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

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 $(u_n,v_n)_{n\in \mathbb {Z}}$. 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 $\mathbb {Q}[t]$, 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 $n \leq 6$.