On the Diophantine equation $f(x)f(y)=f(z)^n$ involving Laurent polynomials

TL;DR

This paper explores rational solutions to the Diophantine equation involving Laurent polynomials, using elliptic curve theory to analyze cases where the exponent n is 1 or 2.

Contribution

It introduces a novel approach applying elliptic curve theory to find rational solutions for specific Laurent polynomial equations.

Findings

Rational solutions exist for certain Laurent polynomials when n=1 or 2.

Elliptic curve methods effectively analyze the solution space.

New parametric solutions are constructed for the equation.

Abstract

By the theory of elliptic curves, we investigate the nontrivial rational parametric solutions of the Diophantine equation $f(x)f(y)=f(z)^n$, where $n=1,2$ and $f(X)$ are some simple Laurent polynomials.