Homogeneous Diophantine Equation of Degree Two in NP-Complete

EasyChair Preprint 9354, version 1

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 prove that this decision problem is actually in $\textit{NP--complete}$ under the constraint that the each variable is required to be evaluated in $\{0, 1\}$.

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

@booklet{EasyChair:9354,
  author    = {Frank Vega},
  title     = {Homogeneous Diophantine Equation of Degree Two in NP-Complete},
  howpublished = {EasyChair Preprint 9354},
  year      = {EasyChair, 2022}}