Applications of the Quillen-Suslin theorem to multidimensional systems theory

French title: Applications du théorème de Quillen-Suslin à la théorie des systèmes multidimensionnels

Keywords: Constructive versions of the Quillen-Suslin theorem, Lin-Bose´s conjecture, multidimensional linear systems, flat systems, (weakly) doubly coprime factorizations of rational transfer matrices, factorization and decomposition of linear functional systems, symbolic computation..

French keywords: Versions constructives du théorème de Quillen-Suslin, conjecture de Lin-Bose, systèmes linéaires multidimensionnels, systèmes plats, factorisations doublement (faiblement) copremières de matrices de transfert rationnelles, factorisation et décomposition des systèmes linéaires fonctionnels, calcul formel..

Abstract

The purpose of this paper is to give four new applications of the Quillen-Suslin theorem to mathematical systems theory. Using a constructive version of the Quillen-Suslin theorem, also known as Serre´s conjecture, we show how to effectively compute flat outputs and injective parametrizations of flat multidimensional linear systems. We prove that a flat multidimensional linear system is algebraically equivalent to the controllable 1-D dimensional linear systems obtained by setting all but one functional operator to zero in the polynomial matrix defining the system. In particular, we show that a flat ordinary differential time-delay linear system is algebraically equivalent to the corresponding ordinary differential system without delay, i.e., the controllable ordinary differential linear system obtained by setting all the delay amplitudes to zero. We also give a constructive proof of a generalization of Serre´s conjecture known as Lin-Bose´s conjecture. Moreover, we show how to constructively compute (weakly) left-/right-/doubly coprime factorizations of rational transfer matrices over a commutative polynomial ring. The Quillen-Suslin theorem also plays a central part in the so-called decomposition problem of linear functional systems studied in the literature of symbolic computation. In particular, we show how the basis computation of certain free modules, coming from projectors of the endomorphism ring of the module associated with the system, allows us to obtain unimodular matrices which transform the system matrix into an equivalent block-triangular or a block-diagonal form. Finally, we demonstrate the package QuillenSuslin which, to our knowledge, contains the first implementation of the Quillen-Suslin theorem in a computer algebra system as well as the different algorithms developed in the paper.

French Abstract

Le but de ce papier est de donner quatre nouvelles applications du théorème de Quillen-Suslin à la théorie mathématique des systèmes. A l´aide d´une version constructive du théorème de Quillen-Suslin, aussi connu sous le nom de conjecture de Serre, nous montrons comment calculer de manière effective les sorties plates et les paramétrisations injectives des systèmes linéaires multidimensionnels plats. Nous prouvons que tout système linéaire multidimensionnel plat est algébriquement équivalent aux systèmes linéaires 1-D contrôlables obtenus par annulation de tous les opérateurs fonctionnels sauf un dans la matrice polynômiale définissant le système. En particulier, nous montrons que tout système linéaire différentiel à retard plat est algébriquement équivalent au système différentiel sans retard, c´est-à-dire, au système linéaire contrôlable d´équations différentielles obtenu en annulant les amplitudes des retards. Nous donnons aussi une preuve constructive d´une généralisation de la conjecture de Serre appelée conjecture de Lin-Bose. De plus, nous montrons comment calculer de manière effective des factorisations (faiblement) copremières à gauche et à droite de matrices de transfert rationnelles sur une algèbre commutative de polynômes. Le théorème de Quillen-Suslin joue aussi un rôle important dans l´étude du problème de décomposition des systèmes linéaires fonctionnels étudié dans la littérature du calcul formel. En particulier, nous montrons comment le calcul de bases de certains modules libres, provenant de projecteurs de l´anneau des endomorphismes du module associé au système, nous permet de calculer des matrices unimodulaires qui transforment la matrice du système en une matrice équivalente ayant une forme bloc-triangulaire ou bloc-diagonale. Finalement, nous décrivons le logiciel QuillenSuslin qui, à notre connaissance, contient la première implémentation du théorème de Quillen-Suslin dans un système de calcul formel, ainsi que les différents algorithmes obtenus dans le papier.

Note: The first author is very grateful to W. Plesken for his support. She would also like to express thanks to F. Castro-Jimenez and J. Gago-Vargas for the opportunity to discuss with them on the Quillen-Suslin theorem and to A. van den Essen for pointing out to her an interesting example that has been included in the paper..

Note: The second author would like to thank Y. Lam, T. Coquand, H. Lombardi and I. Yengui for interesting discussions on the Quillen-Suslin theorem and Z. Lin and H. Park for discussions on applications of the Quillen-Suslin theorem to mathematical systems theory, control theory and signal processing..

Short Table of Contents

1. Introduction

In 1784, Monge studied the integration of certain underdetermined non-linear systems of ordinary differential equations, namely, systems containing more unknown functions than differential independent equations ([31]). He showed how the solutions of these systems could be parametrized by means of a certain number of arbitrary functions of the independent variable. This problem was called the Monge problem and it was studied by famous mathematicians such as Hadamard, Hilbert, Cartan and Goursat. In particular, motivated by problems coming from linear elasticity theory, Hadamard considered the case of linear ordinary differential equations and Goursat investigated underdetermined systems of partial differential equations. We refer the reader to [31] for a historical account on the Monge problem and for the main references.

Within the algebraic analysis approach ([2], [21], [30], [35]), the Monge problem was recently studied for underdetermined systems of linear partial differential equations in [21], [35], [44], [45], [46] and for linear functional systems in [5], [6] (e.g., differential time-delay systems, discrete systems). Depending on the algebraic properties of a certain module M defined over a ring D of functional operators and intrinsically associated with the linear functional system, we can prove or disprove the existence of different kinds of parametrizations of the system (i.e., minimal or injective parametrizations, non-minimal parametrizations, chains of successive parametrizations). Constructive algorithms for checking these algebraic properties (i.e., torsion, existence of torsion elements, torsion-free, reflexive, projective, stably free, free) and computing the different parametrizations were recently developed in [5], [44], [45], [46], implemented in the package OreModules ([5], [6]) and illustrated on numerous examples coming from mathematical physics and control theory ([5], [6]). Finally, we proved in [5], [44], [45], [46] how the Monge problem gave answers for the search of potentials in mathematical physics and image representations in control theory ([41], [42], [65], [66]).

The last results show that the Monge problem is constructively solved for certain classes of linear functional systems up to a last but important point: we can check whether or not a linear functional system admits injective parametrizations but we are generally not able to compute one even if some heuristic methods were presented in [5], [44], [45]. Indeed, the existence of injective parametrizations for a linear functional system was proved to be equivalent to the freeness of the corresponding module M. In the case of a linear functional system with constant coefficients, the corresponding ring D of functional operators is a commutative polynomial ring over a field k of constants. Using the famous Quillen-Suslin theorem ([56], [58]), also known as Serre´s conjecture ([24], [25]), we then know that free D-modules are projective ones. Using Gröbner or Janet bases ([5], [11], [44]), we can check whether or not a module over a commutative polynomial ring is projective. See [3], [11], [20] and the references therein for introductions to Janet and Gröbner bases. Hence, we can constructively prove the existence of an injective parametrization for a linear functional system. However, we need to use a constructive version of the Quillen-Suslin theorem ([15], [19], [23], [27], [29], [37], [61], [62]) to get injective parametrizations of the corresponding system.

The main purpose of this paper is to recall a general algorithm for computing bases of a free module over a commutative polynomial ring, give four new applications of the Quillen-Suslin theorem to mathematical systems theory and demonstrate the implementation of the QuillenSuslin package ([13]) developed in the computer algebra system MAPLE. To our knowledge, the QuillenSuslin package is the first package available which performs basis computation of free modules over a commutative polynomial ring with rational and integer coefficients and is dedicated to different applications coming from mathematical systems theory.

