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

TL;DR

This paper establishes a polynomial parametrization for the group SL_2(R) over the ring of integers R of a quadratic number field, enabling representation of all such elements through polynomial specialization.

Contribution

It provides the first known polynomial parametrization of SL_2(R) for quadratic number rings, extending previous results over simpler rings.

Findings

Existence of polynomial parametrization for SL_2(R) over quadratic rings

Construction of explicit polynomial A in SL_2(\u007fZ[x_1,\u2026,x_n])

Every element of SL_2(R) obtained by specialization of A

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 $\mathbb{Z}[x_1,\ldots,x_n]\to R$.