Computation of the Smith Form for Multivariate Polynomial Matrices Using Maple

In this paper we show how the transformations associated with the reduction to the Smith form of some classes of multivariate 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.


Introduction
Matrices whose elements are polynomials in more than one indeterminate have been studied by many authors.Such matrices arise in the mathematical treatment of the so-called multidimensional systems which can be considered as extensions of the ordinary differential or difference systems.These include delay-differential systems and partial differential systems.In particular the Smith normal form of a matrix plays an important role in many areas of mathematics such as the polynomial approach in control theory see for example Rosenbrock [1] and Kailath [2].The problem of reducing an univariate polynomial matrix to its Smith form is well understood and the relevant algorithm is already implemented in most computer algebra systems.For the multivariate case, however the problem is still open.Despite that some necessary and/or sufficient conditions for the reduction of a matrix to its Smith form have been given in the literature, no algorithm has been given to show how the transformations involved in the reduction are actually computed.These transformations are important if the solutions of the reduced system are to be expressed in terms of the original variables.So far the computations associated with these conditions have been difficult if not impossible to carry out.Recently however, some progress has been made in symbolic computation in particular a QuillenSuslin Maple package [3] has been developed.This package which is a maple implementa-tion of the Quillen-Suslin theorem provides an algorithm which computes a basis of a free module over a polynomial ring.In terms of matrices, this algorithm completes a unimodular rectangular matrix to an invertible matrix over the given polynomial ring with rational or integer coefficients (for more details see http://wwwb.math.rwth-aachen.de/QuillenSuslin/).In this paper, we show how this package can be used to compute the Smith form and the associated transformation for some classes of multivariate polynomial matrices.The classes of matrices considered can be regarded as those associated with linear determined systems of multidimensional equations which can be reduced to a single equation, thereby simplifying the analysis of such systems.The transformation used to obtain the Smith form is that of unimodular equivalence which will be defined later.First we need to introduce some definitions and a theorem which play a key role in this paper.

Definition 1 Let be a ring. The general linear group
is called a unimodular matrix.In the case where , a polynomial ring in the indeterminates ank of T and i r is the r  is the gcd atrix to its Smith form by an equivalence of the type (2).In order to show that any matrix can be brought by an equivalence transformation to its Smith form, it is usually required that D is a principal ideal or a Euclidean domain.In the c e when , the matrices may be treated as havi one of the indeterminates with coefficients that are rational forms in the other indeterminates e.g.

 
ng elements in . This approach which involves a reno as the disadvantage of yielding Smith forms which are not unique, see the work of Frost and Storey [6] and Morf et al. [7].Conditions under which a matrix with elements in is equivalent to its Smith form have been mber of authors.For the ring given by a nu , z z  , Lee and Zak [8] proposed a necessary and sufficient condition in terms of the existence of solutions to certain polynomial equations.Frost and Boudellioua [9] gave a necessary and sufficient condition for a class of multivariate polynomial matrices in terms of the existence of a polynomial vector.Lin et al. [10] proposed a sufficient condition for a class of matrices whose determinant is linear in one of the indeterminates.
Theorem 2 (Boudellioua and Quadrat [11]) if and only if there exists a ZRP vector such that the matrix   T U is ZLP.This is a particular type of Smith form where all the in p U D  variant polynomials are equal to 1 except the last one which is given by the determinant of the matrix.This form is important for simplification considerations, i.e. a system whose matrix is equivalent to the Smith form ( 3) is equivalent to a single equation in one unknown.Thus making it easier to analyse such a system either analyticcally or numerically.In order to express the conclusions and solutions made about the reduced system in terms of the original system, one has to compute the transformations (2) connecting the original system matrix to the Smith form.Suppose that such a vector is obtained then the transformations M and N that reduce T to the Smith form ( 3 from the first r and the first 1 p  columns of the matrix 3 1  Construct the matrix  Check that the m of a sufficient co Smith for = S MTN .Another interesting result in the form ndition for the reduction of class of linear multidimensional systems was given by Lin et al. [10].This is the class of systems where the determinant of the system matrix is linear in one of the indeterminates.Such systems are also equivalent to a single equation.The result is given for the case when the determinant is linear in 1 z but it is equally valid for any indeterminate , =1, ,   IsUnimod(UT, var, true);

Let us check if the the matrix is unimodular
Applying the QSAlgorithm ocedure to the row w orithm(UT, var, true);

Conclusion
have shown that the recent implee of a constructive version of t

Acknowledgements
drat and Anna Fabianska for author with a copy of the , 1970.
In this paper, we mentation in Mapl he Quillen-Suslin Theorem can be used effectively to compute the Smith form and the associated unimodular transformations for a class of multivariate polynomial matrices.The classes of matrices considered are those arising from multidimensional systems amenable to be simplified to a single equation in one unknown.The case of underdetermined systems can also be treated in a similar fashion.


) can b btained via the followin Maple algorithm.Declare the path such as QuillenSuslin are stored and load the packages LinearAlgebra and QuillenSuslin. Declare the ring over which the matrix declaring the indeterminates and the field of coefficients. Enter th T U .Test the zero-primeness of the matrix  

2. Equivalence to the Smith Form is
theorem is at the centre of the proofs of the results presented in this paper and its Maple implementation is used in the computation of the transformations.