More precisely, the plan of the paper is the following one. In the second section, we recall how the structural properties of linear functional systems can be constructively studied within the algebraic analysis approach as well as different results on the Monge problem. A constructive version of the Quillen-Suslin theorem, which is the main tool we use in the paper, is presented in the third section and the implementation is illustrated on many examples in the Appendix of the paper. We also describe some heuristic methods that highly simplify the computation of a basis of a free module over polynomial ring in certain special cases. The constructive version of the Quillen-Suslin theorem and, in particular the patching procedure, gives us the opportunity to make a new observation concerning linear functional systems which admit injective parametrizations also called flat multidimensional systems in mathematical systems theory. In the fourth section, we prove that a flat multidimensional system is algebraically equivalent to a 1-D flat linear system obtained by setting all but one functional operator to zero in the system matrix. This result gives an answer to a natural question on flat multidimensional systems. In particular, we prove that every flat differential time-delay system is algebraically equivalent to the differential system without delays, namely, the system obtained by setting to zero all the time-delay amplitudes. In the fifth section, we consider a generalization of Serre´s conjecture. We recall that Serre´s conjecture conjecture, also known as the Quillen-Suslin theorem, can be expressed in the language of matrices as follows: every matrix R over a commutative polynomial ring

D = k [x_{1}, ..., x_{n}]

whose maximal minors generate D (unimodular matrix) can be completed to a square invertible matrix over D (i.e., its determinant is a non-zero element of the field k). The generalization, stated by Lin and Bose in [26] and first proved by Pommaret in [43] by means of algebraic analysis, can be formulated as the possibility of completing a matrix R whose maximal minors divided by their greatest common divisor d generate D to a square polynomial matrix whose determinant equals d. Serre´s conjecture is then the special case where

d = 1

. Using the Quillen-Suslin theorem, we give a constructive algorithm for computing such a completion. Using the possibility of computing basis of a free module in our implementation QuillenSuslin, this algorithm has been implemented in this package. In the sixth section, we study the existence of (weakly) left-/right-coprime factorizations of rational transfer matrices using recent results developed in [50]. We give algorithms for computing such factorizations using the constructive version of the Quillen-Suslin theorem. These results constructively solve open questions in the literature of multidimensional linear systems (see [63], [64] and the references therein). Finally, we show that the constructive Quillen-Suslin theorem also plays an important role in the decomposition problem of linear functional systems studied in the literature of symbolic computation. See [9] and the references therein for more details. The main idea is to transform the system matrix into an equivalent block-triangular or a block-diagonal form ([9], [10]).

The different algorithms presented in the paper have been implemented in the package QuillenSuslin based on the library Involutive ([3]) (an OreModules ([6]) version will be soon available). The Appendix illustrates the main procedures of the QuillenSuslin package on different examples taken from the literature ([19], [23], [38], [61]). The package QuillenSuslin also contains a completion algorithm for unimodular matrices over Laurent polynomial rings described in [36], [38]. See also [1] for a recent algorithm. In [38], Park explains the importance and the meaning of the completion problem of unimodular matrices over Laurent polynomial rings to signal processing and gives an algorithm for translating this problem to a polynomial case. Park´s results can also be used for computing flat outputs of δ-flat multidimensional linear systems ([32], [33]). See [5] for another constructive algorithm and [6] for illustrations on different explicit examples.

1.1. Notation.

In what follows, we shall denote by k a field,

D = k [x_{1}, ..., x_{n}]

a commutative polynomial ring with coefficients in k,

D^{1 \times p}

the D-module formed by the row vectors of length p with entries in D and

D^{q \times p}

the set of

q \times p

-matrices with entries in D.

F

will always denote a D-module. We denote by

R^{T}

the transpose of the matrix R and by

I_{p}

the

p \times p

identity matrix. Finally, the symbol ≜ means ``by definition´´.

2. A module-theoretic approach to systems theory

Let

D = k [x_{1}, ..., x_{n}]

be a commutative polynomial ring over a field k and

R \in D^{q \times p}

. We recall that a matrix R is said to have full row rank if the first syzygy module of the D-module

D^{1 \times q} R

formed by the D-linear combinations of the rows of R, namely,

is reduced to 0. In other words,

λ R = 0

implies

λ = 0

, i.e., the rows of R are D-linearly independent.

Definition 1 ([34], [62], [66])

Let $D = R [x_{1}, ..., x_{n}]$ be a commutative polynomial ring, $R \in D^{q \times p}$ a full row rank matrix, J the ideal generated by the $q \times q$ minors of R and $V (J)$ the algebraic variety defined by:

V (J) = {ξ \in C^{n} | P (ξ) = 0, \forall P \in J} .

R is called minor left-prime if $\dim_{C} V (J) \leq n - 2,$ i.e., the greatest common divisor of the $q \times q$ minors of R is 1.
R is called weakly zero left-prime if $\dim_{C} V (J) \leq 0,$ i.e., the $q \times q$ minors of R may only vanish simultaneously in a finite number of points of $C^{n}$ .
R is called zero left-prime if $\dim_{C} V (J) = - 1,$ i.e., the $q \times q$ minors of R do not vanish simultaneously in $C^{n}$ .

The previous classification plays an important role in multidimensional systems theory. See [34], [62], [66] and the references therein for more details.

The purpose of this section is twofold. We first recall how we can generalize the previous classification for general multidimensional linear systems, i.e., systems which are not necessarily defined by full row rank matrices. We also explain the duality existing between the behavioural approach to multidimensional systems ([34], [41], [65], [66]) and the module-theoretic one ([44], [45], [46]). See also [65] for a nice introduction.

In what follows, D will denote a commutative polynomial ring with coefficients in a field k. In particular, we shall be interested in commutative polynomial rings of functional operators such as partial differential operators, differential time-delay operators or shift operators. Let us consider a matrix

R \in D^{q \times p}

and a D-module

F

, namely:

where

D^{1 \times p}

denotes the D-module of row vectors of length p with entries in D, then the cokernel of the D-morphism

. R

is defined by:

The D-module M is said to be presented by R or simply finitely presented ([5], [57]). Moreover, we can also define the system or behaviour as follows:

As it was noticed by Malgrange in [30], the D-module M and the system

{ker}_{F} (R .)

are closely related. As this relation will play an important role in what follows, we shall explain it in details. In order to do that, let us first introduce a few classical definitions of homological algebra. We refer the reader to [57] for more details.

Definition 2

A sequence $(M_{i}, d_{i} : M_{i} \to M_{i - 1})_{i \in Z}$ of D-modules $M_{i}$ and D-morphisms $d_{i} : M_{i} \to M_{i - 1}$ is a complex if we have:

$\forall i \in Z, im d_{i} \subseteq ker d_{i - 1} .$

We denote the previous complex by:

$... \overset{d_{i + 2}}{\to} M_{i + 1} \overset{d_{i + 1}}{\to} M_{i} \overset{d_{i}}{\to} M_{i - 1} \overset{d_{i - 1}}{\to} ...$ (1)
The defect of exactness of the complex (1) at $M_{i}$ is defined by:

$H (M_{i}) = ker d_{i} / im d_{i + 1} .$
The complex (1) is said to be exact at $M_{i}$ if we have:

$H (M_{i}) = 0 \Leftrightarrow ker d_{i} = im d_{i + 1} .$
The complex (1) is exact if:

$\forall i \in Z, ker d_{i} = im d_{i + 1} .$
The complex (1) said to be a split exact sequence if (1) is exact and if there exist D-morphisms $s_{i} : M_{i - 1} \to M_{i}$ satisfying the following conditions:

$\forall i \in Z, \{\begin{matrix} s_{i + 1} \circ s_{i} = 0, \\ s_{i} \circ d_{i} + d_{i + 1} \circ s_{i + 1} = i d_{M_{i}} . \end{matrix}$
A finite free resolution of a D-module M is an exact sequence of the form

$0 \to D^{1 \times p_{m}} \overset{. R_{m}}{\to} ... \overset{. R_{2}}{\to} D^{1 \times p_{1}} \overset{. R_{1}}{\to} D^{1 \times p_{0}} \overset{π}{\to} M \to 0 .$ (2)

where $p_{i} \in Z_{+} = {0, 1, 2, ...}$ , $R_{i} \in D^{p_{i} \times p_{i - 1}}$ , and the D-morphism $. R_{i}$ is defined by:

$\begin{matrix} . R_{i} : D^{1 \times p_{i}} & \to & D^{1 \times p_{i - 1}} \\ λ & \mapsto & (. R_{i}) (λ) = λ R_{i} . \end{matrix}$

The next classical result of homological algebra will play a crucial role in what follows.

