6 z0 + z1 + z2 + z3 + z4 + z5 - 1; z0*z1 + z1*z2 + z2*z3 + z3*z4 + z4*z5 + z5*z0 - 1; z0*z1*z2 + z1*z2*z3 + z2*z3*z4 + z3*z4*z5 + z4*z5*z0 + z5*z0*z1 - 1; z0*z1*z2*z3 + z1*z2*z3*z4 + z2*z3*z4*z5 + z3*z4*z5*z0 + z4*z5*z0*z1 + z5*z0*z1*z2 - 1; z0*z1*z2*z3*z4 + z1*z2*z3*z4*z5 + z2*z3*z4*z5*z0 + z3*z4*z5*z0*z1 + z4*z5*z0*z1*z2 + z5*z0*z1*z2*z3 - 1 ; z0*z1*z2*z3*z4*z5 - 1; TITLE : extended cyclic 6-roots problem, to exploit the symmetry ROOT COUNTS : total degree : 6! = 720 mixed volume : 156 REFERENCES : This is the Arnborg's system or Davenport's problem, extended with the constant term to exploit symmetry. For the original problem : G\"oran Bj\"orck and Ralf Fr\"oberg: `A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots', J. Symbolic Computation (1991) 12, pp 329--336. THE SOLUTIONS : 13 6 =========================================================== solution 1 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : -1.09145911498337E-01 -2.82051661977533E-01 z1 : -1.19329529302722E+00 -3.08367868303911E+00 z2 : 6.84847142039119E-01 1.11902672009036E+00 z3 : 6.68044546894126E-01 8.74453630967107E-01 z4 : 5.51668586068374E-01 7.22120404126932E-01 z5 : 3.97880929523935E-01 6.50129589832241E-01 == err : 3.150E-15 = rco : 2.299E-02 = res : 1.404E-15 == solution 2 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : -3.53129181444054E-01 -4.55887812930893E-01 z1 : -1.06193432801704E+00 -1.37095132239199E+00 z2 : -2.50000000000000E-01 -9.68245836551854E-01 z3 : 1.34883344892200E-01 1.74133649482747E-01 z4 : 2.78018016456890E+00 3.58919715894384E+00 z5 : -2.50000000000000E-01 -9.68245836551854E-01 == err : 5.300E-15 = rco : 2.085E-02 = res : 1.256E-15 == solution 3 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : 3.21007829191040E-01 -2.54057382351715E+00 z1 : 3.21007829191040E-01 2.54057382351715E+00 z2 : 1.30039852922610E-01 9.91508767813914E-01 z3 : 4.89523178863507E-02 3.87426617400472E-01 z4 : 4.89523178863507E-02 -3.87426617400472E-01 z5 : 1.30039852922610E-01 -9.91508767813914E-01 == err : 4.347E-16 = rco : 6.306E-02 = res : 9.155E-16 == solution 4 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : 2.66839454551322E-01 -9.63740995026544E-01 z1 : 7.89153930342621E-01 6.14195469068922E-01 z2 : -4.44985611119495E-01 8.95537718857564E-01 z3 : 7.44236290740844E-01 -6.67916419579807E-01 z4 : -4.96471667324206E-01 -8.68052926695327E-01 z5 : 1.41227602808914E-01 9.89977153375192E-01 == err : 8.917E-16 = rco : 1.092E-01 = res : 7.109E-16 == solution 5 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : 7.44236290740845E-01 6.67916419579807E-01 z1 : -4.96471667324207E-01 8.68052926695327E-01 z2 : 1.41227602808914E-01 -9.89977153375192E-01 z3 : 2.66839454551322E-01 9.63740995026544E-01 z4 : 7.89153930342621E-01 -6.14195469068922E-01 z5 : -4.44985611119495E-01 -8.95537718857564E-01 == err : 6.069E-16 = rco : 1.043E-01 = res : 8.951E-16 == solution 6 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : -1.30493744993987E+00 1.10325388562335E+00 z1 : 2.47735189059043E+00 -1.20477380171330E+00 z2 : 8.86709471424394E-01 -4.62327063112546E-01 