AJCM  Vol.2 No.1 , March 2012
Computation of the Smith Form for Multivariate Polynomial Matrices Using Maple
Abstract: In this paper we show how the transformations associated with the reduction to the Smith form of some classes of mul-tivariate polynomial matrices are computed. Using a Maple implementation of a constructive version of the Quillen-Suslin Theorem, we present two algorithms for the reduction to a particular Smith form often associated with the simplification of linear systems of multidimensional equations.
Cite this paper: M. Boudellioua, "Computation of the Smith Form for Multivariate Polynomial Matrices Using Maple," American Journal of Computational Mathematics, Vol. 2 No. 1, 2012, pp. 21-26. doi: 10.4236/ajcm.2012.21003.

[1]   H. H. Rosenbrock, “State Space and Multivariable Theory,” Nelson-Wiley, London, 1970.

[2]   T. Kailath, “Linear Sys-tems,” Prentice-Hall, Englewood Cliffs, 1980.

[3]   A. Fabianska and A. Quadrat, “Applications of the Quillen-Suslin Theo-rem to Multidimensional Systems Theory,” Technical Report 6126, INRIA, Sophia Antipolis, 2007.

[4]   D. Quillen, “Projective Modules over Polynomial Rings,” Inventiones Mathematicae, Vol. 36, 1976, pp. 167-171. doi:10.1007/BF01390008

[5]   A. Suslin, “Projective modules over Polynomial Rings Are Free,” Soviet Mathematics—Doklady, Vol. 17, No. 4, 1976, pp. 1160-1164.

[6]   M. Frost and C. Storey, “Equivalence of a Matrix over R[s; z] with Its Smith Form,” International Journal of Control, Vol. 28, No. 5, 1979, pp. 665-671. doi:10.1080/00207177808922487

[7]   M. Morf, B. Levy and S. Kung, “New Results in 2-D Systems Theory: Part I: 2-D Polynomial Matrices, Factorization and Coprimeness,” Proceedings of the IEEE, Vol. 65, No. 6, 1977, pp. 861-872. doi:10.1109/PROC.1977.10582

[8]   E. Lee and S. Zak, “Smith Forms over R[z1; z2],” IEEE Transactions on Auto-matic Control, Vol. 28, No. 1, 1983, pp. 115-118. doi:10.1109/TAC.1983.1103118

[9]   M. Frost and M. S. Boudellioua, “Some Further Results Concerning Matrices with Elements in a Polynomial Ring,” International Journal of Control, Vol. 43, No. 5, 1986, pp. 1543-1555. doi:10.1080/00207178608933558

[10]   Z. Lin, M. S. Boudel-lioua and L. Xu, “On the Equivalence and Factorization of Multivariate Polynomial Matrices,” Proceedings of the 2006 International Symposium of Circuits and Systems, Island of Kos, 21-24 May 2006, p. 4914.

[11]   M. S. Boudellioua and A. Quadrat, “Serre’s Reduction of Linear Functional Systems,” Mathematics in Computer Science, Vol. 4, No. 2, 2010, pp. 289-312. doi:10.1007/s11786-010-0057-y