We refer the reader to Example 13 for explicit computations of

{ext}_{D}^{i} (N, D)

i \geq 0

Coming back to the D-module M, we have the following beginning of a finite free resolution of M:

where π denotes the D-morphism which sends elements of

D^{1 \times p}

to their residue classes in M. If we ``apply the left-exact contravariant functor´´

{hom}_{D} (\cdot, F)

to (4) (see [57] for more details), by Theorem 1, we obtain the following exact sequence:

For more details, see [5], [30], [34], [46], [65] and the references therein. In particular, (5) gives an intrinsic characterization of the

F

-solutions of the system

{ker}_{F} (R .)

. It only depends on two mathematical objects:

If D is now a ring of functional operators (e.g., differential operators, time-delay operators, difference operators), then the issue of understanding which

F

is suitable for a particular linear system has long been studied in functional analysis and is still nowadays a very active subject of research. It does not seem that constructive algebra and symbolic computation can propose new methods to handle this functional analysis problem. However, they are very useful for classifying

{hom}_{D} (M, F)

by means of the algebraic properties of the D-module M. Indeed, a large classification of the properties of modules is developed in module theory and homological algebra. See [57] for more information. Let us recall a few of them.

Definition 3 ([57])

Let D be a commutative polynomial ring with coefficients in a field k and M a finitely presented D-module. Then, we have:

M is said to be free if it is isomorphic to $D^{1 \times r}$ for a non-negative integer r, i.e.:

$M ≅ D^{1 \times r}, r \in Z_{+} = {0, 1, 2 ...} .$
M is said to be stably free if there exist two non-negative integers r and s such that:

$M \oplus D^{1 \times s} ≅ D^{1 \times r} .$
M is said to be projective if there exist a D-module P and non-negative integer r such that:

$M \oplus P ≅ D^{1 \times r} .$
M is said to be reflexive if the canonical map

$ε_{M} : M \to {hom}_{D} ({hom}_{D} (M, D), D),$

defined by

$\forall m \in M, \forall f \in {hom}_{D} (M, D) : ε_{M} (m) (f) = f (m),$

is an isomorphism, where ${hom}_{D} (M, D)$ denotes the D-module of D-morphisms from M to D.
M is said to be torsion-free if the submodule of M defined by

$t (M) = {m \in M ∣ \exists 0 \neq P \in D : P m = 0}$

is reduced to the zero module. $t (M)$ is called the torsion submodule of M and the elements of $t (M)$ are the torsion elements of M.
M is said to be torsion if $t (M) = M$ , i.e., every element of M is a torsion element.

Let

K = Q (D) = k (x_{1}, ..., x_{n})

be the quotient field of D ([57]) and M a finitely presented D-module. We call the rank of M over D, denoted by

{rank}_{D} (M)

, the dimension of the K-vector space

K \otimes_{D} M

obtained by extending the scalars of M from D to K, i.e.:

We can check that if M is a torsion D-module, we then have

K \otimes_{D} M = 0

, a fact which implies that

{rank}_{D} (M) = 0

. See [57] for more details.

Let us recall a few results about the notions previously introduced in Definition 3.

Theorem 2 ([57])

Let $D = k [x_{1}, ..., x_{n}]$ be a commutative polynomial ring with coefficients in a field k. We have the following results:

We have the implications among the previous concepts:

$\begin{matrix} free \Rightarrow stably free \Rightarrow projective \Rightarrow reflexive \Rightarrow torsion-free . \end{matrix}$
If $D = k [x_{1}]$ , then D is a principal ideal domain – namely, every ideal of D is principal, i.e., it can be generated by one element of D – and every finitely generated torsion-free D-module is free.
(Serre theorem [11]) Every projective module over D is stably free.
(Quillen-Suslin theorem [56], [58]) Every projective module over D is free.

The famous Quillen-Suslin theorem will play an important role in what follows. We refer to [24], [25] for the best introductions nowadays available on this subject.

The next theorem gives some characterizations of the definitions given in Definition 3.

Combining the results of Theorem 3 and the Quillen-Suslin theorem (see 4 of Theorem 2), we then obtain a way to check whether or not a finitely presented D-module M has some torsion elements or is torsion-free, reflexive, projective, stably free or free. We point out that the explicit computation of

{ext}_{D}^{i} (N, D)

can always be done using Gröbner or Janet bases. See [5], [44], [45] for more details and for the description of the corresponding algorithms. We also refer the reader to [4], [6] for the library OreModules in which the different algorithms were implemented as well as to the large library of examples of OreModules which illustrates them. Finally, see also [3], [11], [20] and the references therein for an introduction to Gröbner and Janet bases.

In order to explain why the definitions given in Definition 3 extend the concepts of primeness defined in Definition 1, we first need to introduce some more definitions.

Let us suppose that R has full row rank and let us consider the finitely presented D-module

M = D^{1 \times p} / (D^{1 \times q} R)

. Using the notations of Definition 1 and the fact that

where

N = T (M) = D^{1 \times q} / (D^{1 \times p} R^{T})

is then a torsion D-module, i.e., it satisfies

{ext}_{D}^{0} (M, D) = {hom}_{D} (M, D) = 0

, by Theorem 4, we then obtain:

Hence, by Theorems 3 and 4, we obtain that R is minor left-prime (resp., zero left-prime) iff the D-module M is torsion-free (resp., projective, i.e., free by the Quillen-Suslin theorem stated in 4 of Theorem 2). See [46] for more details and the extension of these results to the case of non-commutative rings of differential operators.

We finally obtain the table given in Figure 1 which sums up the different results previously obtained. We note that the last two columns of this table only hold when the matrix R has full row rank.

To finish, we explain what the system interpretations of the definitions given in Definition 3 are. In particular, these interpretations solve the Monge problem stated in the introduction of the paper. In order to do that, we also need to introduce a few more definitions.

Roughly speaking, an injective cogenerator is a space rich enough for seeking solutions of linear systems of the form

R y = 0

, where

R \in D^{q \times p}

is any matrix and

y \in F^{p}

. In particular, using (5), if

F

is a cogenerator D-module and

M \neq 0

, then

{hom}_{D} (M, F) \neq 0

, meaning that the corresponding system

{ker}_{F} (R .)

is not empty. Finally, if

F

is an injective cogenerator D-module, then we can prove that any complex of the form (3) is exact at

F^{p_{i}}

i \geq 1

, if and only if the corresponding complex (2) is exact. See [34], [41], [65] and the references therein for more details.

Example 1

If Ω is an open convex subset of $R^{n}$ , then the space $C^{\infty} (Ω)$ (resp., $D^{'} (Ω)$ ) of smooth real functions (resp., real distributions) on Ω is an injective cogenerator module over the ring $R [\partial_{1}, ..., \partial_{n}]$ of differential operators with coefficients in $R$ , where we have denoted by $\partial_{i} = \partial / \partial x_{i}$ ([34], [30], [41]).

Example 2

Let k be a field, $F = k^{Z_{+}^{n}}$ be the set of sequences with values in k and $D = k [x_{1}, ..., x_{n}]$ be the ring of shift operators, namely,

\forall f \in F, i = 1, ..., n, (x_{i} f) (μ) = f (μ + 1_{i}),

where $μ = (μ_{1}, ..., μ_{n}) \in Z_{+}^{n}$ and $μ + 1_{i} = (μ_{1}, ..., μ_{i - 1}, μ_{i} + 1, μ_{i + 1}, ..., μ_{n})$ . Then, $F$ is an injective D-module ([34], [65]).

