.10866911063469E+00 -2.63496178734687E-01 x2 : -8.53756643531302E-01 2.02911410611111E-01 x3 : 1.10866911063469E+00 2.63496178734688E-01 x4 : 8.53756643531302E-01 -2.02911410611111E-01 zz1 : 2.19284197820674E-17 3.27639428170521E-17 == err : 1.418E-15 = rco : 1.000E+00 = res : 1.155E-15 == The output from the cascade: 4 + 1*x1 + 1*x2 + 1*x3 + 1*x4; + 1*x1*x2 + 1*x1*x4 + 1*x2*x3 + 1*x3*x4; + 1*x1*x2*x3 + 1*x1*x2*x4 + 1*x1*x3*x4 + 1*x2*x3*x4; + 1*x1*x2*x3*x4 -1; THE SOLUTIONS with zz = 0 : 16 4 =========================================================== solution 1 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 1.81958681465767E-07 9.99999963358102E-01 x2 : 2.06955081351562E-07 -9.99999638288985E-01 x3 : -2.06955081367419E-07 -1.00000036171102E+00 x4 : -1.81958681449911E-07 1.00000003664190E+00 == err : 1.575E-06 = rco : 4.024E-08 = res : 9.359E-13 == solution 2 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 5.95195376712383E-08 -1.00000003143094E+00 x2 : 5.54439299557947E-08 9.99999931626512E-01 x3 : -5.54439301651108E-08 1.00000006837349E+00 x4 : -5.95195374619221E-08 -9.99999968569059E-01 == err : 4.215E-07 = rco : 1.730E-09 = res : 2.958E-14 == solution 3 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 4.29451274680323E-07 -9.99999891027259E-01 x2 : -4.29451274683514E-07 -1.00000010897274E+00 x3 : 1.28807387127659E-07 9.99999983101824E-01 x4 : -1.28807387124468E-07 1.00000001689818E+00 == err : 1.368E-06 = rco : 8.754E-08 = res : 1.193E-12 == solution 4 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 9.99999942698803E-01 4.42032002835034E-09 x2 : -1.00000000346663E+00 -7.23566523888739E-09 x3 : -9.99999996533371E-01 7.23566523708553E-09 x4 : 1.00000005730120E+00 -4.42032002654848E-09 == err : 1.437E-07 = rco : 6.298E-09 = res : 1.541E-14 == solution 5 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -9.99999963760363E-01 -2.18729235295494E-08 x2 : 1.00000002762342E+00 -3.70499791386984E-08 x3 : 9.99999972376578E-01 3.70499791505619E-08 x4 : -1.00000003623964E+00 2.18729235176859E-08 == err : 2.565E-07 = rco : 6.275E-10 = res : 1.052E-14 == solution 6 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 9.99999935849163E-01 4.13291723465630E-07 x2 : 1.00000006415084E+00 -4.13291723460290E-07 x3 : -1.00000005636318E+00 4.18740051199268E-08 x4 : -9.99999943636817E-01 -4.18740051252672E-08 == err : 1.151E-06 = rco : 6.532E-08 = res : 9.582E-13 == solution 7 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -9.99999752469133E-01 1.40234370380481E-07 x2 : -1.00000024753087E+00 -1.40234370427357E-07 x3 : 9.99999976624125E-01 3.69314594203136E-07 x4 : 1.00000002337587E+00 -3.69314594156260E-07 == err : 1.561E-06 = rco : 9.110E-08 = res : 1.113E-12 == solution 8 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -2.63583494577677E-08 -1.00000005314488E+00 x2 : 2.63583494742864E-08 -9.99999946855119E-01 x3 : 1.63357441348949E-08 1.00000022686924E+00 x4 : -1.63357441514136E-08 9.99999773130758E-01 == err : 6.459E-07 = rco : 2.622E-08 = res : 2.053E-13 == solution 9 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -1.00000039180282E+00 3.12632267903306E-07 x2 : 9.99999779489635E-01 1.63388237580416E-07 x3 : 1.00000022051036E+00 -1.63388237581532E-07 x4 : -9.99999608197178E-01 -3.12632267902190E-07 == err : 2.177E-06 = rco : 2.048E-08 = res : 8.665E-13 == solution 10 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 1.00000037539245E+00 2.04006966307699E-08 x2 : -1.00000008954206E+00 6.00742309776093E-08 x3 : -9.99999910457940E-01 -6.00742309864073E-08 x4 : 9.99999624607551E-01 -2.04006966219719E-08 == err : 1.091E-06 = rco : 6.114E-08 = res : 7.252E-13 == solution 11 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 9.99999900273548E-01 -1.52105686962579E-07 x2 : 1.00000009972645E+00 1.52105687078804E-07 x3 : -9.99999746180546E-01 1.39250108656531E-07 x4 : -1.00000025381945E+00 -1.39250108772756E-07 == err : 1.289E-06 = rco : 1.224E-08 = res : 3.579E-13 == solution 12 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -3.15669017932952E-07 9.99999913453304E-01 x2 : 3.15669018002861E-07 1.00000008654670E+00 x3 : 5.82030721500688E-08 -9.99999902390206E-01 x4 : -5.82030722199782E-08 -1.00000009760979E+00 == err : 1.115E-06 = rco : 3.557E-08 = res : 5.071E-13 == solution 13 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -4.20843050044115E-08 -9.99999973002183E-01 x2 : -5.38481188106748E-08 1.00000005796933E+00 x3 : 5.38481185924506E-08 9.99999942030668E-01 x4 : 4.20843052226357E-08 -1.00000002699782E+00 == err : 3.528E-07 = rco : 1.685E-09 = res : 2.185E-14 == solution 14 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -1.00000023895449E+00 -2.20039970117969E-07 x2 : -9.99999761045508E-01 2.20039970070289E-07 x3 : 9.99999994974262E-01 -1.06413635321393E-07 x4 : 1.00000000502574E+00 1.06413635369074E-07 == err : 1.141E-06 = rco : 7.692E-08 = res : 5.633E-13 == solution 15 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : -1.02470127469085E-07 1.00000022675583E+00 x2 : -1.39573224956260E-07 -1.00000013459011E+00 x3 : 1.39573224956997E-07 -9.99999865409888E-01 x4 : 1.02470127468349E-07 9.99999773244165E-01 == err : 1.207E-06 = rco : 4.772E-08 = res : 4.572E-13 == solution 16 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x1 : 4.64517327875020E-08 1.00000001546172E+00 x2 : -4.64517325775751E-08 9.99999984538277E-01 x3 : -1.27386017190946E-08 -1.00000003007889E+00 x4 : 1.27386015091678E-08 -9.99999969921114E-01 == err : 1.920E-07 = rco : 3.663E-09 = res : 1.123E-14 ==