Polynomial Pell’s equation

Webb, William; Yokota, Hisashi

doi:10.1090/S0002-9939-02-06934-4

Polynomial Pell’s equation
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by William A. Webb and Hisashi Yokota PDF

Proc. Amer. Math. Soc. 131 (2003), 993-1006 Request permission

Abstract:

Consider the polynomial Pell’s equation $X^2 -DY^2 = 1$, where $D = A^2 + 2C$ is a monic polynomial in ${\mathcal Z}[x]$ and $\deg {C} < \deg {A}$. Then for $A, C \in {\mathcal Q}[x]$, $\deg {C} < 2$, and $B = A/C \in {\mathcal Q}[x]$, a necessary and sufficient condition for the polynomial Pell’s equation to have a nontrivial solution in ${\mathcal Z}[x]$ is obtained.

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