We have the following important corollary of Theorem 3 which solves the Monge problem in the case of linear functional systems with constant coefficients. See [67] and the references therein and the introduction of the paper.

Corollary 1 ([5], [44])

Let $F$ be an injective cogenerator $D = k [x_{1}, ..., x_{n}]$ -module, $R \in D^{q \times p}$ and $M = D^{1 \times p} / (D^{1 \times q} R)$ . Then, we have the following results:

There exists $Q_{1} \in D^{q_{1} \times q_{2}}$ , where $p = q_{1}$ , such that we have the exact sequence

$F^{q} \overset{R .}{\leftarrow} F^{q_{1}} \overset{Q_{1} .}{\leftarrow} F^{q_{2}},$

i.e., ${ker}_{F} (R .) = Q_{1} F^{q_{2}}$ , iff the D-module M is torsion-free.
There exist $Q_{1} \in D^{q_{1} \times q_{2}}$ and $Q_{2} \in D^{q_{2} \times q_{3}}$ such that we have the exact sequence

$F^{q} \overset{R .}{\leftarrow} F^{q_{1}} \overset{Q_{1} .}{\leftarrow} F^{q_{2}} \overset{Q_{2} .}{\leftarrow} F^{q_{3}},$

i.e., ${ker}_{F} (R .) = Q_{1} F^{q_{2}}$ and ${ker}_{F} (Q_{1} .) = Q_{2} F^{q_{3}}$ , iff the D-module M is reflexive.
There exists a chain of n successive parametrizations, namely, for $i = 1, ..., n$ , there exist $Q_{i} \in D^{q_{i} \times q_{i + 1}}$ such that we have the following exact sequence

$F^{q} \overset{R .}{\leftarrow} F^{q_{1}} \overset{Q_{1} .}{\leftarrow} ... \overset{Q_{n - 1} .}{\leftarrow} F^{q_{n}} \overset{Q_{n} .}{\leftarrow} F^{q_{n + 1}},$

i.e., ${ker}_{F} (R .) = Q_{1} F^{q_{2}}$ and ${ker}_{F} (Q_{i} .) = Q_{i + 1} F^{q_{i + 1}}$ , $i = 1, ..., n - 1$ , iff the D-module M is projective.
There exist $Q \in D^{p \times m}$ and $T \in D^{m \times p}$ such that $T Q = I_{m}$ and the sequence

$F^{q} \overset{R .}{\leftarrow} F^{p} \overset{Q .}{\leftarrow} F^{m} \leftarrow 0,$ (6)

is exact, i.e., ${ker}_{F} (R .) = Q F^{m}$ , and iff the D-module M is free.

We refer the reader to [5], [44], [45], [46], [53], [54] for the solutions of the Monge problem for different classes of linear functional systems with variables coefficients such as partial differential, differential time-delay or difference equations.

The matrices

Q_{i}

defined in Corollary 1 are called parametrizations ([5], [44], [45], [46]). Indeed, from 1 of Corollary 1, if M is torsion-free, then there exists a matrix of operators

Q_{1} \in D^{q_{1} \times q_{2}}

which satisfies

{ker}_{F} (R .) = Q_{1} F^{q_{2}}

. This means that any solution

η \in F^{p}

satisfying

R η = 0

is of the form

η = Q_{1} ξ

for a certain

ξ \in F^{q_{2}}

. In the behaviour approach ([42]), the parametrization is called an image representation of

{ker}_{F} (R .)

([41], [65], [66]). We point out that the parametrizations

Q_{i}

are obtained by computing

{ext}_{D}^{i} (N, D)

(see Theorem 3). Hence, checking whether or not a D-module is torsion-free, reflexive or projective gives the corresponding successive parametrizations. We refer to [5], [44], [45], [46] for more details, the extension of the previous results to non-commutative algebras of functional operators and the implementation of the corresponding algorithms in the library OreModules. Finally, the matrix Q defined in 4 of Corollary 1 is called an injective parametrization of

{ker}_{F} (R .)

as every

F

-solution of

{ker}_{F} (R .)

has the form

η = Q ξ

for a certain

ξ \in F^{m}

and we have

i.e., ξ is uniquely defined by

η \in {ker}_{F} (R .)

. At this stage, it is important to point out that no general algorithm has been developed to get injective parametrizations when the D-module M is free. It is the main purpose of this paper to constructively study this question and to apply the computation of injective parametrizations to some open questions appearing in mathematical systems theory.

Finally, we point out that, if M is a free D-module, then there always exist

Q \in D^{p \times m}

and

T \in D^{m \times p}

such that, for every D-module

F

, we have the exact sequence (6). Indeed, let us recall two standard arguments of homological algebra.

By 1 of Proposition 1, we obtain that

D^{1 \times q} \overset{. R}{\to} D^{1 \times p} \overset{. Q}{\to} D^{1 \times m} \to 0

is a splitting exact sequence and applying the functor

{hom}_{D} (\cdot, F)

to it, by 2 of Proposition 1, we obtain the splitting exact sequence (6). Hence, the assumption that

F

is an injective cogenerator D-module is only important for the converse implication of 6 of Corollary 1.

Explicit examples of computation of parametrizations can be found in [5], [6], [44], [45], [46] as well in the OreModules large library of examples ([4]). We refer the reader to these references and to Section 4 for the computation of injective parametrizations. However, let us give a simple example in order to illustrate the previous results.

Example 3

Let us consider the ring $D = Q [\partial_{1}, \partial_{2}, \partial_{3}]$ of differential operators with rational coefficients ( $\partial_{i} = \partial / \partial x_{i}$ ), the matrix $R = (\partial_{1} \partial_{2} \partial_{3})$ defining the so-called divergent operator in $R^{3}$ and the finitely presented D-module $M = D^{1 \times 3} / (D R)$ . Let us check whether or not the D-module M has some torsion elements or is torsion-free, reflexive or projective, i.e., free by the Quillen-Suslin theorem. In order to do that, we define the D-module $N = D / (D^{1 \times 3} R^{T})$ . A finite free resolution of N can easily be computed by means of Gröbner or Janet bases. We obtain the following exact sequence

0 \to D \overset{. P_{3}}{\to} D^{1 \times 3} \overset{. P_{2}}{\to} D^{1 \times 3} \overset{. R^{T}}{\to} D \overset{σ}{\to} N \to 0,

where σ denotes the canonical projection onto N and:

P_{2} = (\begin{matrix} 0 & - \partial_{3} & \partial_{2} \\ \partial_{3} & 0 & - \partial_{1} \\ - \partial_{2} & \partial_{1} & 0 \end{matrix}), P_{3} = R .

We note that $P_{2}$ corresponds to the so-called curl operator whereas $R^{T}$ is the gradient operator. Then, the defects of exactness of the following complex

0 \leftarrow D \overset{. P_{3}^{T}}{\leftarrow} D^{1 \times 3} \overset{. P_{2}^{T}}{\leftarrow} D^{1 \times 3} \overset{. R}{\leftarrow} D \leftarrow 0

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are defined by:

\{\begin{matrix} {ext}_{D}^{0} (N, D) ≅ {ker}_{D} (. R), \\ {ext}_{D}^{1} (N, D) ≅ {ker}_{D} (. P_{2}^{T}) / (D R), \\ {ext}_{D}^{2} (N, D) ≅ {ker}_{D} (. P_{3}^{T}) / (D^{1 \times 3} P_{2}^{T}), \\ {ext}_{D}^{3} (N, D) ≅ D / (D^{1 \times 3} P_{3}^{T}) . \end{matrix}

Using the fact that R has full row rank, we obtain that ${ext}_{D}^{0} (N, D) ≅ {ker}_{D} (. R) = 0$ , which is equivalent to say that N is a torsion D-module. Now computing the syzygy modules ${ker}_{D} (. P_{2}^{T})$ and ${ker}_{D} (. P_{3}^{T})$ by means of Gröbner or Janet bases, we obtain that

{ker}_{D} (. P_{2}^{T}) = D R, {ker}_{D} (. P_{3}^{T}) = D^{1 \times 3} P_{2}^{T},

which shows that ${ext}_{D}^{1} (N, D) = {ext}_{D}^{2} (N, D) = 0$ . Finally, we can easily check that 1 does not belong to the ideal $I = D \partial_{1} + D \partial_{2} + D \partial_{3}$ of D, and thus, we have:

{ext}_{D}^{3} (N, D) ≅ D / I \neq 0 .

