Computing All Nonsingular Solutions of Cyclic-n Polynomial Using Polyhedral Homotopy Continuation Methods

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.