AN ALGORITHMIC PROOF OF SUSLINS STABILITY THEOREM FOR POLYNOMIAL-RINGS

Let k be a field. Then Gaussian elimination over k and the Euclidean d ivision algorithm for the univariate polynomial ring k[x] allow us to write any matrix in SL(n)(k) or SL(n)(k[x]), n greater than or equal t o 2, as a product of elementary matrices. Suslin's stability theorem s tates that the same is true for SL(n)(k[x(l),...,x(m)]) with n greater than or equal to 3 and m greater than or equal to 1. In this paper, w e present an algorithmic proof of Suslin's stability theorem, thus pro viding a method for finding an explicit factorization of a given polyn omial matrix into elementary matrices. Grobner basis techniques may be used in the implementation of the algorithm. (C) 1995 Academic Press, Inc.