Using Theorem 3, we obtain that M is a reflexive but not a projective, i.e., not a free D-module. This last fact can also be checked as R has full row rank and the dimension $\dim_{D} (N)$ is 0 as the corresponding system is defined by the gradient operator, namely,

\{\begin{matrix} \partial_{1} y = 0, \\ \partial_{2} y = 0, \\ \partial_{3} y = 0, \end{matrix}

whose solution is a constant, i.e., the solution of the system only depends on ``a function of zero independent variables´´. Hence, by Theorem 4, we obtain that $j_{D} (N) = 3$ , meaning that the first non-zero ${ext}_{D}^{i} (N, D)$ has index 3. By Theorem 3, we then get that M is a reflexive D-module but not a projective one.

Finally, if we consider the D-module $F = C^{\infty} (Ω)$ , where Ω is an open convex subset of $R^{3}$ , using Example 1, we obtain that $F$ is an injective cogenerator D-module. Hence, if we apply the functor ${hom}_{D} (\cdot, F)$ to the complex (7), we then obtain the following exact sequence:

F \overset{P_{3}^{T} .}{\to} F^{3} \overset{P_{2}^{T} .}{\to} F^{3} \overset{R .}{\to} F \to 0 .

We find again the classical results in mathematical physics that the smooth solutions on an open convex subset of $R^{3}$ of the divergence operator are parametrized by the curl operator and the solutions of the curl operator are parametrized by the gradient operator.

The only point let open is to constructively compute injective parametrizations of linear functional systems defining free modules over a commutative polynomial ring D. Indeed, checking the vanishing of the

{ext}_{D}^{i} (N, D)

, we generally obtain a successive chain of n parametrizations but not an injective one. In the case of linear systems of partial differential equations with polynomial or rational coefficients, we have recently solved this problem in [53], [54], [55] using a constructive proof of a famous result in non-commutative algebra due to Stafford. However, the same technique cannot be used if we want an injective parametrization Q of

{ker}_{F} (R .)

to have only constant coefficients. The main purpose of this paper is to solve this problem using a constructive proof of the Quillen-Suslin theorem and to show some applications of this result to mathematical systems theory.

3. The Quillen-Suslin theorem

Since Quillen and Suslin independently proved Serre´s conjecture stating that projective modules over commutative polynomial rings with coefficients in a field are free, some algorithmic versions of the proof have been proposed in the literature in order to constructively compute bases of free modules ([15], [19], [27], [29], [37], [59], [60], [61], [62]). We refer the interested reader to Lam´s nice books [24], [25] concerning Serre´s conjecture.

3.1. Projective and stably free modules

In module theory, it is well-known that a finitely presented

D = k [x_{1}, ..., x_{n}]

-module (k is a field)

M = D^{1 \times p} / (D^{1 \times q} R)

, where

R \in D^{q \times p}

, admits a finite free resolution. This is a result is due to Hilbert ([11]). Moreover, if k is a computable field, we can even construct a finite free resolution of M using Gröbner or Janet basis ([3], [11], [20]).

A classical result due to Serre proves that every projective D-module is stably free (a stably free module always being a projective D-module). See [11], [24], [25] for more details. In [53], [55], a constructive proof of this result was given and the corresponding algorithm was implemented in OreModules. Let us recall these useful results.

Proposition 2 ( [53], [55])

Let M be a D-module defined by the finite free resolution:

0 \to D^{1 \times p_{m}} \overset{. R_{m}}{\to} ... \overset{. R_{2}}{\to} D^{1 \times p_{1}} \overset{. R_{1}}{\to} D^{1 \times p_{0}} \overset{π}{\to} M \to 0 .

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If $m \geq 3$ and there exists $S_{m} \in D^{p_{m - 1} \times p_{m}}$ such that $R_{m} S_{m} = I_{p_{m}},$ then we have the following finite free resolution of M

$0 \to D^{1 \times p_{m - 1}} \overset{. T_{m - 1}}{\to} D^{1 \times (p_{m - 2} + p_{m})} \overset{. T_{m - 2}}{\to} D^{1 \times p_{m - 3}} \overset{. R_{m - 3}}{\to} ... \overset{π}{\to} M \to 0,$ (9)

with the following notations:

$\{\begin{matrix} T_{m - 1} = (R_{m - 1} S_{m}) \in D^{p_{m - 1} \times (p_{m - 2} + p_{m})}, \\ T_{m - 2} = (\begin{matrix} R_{m - 2} \\ 0 \end{matrix}) \in D^{(p_{m - 2} + p_{m}) \times p_{m - 3}} . \end{matrix}$
If $m = 2$ and there exists $S_{2} \in D^{p_{1} \times p_{2}}$ such that $R_{2} S_{2} = I_{p_{2}},$ then we have the following finite free resolution of M

$0 \to D^{1 \times p_{1}} \overset{. T_{1}}{\to} D^{1 \times (p_{0} + p_{2})} \overset{τ}{\to} M \to 0,$ (10)

with the notations $T_{1} = (R_{1} S_{2}) \in D^{p_{1} \times (p_{0} + p_{2})}$ and:

$\begin{matrix} τ = π \oplus 0 : D^{1 \times (p_{0} + p_{2})} & \to & M \\ λ = (λ_{1} λ_{2}) & \mapsto & τ (λ) = π (λ_{1}) . \end{matrix}$

Let

R \in D^{q \times p}

and let us suppose that the D-module

M = D^{1 \times p} / (D^{1 \times q} R)

is projective (using the results summed up in Figure 1, we can constructively check this result). Using 1 of Proposition 1, we obtain that the exact sequence (8) splits (see 5 of Definition 2), and thus, there exists

S_{m} \in D^{p_{m - 1} \times p_{m}}

such that

R_{m} S_{m} = I_{p_{m}}

. Repeating inductively the same method with the new finite free resolution of M, we can assume that we have the finite free resolution of M:

As M is a projective D-module, by 1 of Proposition 1, the previous exact sequence splits and thus, there exists a matrix

S_{3}^{'} \in D^{p_{2}^{'} \times p_{3}^{'}}

satisfying

R_{3}^{'} S_{3}^{'} = I_{p_{3}^{'}}

. By 1 of Proposition 2, we then get the finite free resolution of M:

Let us denote by

T_{1} = (R_{1}^{T} 0^{T})^{T}

. Again, as M is a projective D-module, by 1 of Proposition 1, the previous exact sequence splits and there exists

S_{2}^{'} \in D^{(p_{1} + p_{3}^{'}) \times p_{2}^{'}}

such that

(R_{2}^{'} S_{3}^{'}) S_{2}^{'} = I_{p_{2}^{'}}

. Using 2 of Proposition 2, we obtain the following finite free presentation of the D-module

M^{'} = D^{1 \times (p_{0} + p_{2}^{'})} / (D^{1 \times (p_{1} + p_{3}^{'})} (T_{1} S_{2}^{'}))

where

π^{'}

denotes the standard projection on

M^{'}

and

τ : M^{'} \to M

is defined by

τ (m) = π (λ_{1})

, for all

λ = (λ_{1} λ_{2}) \in D^{1 \times (p_{0} + p_{2})}

satisfying

m = π^{'} (λ)

. Moreover, 2 of Proposition 1 says that τ is an isomorphism, i.e.,

M^{'} ≅ M

, a fact that can be also directly checked. We then obtain the following result.

