A note on the use of Rédei polynomials for solving the polynomial Pell equation and its generalization to higher degrees

Abstract

The polynomial Pell equation is P2DQ2=1P^2 - D Q^2 = 1 where DD is a given integer polynomial and the solutions P,QP, Q must be integer polynomials. A classical paper of Nathanson \cite{Nat} solved it when D(x)=x2+dD(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)=f2(x)+dD(x) = f^2(x) + d, for any fZ[X]f \in \mathbb Z[X] and dZd \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

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