PROPERTIES OF SOLUTIONS TO PELL'S EQUATION OVER THE POLYNOMIAL RING

In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers 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 (un, vn)n is an element of 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 vn(t). In particular, we show that over the complex polynomials, there are only finitely many values of n for which vn(t) has a repeated root. Restricting our analysis to Q[t], we give an upper bound on the number of "new" factors of vn(t) of degree at most N. Furthermore, we show that all "new" linear rational factors of vn(t) can be found when n < 3, and all "new" quadratic rational factors when n < 6.