We refer to Example 14 for an illustration of Corollary 2. See also [53], [54], [55].

and using the fact that

M^{'} ≅ M

and M is a projective D-module, by 1 of Proposition 1, we obtain that the previous exact sequence splits and we then get ([5], [57])

which, by 2 of Definition 3, shows that

M ≅ M^{'}

is a stably free D-module.

3.2. Stably free and free modules

Let M be a stably free module over

D = k [x_{1}, ..., x_{n}]

, where k is a field. Using Corollary 2, we can always suppose that M has the form

M = D^{1 \times p} / (D^{1 \times q} R)

, where

R \in D^{q \times p}

admits a right-inverse

S \in D^{p \times q}

. We note that R has then full row rank (

λ R = 0 \Rightarrow λ = λ R S = 0

). Let us characterize when M is a free D-module.

In the case where

D = k [x_{1}, ..., x_{n}]

, we note that

U \in {GL}_{p} (D)

iff the determinant

det U

of U is invertible in D, i.e., is a non-zero element of k. The following result holds for every (commutative) ring D.

Indeed, let us suppose that there exists

U \in {GL}_{p} (D)

such that

R U = (I_{q} 0)

and let us denote by

J = (I_{q} 0) \in D^{q \times p}

. We easily check that

D^{1 \times p} / (D^{1 \times q} J) = D^{1 \times (p - q)}

. Moreover, using the facts that

R U = J

and

U \in {GL}_{p} (D)

, we obtain the following commutative exact diagram

which proves that

M ≅ D^{1 \times (p - q)}

, i.e., M is a free D-module of rank

p - q

Conversely, let us suppose that

M ≅ D^{1 \times (p - q)}

. Combining the isomorphism

ψ : M \to D^{1 \times (p - q)}

and the short exact sequence

If we consider the matrix which corresponds to the D-morphism

ψ \circ π

in the canonical bases of

D^{1 \times p}

and

D^{1 \times (p - q)}

, we then obtain a matrix

Q \in D^{p \times (p - q)}

such that:

or, equivalently, there exists a matrix

T \in D^{(p - q) \times p}

such that the following Bézout identities hold (see [5], [44], [50], [52], [57] for more details):

In particular, we obtain that there exists a matrix

U = (S Q) \in {GL}_{p} (D)

satisfying:

Finally, the family

{π (T_{i})}_{1 \leq i \leq p - q}

forms a basis of the free D-module M, where

T_{i}

denotes the i^th row of

T \in D^{(p - q) \times p}

We are now in position to state the famous Quillen-Suslin theorem ([24], [25], [57]).

Moreover, the problem of finding a basis of a free finitely generated D-module M can be reformulated in terms of matrices as follows:

The previous problem is equivalent to completing R to a square invertible matrix:

The Quillen-Suslin theorem states that Problem 1 has always a solution over a polynomial ring

D = A [x_{1}, ..., x_{n}]

with coefficients in a principal ring A and, in particular, in a field k. In what follows, an algorithm which computes such a matrix U will be called a QS-algorithm.

Let us consider a matrix

R \in D^{q \times p}

which admits a right-inverse over D and let us denote by

R_{i}

the i^th row of R. We note that the row

R_{1} \in D^{1 \times p}

admits a right-inverse over D. Let us suppose that we can find a matrix

U_{1} \in {GL}_{q} (D)

satisfying

R_{1} U = (1 0 ... 0) .

Then, we have

where

R_{2} \in D^{(q - 1) \times (p - 1)}

and ☆ denotes a certain element of

D^{(q - 1) \times 1}

. Hence, we restrict our considerations to the new matrix

R_{2}

, which can easily be shown to admit a right-inverse over D, and reduce Problem 1 to the following one:

The purpose of the next paragraphs is to recall a QS-algorithm solving Problem 2 over a commutative polynomial ring

D = k [x_{1}, ..., x_{n}]

over a computable field k (for instance,

k = Q

). This algorithm was implemented in the package QuillenSuslin ([13]). See also the Appendix. We also point out that a QS-algorithm has also been implemented in QuillenSuslin for the case

D = Z [x_{1}, ..., x_{n}]

. Even though there are some differences in the constructive proofs of the Quillen-Suslin theorem developed in [15], [19], [24], [27], [37], [59], [60], [62], we note that our algorithm is based on the same main idea, i.e., it proceeds by induction on the number of variables

x_{i}

D = k [x_{1}, ..., x_{n}]

. Each inductive step of the general QS-algorithm reduces the problem to the case with one variable less. A more global and interesting approach has recently been developed in [29], [61] which needs to be studied with care in the future.

3.3. Solution of Problem 2 in some special cases

Although the tedious inductive method, which will be explained in the next section, cannot generally be avoided, there are cases where simpler and faster heuristic methods can be used. We shall first consider such cases.

3.3.1. Matrices over a principal ideal domain D

We first consider the special case of matrices over a principal ideal domain D (e.g.,

D = k [x_{1}]

, k a field,

Z

). Let

R \in D^{q \times p}

be a matrix which admits a right-inverse over D. Then, computing the Smith normal form of R ([42]), we obtain two matrices

F \in {GL}_{q} (D)

and

G \in {GL}_{p} (D)

satisfying:

If we denote by

r = p - q

G = (G_{1}^{T} G_{2}^{T})^{T}

, where

G_{1} \in D^{q \times p}

G_{2} \in D^{r \times p}

and

G^{- 1} = (H_{1} H_{2})

, where

H_{1} \in D^{p \times q}

H_{2} \in D^{p \times r}

, then we get

R = F G_{1}

, i.e.,

G_{1} = F^{- 1} R

, and thus, we get

which solves Problem 1 as we can take

U = (H_{1} F H_{2}) \in {GL}_{p} (D)

and

T = G_{2}

3.3.2.

(p - 1) \times p

matrices over an arbitrary commutative ring D

Let us consider the case of a matrix

R \in D^{(p - 1) \times p}

which admits a right-inverse over a commutative ring D. If we denote by

m_{i}

the

(p - 1) \times (p - 1)

minor of R obtained by removing the i ^th column of R, then, using the fact R admits right-inverse, we get that the family

{m_{i}}_{1 \leq i \leq p}

satisfies a Bézout identity

\sum_{i = 1}^{p} n_{i} m_{i} = 1

for certain

n_{i} \in D

and

i = 1, ..., p

. Let us denote by:

Expand the determinant of V along the last row, using the Laplace´s formula, we then get

det V = 1

. Hence, if we denote by

U \in D^{p \times p}

the inverse of the matrix V, we then obtain

R U = (I_{p - 1} 0)

, which solves Problem 1.

3.3.3.

1 \times p

rows over an arbitrary commutative ring D

We now consider Problem 2, i.e., the case of a row vector

f = (f_{1} ... f_{p}) \in D^{1 \times p}

which admits a right-inverse over an arbitrary commutative ring D.

Remark 3 (Special form of the row)

We note that if one of the components of f is an invertible element of D, we can then transform the row f into $(1 0 ... 0)$ by means of trivial elementary operations. For instance, if $f_{1}^{- 1} \in D$ , then the matrix defined by

$W = (\begin{matrix} f_{1}^{- 1} & 0 \\ 0 & I_{p - 1} \end{matrix})$

satisfies $det W = f_{1}^{- 1} \in D$ and $f W = (1 f_{2} ... f_{p})$ . Then, simple elementary operations transform $f W$ into the vector $(1 0 ... 0)$ .
Another simple case is when two components of f generate D. Let us suppose that there exist $h_{1}$ and $h_{2} \in D$ such that we have the Bézout identity $f_{1} h_{1} + f_{2} h_{2} = 1$ and let us define the following matrix:

$W = (\begin{matrix} h_{1} & - f_{2} & 0 \\ h_{2} & f_{1} & 0 \\ 0 & 0 & I_{p - 2} \end{matrix}) .$

We easily check that $det W = 1$ and $f W = (1 0 f_{3} ... f_{p})$ . Then, we can reduce $f W$ to $(1 0 ... 0)$ by means of elementary operations.
If the i^th component of f is 0 or the ideal generated by the elements $f_{1}, ..., f_{i - 1},$ $f_{i + 1}, ..., f_{p}$ is already D, then we can follow an idea analogous to the one developed in [53], [55]. Let us suppose that $i = 1$ , i.e., $f_{1}$ is a redundant component in the sense that $(f_{2}, ..., f_{n}) = D$ . Then, there exist $h_{2}, ..., h_{p} \in D$ satisfying the Bézout equation $\sum_{i = 2}^{p} f_{i} h_{i} = 1$ . Then, the matrix

$W = (\begin{matrix} 1 \\ (1 - f_{1}) h_{2} & 1 \\ ⋮ & ⋱ \\ (1 - f_{1}) h_{p} & 1 \end{matrix}) .$

satisfies $f W = (1 f_{2} ... f_{p})$ and $det W = 1$ . We can now reduce $f W$ to $(1 0 ... 0)$ by means of elementary operations.

