Polynomial Pell's equation–II

Abstract

The polynomial Pell's equation is X 2− DY 2=1, where D is a polynomial with integer coefficients and the solutions X, Y must be polynomials with integer coefficients. Let D= A 2+2 C be a polynomial in Z[x] , where deg C< deg A . Then for pB=pA/C∈ Z[x],p a prime, a necessary and sufficient condition for which the polynomial Pell's equation has a nontrivial solution is obtained. Furthermore, all solutions to the polynomial Pell's equation satisfying the above condition are determined.