The ring of real-valued multivariate polynomials: an analyst's perspective

In this survey we determine an explicit set of generators of the maximal ideals in the ring $\mathbb{R}[x_1,\dots ,x_n]$ of polynomials in $n$ variables with real coefficients and give an easy analytic proof of the Bass-Vasershtein theorem on the Bass stable rank of $\mathbb{R}[x_1,\dots ,x_n]$. The ingredients of the proof stem from different publications by Coquand, Lombardi, Estes and Ohm. We conclude with a calculation of the topological stable rank of $\mathbb R[x_1,\dots,x_n]$, which seems to be unknown so far.