In particular, this strategy is always successful when the length p of the row f exceeds the stable range of the ring D. We note that the stable range of $D = R [x_{1}, ..., x_{n}]$ is equal to $n + 1$ . We refer the reader to [53], [55] for more details.

We note that all the conditions given in Remark 3 can be checked using Gröbner or Janet bases.

The matrix U can also be easily computed in cases where a right-inverse g of the row f has a special form.

Remark 4 (Special form of the right-inverse)

Let $g \in D^{p \times 1}$ be the right-inverse of the unimodular row $f \in D^{1 \times p}$ , i.e., $f g$ =1.

Let us suppose that one of the entries of a right-inverse g of f, say $g_{1}$ , is invertible in D. Then, the following matrix

$W = (\begin{matrix} g_{1} \\ g_{2} & 1 \\ ⋮ & ⋱ \\ g_{p} & 1 \end{matrix})$

satisfies $det W = g_{1}$ and $f W = (1 f_{2} ... f_{p})$ . As $g_{1}$ is an invertible element of D, then W is a unimodular matrix and $f W$ can easily be transformed into $(1 0 ... 0)$ by means of elementary operations.
If two components $g_{1}, g_{2}$ of g generate the whole ring D, then there exist elements $h_{1}, h_{2} \in D$ such that $g_{1} h_{1} + g_{2} h_{2} = 1$ . Then, the matrix defined by

$W = (\begin{matrix} g_{1} & - h_{2} \\ g_{2} & h_{1} \\ g_{3} & 1 \\ ⋮ & ⋱ \\ g_{p} & 1 \end{matrix})$

satisfies $det W = 1$ and $f W = (1 ☆ f_{3} ... f_{p})$ , where ☆ denotes a certain element of D. We can then reduce $f W$ to $(1 0 ... 0)$ by means of elementary operations.

Finally, we also note that if

f \in D^{1 \times p}

admits a right-inverse g over D for which any of the heuristic methods explained in Remark 3 may be used for

g^{T}

, then a unimodular matrix V having

g^{T}

as a first row can be easily computed. Then, the product

f V^{T} = (1 ☆ ... ☆)

can be reduced to the first standard basis vector by elementary column operations.

For instance, let us illustrate 1 of Remark 4. In some of the illustrating examples, we shall also use the notation

D = k [z_{1}, ..., z_{n}]

as these examples come from the control theory and signal processing literatures where

z_{i}

is commonly used. The independent variables

z_{i}

i = 1, ..., n

, usually denote the variables appearing in the discrete Laplace transform.

Example 4

Let us consider $D = Q [z_{1}, z_{2}, z_{3}]$ and the following row vector:

R = (z_{1}^{2} z_{2}^{2} + 1 z_{1}^{2} z_{3} + 1 z_{1} z_{2}^{2} z_{3}) .

We can easily check that R admits the following right-inverse $S = (- z_{1}^{2} z_{3} 1 z_{1}^{3})^{T}$ . As the second component of S is invertible over D, we can apply 1 of Remark 4 in order to find a unimodular matrix U over D which satisfies $R U = (1 0 0)$ . Let us define the following elementary matrices:

U_{1} = (\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}), U_{2} = (\begin{matrix} 1 & 0 & 0 \\ - z_{1}^{2} z_{3} & 1 & 0 \\ z_{1}^{3} & 0 & 1 \end{matrix}) .

We then have $R (U_{1} U_{2}) = (1 z_{1}^{2} z_{2}^{2} + 1 z_{1} z_{2}^{2} z_{3})$ . Finally, if we denote by

U_{3} = (\begin{matrix} 1 & - z_{1}^{2} z_{2}^{2} - 1 & - z_{1} z_{2}^{2} z_{3} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}),

we then have $R U = (1 0 0)$ , where the unimodular $U = U_{1} U_{2} U_{3}$ is defined by:

U = (\begin{matrix} - z_{1}^{2} z_{3} & z_{1}^{4} z_{2}^{2} z_{3} + z_{1}^{2} z_{3} + 1 & z_{1}^{3} z_{2}^{2} z_{3}^{2} \\ 1 & - z_{1}^{2} z_{2}^{2} - 1 & - z_{1} z_{2}^{2} z_{3} \\ z_{1}^{3} & - z_{1}^{3} (z_{1}^{2} z_{2}^{2} + 1) & - z_{1}^{4} z_{2}^{2} z_{3} + 1 \end{matrix}) .

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3.4. A QS-algorithm for commutative polynomial rings

Over an arbitrary commutative ring A, not every row admitting a right-inverse can be completed to a unimodular matrix over A. The module-theoretic interpretation of this result is that, over certain rings, there exist stably free modules which are not free. For instance, using a classical topological theorem on vector fields on the sphere

S_{2} (R)

, we can prove that the row vector

R = (x_{1} x_{2} x_{3})

with entries in the commutative ring

D = R [x_{1}, x_{2}, x_{3}] / (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} - 1)

, which admits the right-inverse

R^{T}

, cannot be completed to a unimodular matrix over D. For more details, see [25].

However, it is always possible over a polynomial ring with coefficients on a field k or a principal ideal domain A. See Quillen-Suslin Theorem 6. We shortly describe a QS-algorithm which has recently been implemented in a package called QuillenSuslin ([13]). See the Appendix for more details. In what follows, we shall only consider a commutative polynomial ring

D = k [x_{1}, ..., x_{n}]

over a field k even if the extension of the algorithms exists when k is replaced by a principal ideal ring A. For instance, the case of

A = Z

has also been implemented in QuillenSuslin. Let

f \in D^{1 \times p}

be a row vector which admits a right-inverse g over D. When no method explained in Section 3.3 can be applied to f, we then need to consider a general algorithm. However, we point out that most of the examples we know do not require the general algorithm as the previous heuristic methods are generally enough to get the result.

The QS-algorithm proceeds by induction on the number n of independent variables

x_{i}

of the ring

D = k [x_{1}, ..., x_{n}]

. Each inductive step, which simplifies the problem to the case of a polynomial ring containing one variable less, consists of three main parts:

3.4.1. Normalization Step

The next lemma is essential for Horrocks´ theorem which is used in the local loop.

In the case where k is an infinite field, we can achieve the same result by using only a linear transformation whose coefficients are appropriately chosen ([60], [62]). The normalization step can also be generalized to the case

D = A [x_{1}, ..., x_{n}]

, where A is is a principal ideal domain. See [59] for more details.

3.4.2. Local Loop

In the second step, we need to compute a finite number of local solutions of Problem 2 over a local ring ring A, namely, a ring A which has only one maximal ideal, i.e., a proper ideal

M

of A which is not properly contained in any ideal of A other than A itself. In order to do that, we use the so-called Horrocks´ theorem. Let us recall it.

Horrocks´ theorem can easily be implemented using, for instance, the approaches developed in [27], [57], [62]. In particular, the implementation in QuillenSuslin of this theorem follows [57]. If

M

is a maximal ideal of D, we then denote by

D_{M}

the local ring, which is a standard localization of D with respect to the multiplicative closed subset

S = D \ M

of D, namely,

D_{M} = {a / b | a \in D, b \notin M}

([57]).

