Polynomial Parametrization for ${\text{SL}}_2$ over Quadratic Number Rings

Abstract

If $R$ is the ring of integers of a number field, then there exists a polynomial parametrization of the set ${\text{SL}}_2(R)$, i.e., an element $A\in {\text{SL}}_2(\mathbb{Z}[x_1,\dots ,x_n])$ such that every element of ${\text{SL}}_2(R)$ is obtained by specializing $A$ via some homomorphism $\mathbb{Z}[x_1,\dots ,x_n]\to R$.