Notes on Generalised Integral Polynomial Pell Equations

Given a nonzero integer n , Gupta and Saha [‘Integer solutions of the generalised polynomial Pell equations and their finiteness: the quadratic case’, Canad. Math. Bull. , to appear] classified all polynomials x²+ax+b∈ Z[x] for which the polynomial Pell equation P²-(x²+ax+b)Q²=n has solutions P,Q∈ Z[x] with Q≠ 0 . We generalise their work to the equation P²-(f²+af+b)Q²=nR , where f is a fixed polynomial in Z[x] . As an application of our results, we study the equation P²-D(f)Q²=n , where D is a monic, quartic and non square-free polynomial in Z[x] . This extends Theorem 1.4 of Scherr and Thompson [‘Quartic integral polynomial Pell equations’, J. Number Theory 259 (2024), 38–56].