An Algorithmic Proof of Suslin's Stability Theorem over Polynomial Rings

Abstract

Let $k$ be a field. Then Gaussian elimination over $k$ and the Euclidean division algorithm for the univariate polynomial ring $k[x]$ allow us to write any matrix in $S{L}_n(k)$ or $S{L}_n(k[x])$, $n\ge 2$, as a product of elementary matrices. Suslin's stability theorem states that the same is true for the multivariate polynomial ring $S{L}_n(k[x_1,\dots ,x_m])$ with $n\ge 3$. As Gaussian elimination gives us an algorithmic way of finding an explicit factorization of the given matrix into elementary matrices over a field, we develop a similar algorithm over polynomial rings.

arXiv comment23 pages, LaTex