Integer solutions of the generalized polynomial Pell equations and their finiteness: The quadratic case

Abstract Determining the polynomials $D \in {\mathbb Z}[x]$ such that the polynomial Pell equation ${P^2-DQ^2=1}$ has nontrivial solutions $P,Q$ in ${\mathbb Q}[x]$ (and in ${\mathbb Z}[x]$ ) is an open question. In this article, we consider the generalized polynomial Pell equation $P^2-DQ^2=n$ , where $D \in {\mathbb Z}[x]$ is a monic quadratic polynomial and n is a nonzero integer. For $n=1$ , such an equation always has nontrivial solutions in ${\mathbb Q}[x]$ , but for a non-square integer n , the generalized polynomial Pell equation $P^2-DQ^2=n$ may not always have a solution in ${\mathbb Q}[x]$ . Depending on n , we determine the polynomials $D=x^2+cx+d$ , for which the equation $P^2-DQ^2=n$ has nontrivial solutions in ${\mathbb Q}[x]$ and in ${\mathbb Z}[x]$ . Taking $n=-1$ , this allows us to solve the negative polynomial Pell equation completely for any such D . An interesting feature is that there are certain polynomials D for which the generalized polynomial Pell equation has nontrivial solutions in ${\mathbb Z}[x]$ , but only finitely many, whereas the solutions in ${\mathbb Q}[x]$ are infinitely many. Finally, we determine the monic quadratic polynomials D for which the solutions of $P^2-DQ^2=n$ in ${\mathbb Z}[x]$ exhibit this finiteness phenomenon.

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