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

2019/11/05 by Murru, Nadir
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1911.01837

Abstract

The polynomial Pell equation is P² - D Q² = 1 where D is a given integer polynomial and the solutions P, Q must be integer polynomials. A classical paper of Nathanson \citeNat solved it when D(x) = x² + d. We show that the Rédei 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) + d, for any f ∈ \mathbb Z[X] and d ∈ \mathbb Z, generalizing the result of Nathanson. We are also able to find solutions of some generalized polynomial Pell equations introducing an extension of Rédei polynomials to higher degrees.

Related

• Parametric solutions of Pell equations
• On some general solutions of the simple Pell equation
• Some new results on negative polynomial Pell's equation
• Properties of solutions to Pell's equation over the polynomial ring
• Quartic Integral Polynomial Pell Equations