Reduction of Linear Functional Systems using Fuhrmann ' s Equivalence

Functional systems arise in the treatment of systems of partial differential equations, delay-differential equations, multidimensional equations, etc. The problem of reducing a linear functional system to a system containing fewer equations and unknowns was first studied by Serre. Finding an equivalent presentation of a linear functional system containing fewer equations and fewer unknowns can generally simplify both the study of the structural properties of the linear functional system and of different numerical analysis issues, and it can sometimes help in solving the linear functional system. In this paper, Fuhrmann's equivalence is used to present a constructive result on the reduction of under-determined linear functional systems to a single equation involving a single unknown. This equivalence transformation has been studied by a number of authors and has been shown to play an important role in the theory of linear functional systems.


Introduction
olynomial matrices play an important role in the theory of linear systems described by ordinary differential equations (see for example Rosembrock [1] and Kailath [2]).In this case, the polynomial matrices involve a single indeterminate with coefficients in the field of real or complex numbers.The study of linear systems of ordinary differential equations is thus reduced to the study of matrices over the ring [] s or [] s .These rings have the Euclidean division property which makes it possible to establish canonical forms such as the Smith normal form.In fact, the polynomial theory of such systems can be regarded as more or less complete.In the case of linear functional systems, arising for example from partial differential equations or delay-differential equations, the resulting system matrices involve polynomials in more than one indeterminate.Since multivariate polynomial rings do not have the Euclidean division property, it is no longer possible to extend most of the results obtained for the single indeterminate case.Throughout this paper unless specified otherwise, [ ] denotes the polynomial ring in the indeterminates with coefficients in an arbitrary but fixed field K. First we present a few definitions that will be needed later in the paper. , the i th order invariant polynomial i  of T is defined by : 1 ;1 = 0; min( , ) where r is the normal rank of T , 0 =1  and i  is the greatest common divisor of all the ii  minors of .T As in the single variable case, the zero structure of a multivariate polynomial matrix is a crucial indicator of its properties.However, unlike the single variable case, the zero structure of a multivariate polynomial matrix is not completely captured by the invariant polynomials but by the invariant zeros as defined by the following.
pq be a full row rank matrix and i the ideal generated by the ii  minors of R and () i V the algebraic variety defined by: The i th order invariant zeros of R are the elements of the variety () i V .
The general linear group () T and 1 S of appropriate dimensions, are said to be zero-left-coprime T and 2 S , of appropriate dimensions, are said to be zero-right-coprime ( ZRC ) if the matrix  

22
T TT

TS
has left-inverse in D .
One of standard tasks carried out in systems theory is to transform a given system representation into a simpler form.An equivalence transformation used in the context of multidimensional systems is Fuhrmann's equivalence [6] and is defined by the following.

T and 2
T are said to be Fuhrmann-equivalent (F-E) if there exist matrices 1 S , 2 S of appropriate dimensions with elements in D such that where 11 , TS are ZLC and 22 , TS are ZRC.
In the case when 1 T and 2 T have the same size and 1 S and 2 S are square, the transformation in (2) reduces to the classical unimodular equivalence.F-E has been studied by a number of authors.For instance, Pugh et al. [7,8] have shown that it exhibits fundamental algebraic properties amongst its invariants.In particular, they have shown that it preserves the invariant polynomials as well as the the invariant ideals.Lemma 1.1 [7] Suppose that two matrices = for = 0,1, , max( 1, 1) = 1 or = 1 in case < or < .

TT h i k i i h i k     
The problem of reducing a linear functional system to a system containing fewer equations and unknowns was first studied by Serre [9].Such a process is based on the application of the well known Quillen-Suslin Theorem.= q RR I .Then there exists a unimodular matrix

Reduction using Fuhrmann's Equivalence
The reduction of a multivariate polynomial matrix in the context of multidimensional systems theory was first studied by Frost and Boudellioua [12].They obtained necessary and sufficient conditions under which a square multivariate polynomial matrix is unimodular equivalent to a simpler form.This corresponds to the reduction of a determined linear functional system into a single equation in one unknown function.Boudellioua and Quadrat [13] generalized this result to underdetermined systems using a module theoretic approach.Their result gives necessary and sufficient conditions for the reduction of a linear system to a system containing a single equation with several unknown functions.Boudellioua [14] generalized the earlier result to under-determined systems thereby giving necessary and sufficient conditions for the unimodular equivalence of an under-determined linear system to one containing a single equation involving only one unknown function.The results mentioned so far are based mainly on unimodular equivalence.This transformation has the disadvantage of establishing a connection between matrices and hence systems which have the same size.Boudellioua [15] used Fuhrmann's equivalence to reduce a class of determined linear functional systems to a system containing a single equation in one unknown.In this paper, we extend this latter result to the under-determined case.Before presenting the main result of this paper, we first state the following result which is a statement of the positive answer of the Lin-Bose conjecture [16].This theorem which will be used later is given by Fabianska and Quadrat [17].

Theorem 2.1 [Section 5 of [17] ]
x be a commutative polynomial ring over a field K and qp RD   a full row rank matrix.Then the following two assertions are equivalent: 1.The ideal () q IR generated by the qq  minors of R is principal, i.e. can be generated by the greatest common divisor  of these minors.
2. There exist , pq with full row rank, then T is Fuhrmann- equivalent to the row vector where D  is the gcd of the qq  minors of T , if and only if there exist a vector with coefficients in a computable field K (e.g., = K ), one may take a generic vector q UD  with a fixed total degree in the i x 's and compute the D -module   / ( ) by means of a Gröbner basis computation and check whether or not this D -module vanishes on certain branches of the corresponding tree of integrability conditions (see Pommaret and Quadrat [18]) or on certain obstructions to genericity ( see Levandoskyy and Zerz [19]).

Conclusion
We have presented a constructive result for the simplification of a class of linear functional systems.More specifically, we have given necessary and sufficient conditions under which a rectangular multivariate polynomial matrix can be reduced by Fuhrmann's equivalence to a form that corresponds to the reduction of a linear functional system to a single equation with only one unknown function.The result can be easily implemented on a computer algebra system such as Maple using the OreModules package (see Chyzak et al. [20]).

Acknowledgment
The author wishes to express his thanks to Sultan Qaboos University (Oman) for supporting this research (Grant No. IG/SCI/DOMS/15/01).

P Definition 1 . 1
Given a matrix qp TD  matrices with elements in D where , mn are fixed positive integers and > min( , ) r m n  . 1

Theorem 1 . 1 [ 10 , 11 ]
Let K be a principal ideal domain and inverse in D such that the matrix   TU has a right inverse over D and the ideal generated by the qq  minors of T is principal.