Homogeneous Diophantine Equation of Degree Two in NP-Complete

EasyChair Preprint no. 9354, version 2

3 pagesDate: December 3, 2022

Abstract

In mathematics, a Diophantine equation is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones. A homogeneous Diophantine equation is a Diophantine equation that is defined by a homogeneous polynomial. Solving a homogeneous Diophantine equation is generally a very difficult problem. However, homogeneous Diophantine equations of degree two are considered easier to solve. Certainly, using the Hasse principle we may able to decide whether a homogeneous Diophantine equation of degree two has an integer solution: we are able to reject an instance when there is no solution reducing the equation modulo p. We prove that this decision problem is actually in NP-complete under the constraints that all solutions contain only positive integers which are actually a residue of modulo a single positive integer. This problem remains in NP-complete even when all the coefficients are non-negative.

Keyphrases: Boolean formula, completeness, complexity classes, polynomial time