<< Solution of the n dimensional cyclic polynomial system >> 1. The n dimensional cyclic polynomial f(x) = (f1(x),f2(x),...,fn(x)) in n dimensional compex variable vector x = (x1,x2,...,xn) is described as f1(x) = x1 + x2 + ... + x2, f2(x) = x1 x2 + x2 x3 + ... + x(n-1) xn + xn x1, ... f(n-1)(x) = x1 x2 ... x(n-1) + ... + xn x1 ... x(n-2) fn(x) = x1 x2 ... xn - 1. 2. This directory contains readme.cyc --- this file, cyclic8, cyclic9, cyclic10, cyclic11, cyclic12 and cyclic13 --- lists of approximate nonsingular solutions of cyclic polynomials f(x) with dimensions n = 8, 9, 10, 11, 12 and 13, respectively. cyclic9.singular --- a list of approximate isoldated singular solutions of cyclic 9 polynomial f(x) with multiplicity 4. cyclic12.singular5 --- a list of approximate isolated singular solutions of cyclic 12 polynomial f(x) with multiplicity 5. cyclic12.singular10 --- a list of approximate isolated singular solutions of cyclic 12 polynomial f(x) with multiplicity 10. cyclic13.singular --- a list of approximate isoldated singular solutions of cyclic 13 polynomial f(x) with multiplicity 4. 3. For every solution x = (x1,x2,...,xn), we assigned a key number key(x) = \sum_{p=1}^n ( \gamma1^{p-1} real(x_p) + gamma2^{p-1} imag(x_p) ). gamma1 = 0.58 and gamma2 = 0.60 are used in the computation. 4. In each orbit consisting of 2*n solutions, we picked up a solution x with the smallest key number value key(x) among them. We can expand the solution to all other solutions in the orbit by reversing the order of the subscript indices 1,2,...,n of x1,x2,...,xn and applying cyclic permutaions to the subscript indices. 5. Then the solutions selected from the orbits are listed in ascending order of their key values. 6. Each line of the cyclic-$n$ solution file consists of 2*n + 2 real numbers, real(x1), imag(x1), ... , real(xn), imag(xn), norm of f(x) and key(x). Here norm of f(x) denotes the infinity norm of the complex value of the the polynomial f(x) at an approximate solution x, i.e., max { |f1(x)|, |f2(x)|, ... , |fn(x)| } 7. The number of isolated solutions computed. cyclic8 # of nonsingular solutions = 72*16 = 1152 # of isolated singular solutions = 0 the mixed volume = 2560 cyclic9 # of nonsingular solutions = 333*18 = 5994 # of isolated singular solutions with multiplicity 4 = 9*18 = 162 the mixed volume = 11016 cyclic10 # of nonsingular solutions = 1747*20 = 34940 # of singular isoltated solutions = 0 the mixed volume = 35940 cyclic11 # of nonsingular solutions = 8398*22 = 184756 = the mixed volume # of isolated singular solutions = 0 the mixed volume = 184756 Hence all solutions were obtained. cyclic12 # of nonsingular solutions = 15312*24 = 367488 # of isolated singular solutions with multiplicity 5 = 24*24 = 576 # of isolated singular solutions with multiplicity 10 = 2*24 = 48 the mixed volume = 500352 cyclic13 # of nonsinglar solutions = 103694*26 = 2696044 # of isolated singular with multiplicity 4 = 78*26 = 2028 the mixed volume = 2704156 = 103694*26 + 78*26*4 Hence all solutions were obtained.