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

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 SLn(k) or SLn(k[x]), n ≥ 2, as a product of elementary matrices. Suslin’s stability theorem states that the same is true for the multivariate polynomial ring SLn(k[x1, . . . , xm]) with n ≥ 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.