All isolated solutions of the cyclic-n polynomial equations are not
known for larger dimensions than 8 except the dimensions 10 and 11.
We exploit two types of symmetric structures in the cyclic-n polynomial
to compute all isolated nonsingular solutions of the equations efficiently
by the polyhedral homotopy continuation method and to verify the
correctness of the generated approximate solutions. Numerical results on
the cyclic-8 to the cyclic-12 polynomial equations, including their
solution information, are given.