An Algorithmic Proof of Suslin′s Stability Theorem for 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 SL n ( k) or SL n ( k[ x]), n ≥ 2, as a product of elementary matrices. Suslin′s stability theorem states that the same is true for SL n ( k[ x l,..., x m ]) with n ≥ 3 and m ≥ 1. In this paper, we present an algorithmic proof of Suslin′s stability theorem, thus providing a method for finding an explicit factorization of a given polynomial matrix into elementary matrices. Gröbner basis techniques may be used in the implementation of the algorithm.