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

## TL;DR

This paper investigates rational solutions to specific Diophantine equations involving Laurent polynomials using elliptic curve theory, providing insights into their parametric solutions.

## Contribution

It applies elliptic curve theory to analyze rational solutions of equations involving Laurent polynomials, a novel approach for these types of Diophantine equations.

## Key findings

- Identification of conditions for rational solutions
- Explicit parametric solutions for certain Laurent polynomials
- Extension of elliptic curve methods to Laurent polynomial equations

## Abstract

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