The stable rank of $\mathbb{Z}[x]$ is $3$

Abstract

Grunewald, Mennicke and Vaserstein proved that the Bass stable rank of $\mathbb{Z}[x]$, the ring of the univariate polynomials over $\mathbb{Z}$, is $3$. This note addresses minor errors found in their proof. Using their method, we show in addition that the unimodular row $(3, x + 1, x^2 + 16)$ is not stable.