10 1 + y1 + y2 + y3 + y4 + y5 + y6 + y7 + y8 +( 3.95753693421220E-01 + 9.18356692219022E-01*i)*zz1 +( 7.69042058822239E-01 + 6.39198178785306E-01*i)*zz2; y1 + y1*y2 + y2*y3 + y3*y4 + y4*y5 + y5*y6 + y6*y7 + y7*y8 + y8 +(-1.96500593957898E-01 + 9.80503705538227E-01*i)*zz1 +(-7.88287439413538E-01 - 6.15307169520109E-01*i)*zz2; y1*y2 + y1*y2*y3 + y2*y3*y4 + y3*y4*y5 + y4*y5*y6 + y5*y6*y7 + y6*y7*y8 + y7*y8 + y8*y1 +( 9.04235091530080E-01 + 4.27035009391020E-01*i)*zz1 +( 1.85960951032686E-01 + 9.82557135586028E-01*i)*zz2; y1*y2*y3 + y1*y2*y3*y4 + y2*y3*y4*y5 + y3*y4*y5*y6 + y4*y5*y6*y7 + y5*y6*y7*y8 + y6*y7*y8 + y7*y8*y1 + y8*y1*y2 +(-6.32984114878484E-01 + 7.74164782401978E-01*i)*zz1 +(-5.08499966798541E-01 + 8.61062009245492E-01*i)*zz2; y1*y2*y3*y4 + y1*y2*y3*y4*y5 + y2*y3*y4*y5*y6 + y3*y4*y5*y6*y7 + y4*y5*y6*y7*y8 + y5*y6*y7*y8 + y6*y7*y8*y1 + y7*y8*y1*y2 + y8*y1*y2*y3 +( 5.27850723422555E-01 - 8.49337161427831E-01*i)*zz1 +(-9.36971044275055E-01 + 3.49407015084289E-01*i)*zz2; y1*y2*y3*y4*y5 + y1*y2*y3*y4*y5*y6 + y2*y3*y4*y5*y6*y7 + y3*y4*y5*y6*y7*y8 + y4*y5*y6*y7*y8 + y5*y6*y7*y8*y1 + y6*y7*y8*y1*y2 + y7*y8*y1*y2*y3 + y8*y1*y2*y3*y4 +(-1.20412219992003E-01 + 9.92723978393087E-01*i)*zz1 +(-8.31469612302545E-01 - 5.55570233019602E-01*i)*zz2; y1*y2*y3*y4*y5*y6 + y1*y2*y3*y4*y5*y6*y7 + y2*y3*y4*y5*y6*y7*y8 + y3*y4*y5*y6*y7*y8 + y4*y5*y6*y7*y8*y1 + y5*y6*y7*y8*y1*y2 + y6*y7*y8*y1*y2*y3 + y7*y8*y1*y2*y3*y4 + y8*y1*y2*y3*y4*y5 +( 4.83023472027315E-01 + 8.75607403732219E-01*i)*zz1 +(-9.93564135520595E-01 + 1.13270952177564E-01*i)*zz2; y1*y2*y3*y4*y5*y6*y7 + y1*y2*y3*y4*y5*y6*y7*y8 + y2*y3*y4*y5*y6*y7*y8 + y3*y4*y5*y6*y7*y8*y1 + y4*y5*y6*y7*y8*y1*y2 + y5*y6*y7*y8*y1*y2*y3 + y6*y7*y8*y1*y2*y3*y4 + y7*y8*y1*y2*y3*y4*y5 + y8*y1*y2*y3*y4*y5*y6 +( 9.99677270796318E-01 + 2.54038235945990E-02*i)*zz1 +( 4.38616238538527E-01 + 8.98674465693954E-01*i)*zz2; +(-7.61388880614519E-01 + 6.48295436106541E-01*i)*y1 +(-3.35625778609476E-01 - 9.41995401651613E-01*i)*y2 +( 2.45862120530986E-01 + 9.69304811546918E-01*i)*y3 +(-8.38433773425308E-01 - 5.45003493181281E-01*i)*y4 +( 4.19129843945414E-01 - 9.07926304230852E-01*i)*y5 +( 7.24907838711588E-01 - 6.88845864743699E-01*i)*y6 +( 9.60350371467994E-01 - 2.78795918229240E-01*i)*y7 +(-8.25942904361688E-01 + 5.63753774918253E-01*i)*y8 +( 7.61699565853535E-01 + 6.47930375409685E-01*i)*zz1 +(-4.72326559073175E-01 + 8.81423633444267E-01*i)*zz2 +(-9.96271369139213E-01 + 8.62749038450850E-02*i); +(-6.46614924686512E-01 + 7.62816582916664E-01*i)*y1 +( 8.23992216900489E-01 + 5.66601117619280E-01*i)*y2 +( 9.80878687961889E-01 + 1.94620141563412E-01*i)*y3 +( 9.59867815648877E-01 - 2.80452806157211E-01*i)*y4 +( 9.49634953296390E-02 - 9.95480755491927E-01*i)*y5 +(-3.75586178489217E-01 - 9.26787474304582E-01*i)*y6 +(-7.11432195745217E-01 - 7.02754744457225E-01*i)*y7 +(-6.27625083901071E-01 + 7.78515737835898E-01*i)*y8 +(-6.70848268320170E-01 + 7.41594633807331E-01*i)*zz1 +( 2.09080650157769E-01 + 9.77898400514903E-01*i)*zz2 +(-6.73049386519805E-01 - 7.39597541440826E-01*i); z0**9*y1*y2*y3*y4*y5*y6*y7*y8 - 1; TITLE : embedding and slicing of the system redcyc9, zz1 and zz2 added ROOT COUNTS : The total degree equals 8! = 40320 mixed volume : 4044 REFERENCES : Andrew J. Sommese and Jan Verschelde : "Numerical Homotopies to compute generic Points on positive dimensional Algebraic Sets", Journal of Complexity 16(3):572-602, 2000. J.C. Faug\`ere: "A new efficient algorithm for computing Gr\"obner bases (F_4)", Journal of Pure and Applied Algebra vol. 139, Nos 1-3, pages 61-88, 1999. Proceedings of MEGA'98, 22-27 June 1998, Saint-Malo, France. TIMINGS on alice (machine at MSRI 450MHz pentium) : --------------------------------------------------------------------- | TIMING INFORMATION SUMMARY | --------------------------------------------------------------------- | root counts | start system | continuation | total time | --------------------------------------------------------------------- | 2h48m 4s320ms | 1h 4m 5s130ms | 2h15m 4s540ms | 6h 7m45s270ms | --------------------------------------------------------------------- THE SOLUTIONS with zz1 = 0 = zz2 : 2 10 =========================================================================== solution 1 : start residual : 6.718E-14 t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : y1 : -1.11794056987514E+00 -1.21236488275722E+00 y2 : 6.80961537645904E-01 -1.81121436452074E+00 y3 : -4.99999999999999E-01 -8.66025403784439E-01 y4 : -4.90968502186322E-01 1.57434737481173E+00 y5 : -1.90903842019720E+00 3.15877191658902E-01 y6 : -5.00000000000000E-01 8.66025403784437E-01 y7 : 1.60890907206146E+00 -3.61982492054513E-01 y8 : 1.22807688255130E+00 1.49533717286184E+00 zz1 : 1.76597593979736E-15 -1.62491090343090E-17 zz2 : -8.54015313659796E-16 1.32987936063159E-15 == err : 3.946E-14 = rco : 1.044E-03 = res : 6.718E-14 = complex regular == solution 2 : start residual : 3.072E-14 t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : y1 : -8.14978730956262E-01 1.15955961229130E+00 y2 : 1.68146071161365E+00 2.02254558564662E-01 y3 : -4.99999999999999E-01 8.66025403784439E-01 y4 : -5.96718715968567E-01 -1.28557209069778E+00 y5 : -1.01588794155503E+00 1.35506041244055E+00 y6 : -5.00000000000000E-01 -8.66025403784437E-01 y7 : 1.41169744692483E+00 1.26012478406477E-01 y8 : -6.65572770058619E-01 -1.55731497100521E+00 zz1 : -7.59753820340496E-16 1.59576261747253E-15 zz2 : -1.35770442035932E-15 -1.16265622428361E-15 == err : 2.793E-14 = rco : 3.533E-03 = res : 3.072E-14 = complex regular ==