On the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ involving Laurent polynomials

TL;DR

This paper investigates rational solutions to specific Diophantine equations involving Laurent polynomials and elliptic curves, providing new parametric solutions and insights into their structure.

Contribution

It introduces a method to find rational solutions to equations involving Laurent polynomials using elliptic curve theory, expanding understanding of these Diophantine equations.

Findings

Derived parametric solutions for certain Laurent polynomial equations

Established conditions for the existence of rational solutions

Connected solutions to elliptic curve properties

Abstract

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f$.