Properties of solutions to Pell's equation over the polynomial ring

TL;DR

This paper explores the polynomial analogue of Pell's equation over complex and rational polynomials, analyzing the roots and factors of solutions, and establishing bounds and classifications for factors of solutions.

Contribution

It introduces the polynomial version of Pell's equation, investigates the roots of solutions, and provides bounds and classifications for factors over complex and rational polynomials.

Findings

Finitely many $n$ with repeated roots of $v_n(t)$ over complex polynomials.

Upper bounds on new factors of $v_n(t)$ of degree at most $N$ over $ ext{Q}[t]$.

Complete characterization of linear and quadratic factors for small $n$.

Abstract

In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers $\sqrt{D}$ is the solution to Pell's equation for $D$. It is well-known that, once an integer solution to Pell's equation exists, we can use it to generate all other solutions $(u_n,v_n)_{n\in\Zee}$. Our object of interest is the polynomial version of Pell's equation, where the integers are replaced by polynomials with complex coefficients. We then investigate the factors of $v_n(t)$. In particular, we show that over the complex polynomials, there are only finitely many values of $n$ for which $v_n(t)$ has a repeated root. Restricting our analysis to $\Qee[t]$, we give an upper bound on the number of "new" factors of $v_n(t)$ of degree at most $N$. Furthermore, we show that all "new" linear rational factors of $v_n(t)$ can be found when $n\leq 3$, and all "new" quadratic rational factors when $n\leq 6$.