5 a + b + c + d + e - 1; a*b + b*c + c*d + d*e + e*a - 0.30901699437495 - 0.95105651629515*i; a*b*c + b*c*d + c*d*e + d*e*a + e*a*b + 0.80901699437495 + 0.58778525229247*i; a*b*c*d + b*c*d*e + c*d*e*a + d*e*a*b + e*a*b*c - 0.30901699437495 - 0.95105651629515*i; a*b*c*d*e - 1; TITLE : extended cyclic 5-roots problem, to exploit the symmetry ROOT COUNTS : total degree : 120 5-homogeneous Bezout number : 120 with partition : {a }{b }{c }{d }{e } generalized Bezout number : 106 based on the set structure : {a b c d e } {a c e }{b d e } {a d }{b d e }{c e } {a e }{b e }{c e }{d e } {a }{b }{c }{d }{e } mixed volume : 70 = 14*5 = 7*10 REFERENCES : Jan Verschelde and Karin Gatermann: `Symmetric Newton Polytopes for Solving Sparse Polynomial Systems', Adv. Appl. Math., 16(1): 95-127, 1995. 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. NOTE : EXPLOITATION OF SYMMETRY AND CHOICE OF CONSTANTS : By extending the equations of the original system with a random complex constant, we add a fixed point to the symmetry. The two generating elements of the symmetry group are b c d e a e d c b a which are respectively the cyclic permutation and the reading backwards operation. The fifth root of unity w : w = 0.30901699437495 + 0.95105651629515i w^2 = -0.80901699437495 + 0.58778525229247i w^3 = -0.80901699437495 - 0.58778525229247i w^4 = 0.30901699437495 - 0.95105651629515i w^5 = 1.0 Note however that : 1 + w + w^2 + w^3 + w^4 = -1.110223024625157e-16 + 3.330669073875470e-16i. When (w,w^2,w^3,w^4,1) is the vector of the right hand sides, then (w,w,w,w,w) is a solution of all subsystems of a certain type. Therefore, (1,w,w^3,w,1) seems to be a better choice. THE GENERATING SOLUTIONS : 7 5 =========================================================== solution 1 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 10 the solution for t : a : 3.51566035447901E-02 -8.27261453800970E-01 b : 1.01623041236047E+00 -1.24241632490855E+00 c : -9.20411830092637E-02 2.16579860363359E+00 d : -2.10069650118794E-01 4.96678926154659E-01 e : 2.50723817222800E-01 -5.92799751078729E-01 == err : 3.507E-15 = rco : 7.423E-02 = res : 4.003E-16 == solution 2 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 10 the solution for t : a : 1.16341787215603E+00 3.24099681249188E-01 b : 1.54792244147177E+00 -1.33407662151979E-01 c : -1.84748735988833E+00 1.59225658169080E-01 d : -4.44386084044522E-01 -1.23795062494183E-01 e : 5.80533130305057E-01 -2.26122614772107E-01 == err : 4.843E-15 = rco : 7.975E-02 = res : 4.475E-16 == solution 3 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 10 the solution for t : a : 4.10014793736554E-01 7.50751897540148E-01 b : 1.60100166102993E-01 7.35198384573590E-01 c : -4.08340693440269E-01 -2.85192410929864E+00 d : -6.11528218468393E-02 -2.80820794433095E-01 e : 8.99378555447562E-01 1.64679462161799E+00 == err : 5.729E-15 = rco : 3.872E-02 = res : 8.006E-16 == solution 4 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 10 the solution for t : a : -8.09016994374948E-01 5.87785252292474E-01 b : -5.37688913226986E-01 8.43143304897086E-01 c : 3.11581670732691E+00 -1.47039386691358E+00 d : 1.98921146442162E-01 -2.11361625033512E-01 e : -9.68031946167142E-01 2.50826934757527E-01 == err : 5.436E-15 = rco : 4.732E-02 = res : 9.930E-16 == solution 5 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 10 the solution for t : a : -8.02615012782033E-01 -8.49949264705670E-01 b : -3.65901573839422E-01 -3.87480633537469E-01 c : 3.00459112225004E+00 1.75469990030919E+00 d : 3.11577150774741E-01 1.52965719796006E-01 e : -1.14765168640332E+00 -6.70235721862056E-01 == err : 3.466E-15 = rco : 4.368E-02 = res : 8.473E-16 == solution 6 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 10 the solution for t : a : 8.91931221365206E-01 -4.52171091904345E-01 b : -4.86191902445739E-01 -5.68082696200543E-01 c : 6.97615506804102E-01 1.14101698421537E+00 d : 7.05662168651377E-01 -7.08548448402957E-01 e : -8.09016994374947E-01 5.87785252292474E-01 == err : 2.209E-15 = rco : 1.096E-01 = res : 7.109E-16 == solution 7 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 10 the solution for t : a : -8.09016994374947E-01 5.87785252292475E-01 b : -3.64715822016015E-01 -9.31118880257073E-01 c : 2.05677941924575E-02 -6.95027127677804E-01 d : 1.38032173080495E+00 4.03763838735321E-01 e : 7.72843291393555E-01 6.34596916907081E-01 == err : 9.805E-16 = rco : 2.227E-01 = res : 5.579E-16 ==