A Realization Algorithm for SL 2 ( R [ x 1 , . . . , x m ]) over the Euclidean Domain
Hyungju Park · SIAM Journal on Matrix Analysis and Applications · 1999
Let R be an arbitrary Euclidean domain and A be an n x 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 x n multivariate polynomial matrix of determinant 1 with $n\geq 3$ 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 Études Sci. Publ. Math. 30 (1996), pp. 365--413]. In this paper, an algorithm is developed that determines precisely when a given matrix in SL 2 (R[x 1 , . . . ,x m ]) 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 $\Bbb Z$ of integers. Extending the idea used in this algorithm, we disprove a conjecture of Tolhuizen--Hollmann--Kalker regarding the realizability of a certain FIR filter.