z3 : 3.26450487727244E-01 -1.58757823894198E-01 z4 : -4.46891660260174E-01 3.77822676985343E-01 z5 : -9.38682739542032E-01 3.44782126111355E-01 == err : 6.011E-15 = rco : 6.878E-02 = res : 7.216E-16 == solution 7 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : -3.81966011250105E-01 6.04048609987806E-92 z1 : 6.85410196624969E+00 8.43750439348047E-91 z2 : 1.45898033750315E-01 -4.45845402610047E-92 z3 : -2.61803398874990E+00 -5.44602556306467E-91 z4 : -3.81966011250105E-01 6.61577049034264E-92 z5 : -2.61803398874990E+00 -3.83522926976385E-91 == err : 4.788E-15 = rco : 1.766E-03 = res : 8.882E-16 == solution 8 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : 5.00000000000000E-01 -8.66025403784439E-01 z1 : -9.57427107756338E-01 -2.88675134594813E-01 z2 : -9.57427107756338E-01 2.88675134594813E-01 z3 : 5.00000000000000E-01 8.66025403784439E-01 z4 : 9.57427107756338E-01 2.88675134594813E-01 z5 : 9.57427107756338E-01 -2.88675134594813E-01 == err : 4.711E-16 = rco : 1.213E-01 = res : 7.109E-16 == solution 9 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : 3.26450487727244E-01 1.58757823894198E-01 z1 : 8.86709471424394E-01 4.62327063112546E-01 z2 : 2.47735189059043E+00 1.20477380171330E+00 z3 : -1.30493744993987E+00 -1.10325388562335E+00 z4 : -9.38682739542032E-01 -3.44782126111355E-01 z5 : -4.46891660260174E-01 -3.77822676985343E-01 == err : 5.690E-15 = rco : 5.844E-02 = res : 6.713E-16 == solution 10 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : 1.00000000000000E+00 3.41829996175611E-86 z1 : 1.00000000000000E+00 -3.01614702507892E-86 z2 : 1.00000000000000E+00 3.61937643009471E-86 z3 : 1.00000000000000E+00 -2.61399408840173E-86 z4 : -3.81966011250105E-01 -3.21722349341752E-86 z5 : -2.61803398874990E+00 4.12206760094120E-86 == err : 4.720E-15 = rco : 3.715E-02 = res : 2.220E-16 == solution 11 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : 5.51668586068374E-01 -7.22120404126932E-01 z1 : 3.97880929523935E-01 -6.50129589832240E-01 z2 : -1.09145911498336E-01 2.82051661977533E-01 z3 : -1.19329529302722E+00 3.08367868303911E+00 z4 : 6.84847142039118E-01 -1.11902672009036E+00 z5 : 6.68044546894127E-01 -8.74453630967107E-01 == err : 3.182E-15 = rco : 2.489E-02 = res : 1.201E-15 == solution 12 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : -6.04864265798915E-01 -7.96328587933109E-01 z1 : 1.15709953428924E-01 9.93283044593774E-01 z2 : 9.89154312369991E-01 -1.46880040576826E-01 z3 : -6.04864265798915E-01 7.96328587933109E-01 z4 : 1.15709953428924E-01 -9.93283044593774E-01 z5 : 9.89154312369991E-01 1.46880040576826E-01 == err : 6.660E-16 = rco : 1.458E-01 = res : 8.006E-16 == solution 13 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 12 the solution for t : z0 : 1.34883344892200E-01 -1.74133649482747E-01 z1 : -2.50000000000000E-01 9.68245836551854E-01 z2 : -1.06193432801704E+00 1.37095132239199E+00 z3 : -3.53129181444054E-01 4.55887812930893E-01 z4 : -2.50000000000000E-01 9.68245836551854E-01 z5 : 2.78018016456890E+00 -3.58919715894384E+00 == err : 5.011E-15 = rco : 1.055E-02 = res : 9.155E-16 ==