Expressing an NP-Complete Problem as the Solvability of a Polynomial Equation

TL;DR

This paper presents a polynomial method to encode the NP-Complete Hamiltonian Cycle Problem as a polynomial equation solvability problem with a fixed number of variables, aiming to shed light on the P vs NP question.

Contribution

It introduces four new theorems for periodic functions and uses trigonometric substitution to express HCP as a polynomial equation with bounded variables.

Findings

HCP can be represented as a polynomial equation solvability problem.

The approach uses a fixed number of variables within bounded real numbers.

Future work is needed to prove whether P=NP.

Abstract

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.