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 D \sqrt {D} is the solution to Pell’s equation for D 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 ∈ Z (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 ) 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 ) v_n(t) has a repeated root. Restricting our analysis to Q [ t ] \mathbb {Q}[t] , we give an upper bound on the number of “new” factors of v n ( t ) v_n(t) of degree at most N N . Furthermore, we show that all “new” linear rational factors of v n ( t ) v_n(t) can be found when n ≤ 3 n \leq 3 , and all “new” quadratic rational factors when n ≤ 6 n \leq 6 .