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 $A \in \text{SL}_2(\mathbb{Z}[x_1,\ldots,x_n])$ such that every element of $\text{SL}_2(R)$ is obtained by specializing $A$ via some homomorphism $\mathbb{Z}[x_1,\ldots,x_n]\to R$.