A REALIZATION ALGORITHM FOR SL2(R[x1, . . . , xm]) OVER THE EUCLIDEAN DOMAIN.

Let R be an arbitrary Euclidean domain and A be an n×n matrix of determinant 1 whose entries are elements of R. Gaussian elimination process based on Euclidean division algorithm (on R) allows one to write A as a product of elementary matrices over R. Suslin's stability theorem states that any n×n multivariate polynomial matrix of determinant 1 with n 3 can be written as a product of elementary matrices. This result fails when n = 2, and a counter-example was constructed by P. M. Cohn [Inst. Hautes ´ Etudes Sci. Publ. Math. 30 (1996), pp. 365­413]. In this paper, an algorithm is developed that determines precisely when a given matrix in SL2(R[x1,...,xm]) allows a factorization into elementary matrices and, if it does, expresses it as a product of elementary matrices. This algorithm has potential applications to signal processing, in which case the coefficient ring R is usually taken as a field or a ring Z of integers. Extending the idea used in this algorithm, we disprove a conjecture of Tolhuizen­Hollmann­Kalker regarding the realizability of a certain FIR filter. [ABSTRACT FROM AUTHOR]