  We demonstrate a polynomial approach to express the decision version of the
directed Hamiltonian Cycle Problem (HCP), which is NP-Complete, as the
Solvability of a Polynomial Equation with a constant number of variables,
within a bounded real space. We first introduce four new Theorems for a set of
periodic Functions with irrational periods, based on which we then use a
trigonometric substitution, to show how the HCP can be expressed as the
Solvability of a single polynomial Equation with a constant number of
variables. The feasible solution of each of these variables is bounded within
two real numbers. We point out what future work is necessary to prove that
P=NP.
