A note on the use of R\'edei polynomials for solving the polynomial Pell equation and its generalization to higher degrees

TL;DR

This paper introduces a novel application of Rédéi polynomials to efficiently solve the polynomial Pell equation for specific classes of polynomials, extending solutions to more general cases and higher degrees.

Contribution

It demonstrates how Rédéi polynomials can be used to find all solutions for D(x) = f^2(x) + d and extends this approach to higher-degree Pell equations.

Findings

Solved polynomial Pell equations for D(x) = f^2(x) + d

Extended Rédéi polynomial methods to higher degrees

Provided explicit solutions for generalized polynomial Pell equations

Abstract

The polynomial Pell equation is \[P^2 - D Q^2 = 1\] where $D$ is a given integer polynomial and the solutions $P, Q$ must be integer polynomials. A classical paper of Nathanson \cite{Nat} solved it when $D(x) = x^2 + d$. We show that the R\'edei polynomials can be used in a very simple and direct way for providing these solutions. Moreover, this approach allows to find all the integer polynomial solutions when $D(x) = f^2(x) + d$, for any $f \in \mathbb Z[X]$ and $d \in \mathbb Z$, generalizing the result of Nathanson. We are also able to find solutions of some generalized polynomial Pell equations introducing an extension of R\'edei polynomials to higher degrees.