Algorithm 1

Input: Let $D = k [x_{1}, ..., x_{n}]$ and $f \in D^{1 \times p}$ a row vector which admits a right-inverse over D and a monic component in the last variable $x_{n}$ .
Output: A finite number of maximal ideals ${M_{i}}_{i \in I}$ of $E = k [x_{1}, ..., x_{n - 1}]$ and unimodular matrices ${H_{i}}_{i \in I}$ over the ring $E_{M_{i}} [x_{n}]$ which satisfy $f H_{i} = (1 0 ... 0),$ and such that the ideal generated by the denominators of the matrices $H_{i}$ , $i \in I$ , generate the ring E.

Let $M_{1}$ be an arbitrary maximal ideal of the ring E. Using Horrocks´ theorem, compute a unimodular matrix $H_{1}$ over $E_{M_{1}} [x_{n}]$ which satisfies $f H_{1} = (1 0 ... 0)$ .
Let $d_{1} \in E$ be the denominator of $H_{1}$ and J the ideal in E generated by $d_{1}$ . Set $i = 1$ .
While $J \neq E$ , do:
1. For $i : = i + 1$ , compute a maximal ideal $M_{i}$ of E such that $J \subset M_{i}$ .
2. Using Horrocks´ theorem, compute a matrix $H_{i}$ over the ring $E_{M_{i}} [x_{n}]$ such that $det H_{i}$ is invertible in $E_{M_{i}} [x_{n}]$ and $f H_{i} = (1 0 ... 0)$ .
3. Let $d_{i}$ be the denominator of the matrix $H_{i}$ and consider the ideal $J = (d_{1}, ..., d_{i})$ .
Return ${M_{i}}_{i \in I}$ , ${H_{i}}_{i \in I}$ and ${d_{i}}_{i \in I}$ .

The local loop stops when all the denominators

d_{i}

generate E. As the ring E is noetherian ([57]), the number of the local solutions, i.e., the cardinal of the set I, is finite.

3.4.3. Patching

(14) is clear as the identity

f (x_{1}, ..., x_{n}) U (x_{1}, ..., x_{n}) = (1 0 ... 0)

implies that we have

f (x_{1}, ..., x_{n} + z) U (x_{1}, ..., x_{n} + z) = (1 0 ... 0)

and then:

Moreover, using the standard formula

U^{- 1} = (det U)^{- 1} adj (U)

, where

adj (U)

denotes the adjugate of U, we can also prove that the common denominator of

Δ (x_{n}, z)

d^{α}

, where

0 \leq α \leq p

Let

{M_{i}}_{i \in I}

{H_{i}}_{i \in I}

and

{d_{i}}_{i \in I}

be the output of Algorithm 1, where I is a finite set. Let us set

I = {1, ..., l}

. The ideal of

E = k [x_{1}, ..., x_{n - 1}]

defined by

{d_{i}}_{i \in I}

generates E. Hence, there exists

c_{i} \in E

i \in I

, such that the Bézout identity holds:

and, in order to simplify the notations, we denote by

\tilde{f} (x_{n})

the function

f (x_{1}, ..., x_{n})

. Then, we have:

Finally, we can prove that we have

Δ_{i} (x_{n}, d_{i}^{p} z) \in {GL}_{p} (D)

i = 1, ..., l

, ([27]) and:

The previous computations then show that

f (x_{1}, ..., x_{n}) U_{1} = f (x_{1}, ..., x_{n - 1}, a_{n})

We consider a row vector

f (x_{1}, ..., x_{n}) \in D^{1 \times p}

admitting a right-inverse

g (x_{1}, ..., x_{n}) \in D^{p \times 1}

. Applying inductively Theorem 8 to

f (x_{1}, ..., x_{n})

for the values

a_{2}, ..., a_{n} \in k

, we then obtain

U_{1}, ..., U_{n - 1} \in {GL}_{p} (D)

such that

Hence, we get

f (x_{1}, ..., x_{n}) (U_{1} ... U_{n - 1}) = f (x_{1}, a_{2}, ..., a_{n})

and we have simplified Problem 2 to the case of a row vector

f (x_{1}, a_{2}, ..., a_{n})

over a principal ideal domain

k [x_{1}]

which admits a right-inverse

g (x_{1}, a_{2}, ..., a_{n})

over

k [x_{1}]

. Using the first result of Section 3.3, we can find a matrix

U_{n} \in {GL}_{p} (D)

such that:

Hence, Problem 2 is then solved if we take

U = U_{1} ... U_{n} \in {GL}_{p} (D)

. We also note that it is generally simpler to take the particular values

a_{2} = ... = a_{n} = 0

Now, let us find a matrix

U^{'}

satisfying

f (x_{1}, ..., x_{n}) U^{'} = f (a_{1}, ..., a_{n})

, where

a_{1} \in k

. Let us define by

U_{n}^{'} (x_{1}) = U_{n} (x_{1}) U_{n}^{- 1} (a_{1}) \in {GL}_{p} (D)

. Then, we have:

Hence, the matrix

U^{'} = U_{1} ... U_{n - 1} U_{n}^{'} \in {GL}_{p} (D)

satisfies:

Example 5

Let us consider the commutative polynomial ring $D = Q [x_{1}, x_{2}]$ and the row vector $R = (x_{1} x_{2}^{2} + 1 3 x_{2} / 2 + x_{1} - 1 2 x_{1} x_{2}) \in D^{1 \times 3}$ . We can check that $S = (1 0 - x_{1} / 2)^{T}$ is a right-inverse of R, a fact implying that the D-module $M = D^{1 \times 3} / (D R)$ is projective, and thus, free by the Quillen-Suslin theorem. Let us compute a matrix $U \in {GL}_{3} (D)$ such that $R U = (1 0 0)$ . As the first component of S is 1, we can easily find such a matrix U using the heuristic methods explained in Section 3.3. However, let us illustrate the main algorithm previously described.

We first note that R contains the normalized component $3 x_{2} / 2 + x_{1} - 1$ over $D = E [x_{2}]$ , where $E = Q [x_{1}]$ . The second step consists in computing certain local solutions. Let us consider the maximal ideal $M_{1} = (x_{1})$ of E. Using an effective version of Horrocks´ theorem, we obtain that

H_{1} = \frac{1}{d_{1}} (\begin{matrix} 4 & - 2 (3 x_{1} + 2 x_{2} - 2) & 4 x_{1} (3 x_{1} - 2) \\ 2 x_{1} (3 x_{1} - 2 x_{2} - 2) & 4 (x_{1} x_{2}^{2} + 1) & - 4 x_{1} (3 x_{1}^{2} x_{2} - 2 x_{1} x_{2} + 2) \\ 0 & 0 & 9 x_{1}^{3} - 12 x_{1}^{2} + 4 x_{1} + 4 \end{matrix}),

where $d_{1} = 9 x_{1}^{3} - 12 x_{1}^{2} + 4 x_{1} + 4 \notin M_{1}$ . We can check that $det H_{1} = 4 / d_{1}$ , i.e., $H_{1} \in {GL}_{3} (E_{M_{1}} [x_{2}])$ , and $R H_{1} = (1 0 0)$ , showing that $H_{1}$ is a local solution.

The ideal $J = (d_{1})$ is strictly contained in E. Therefore, we consider another maximal ideal $M_{2}$ such that $J \subseteq M_{2}$ . For instance, we can take $M_{2} = (9 x_{1}^{3} - 12 x_{1}^{2} + 4 x_{1} + 4)$ . Using an effective version of Horrocks´ theorem, we obtain the matrix

H_{2} = \frac{1}{d_{2}} (\begin{matrix} 0 & 0 & 4 x_{1} (3 x_{1} - 2) \\ 8 x_{1} & - 8 x_{1} x_{2} & - 4 x_{1} (3 x_{1}^{2} x_{2} - 2 x_{1} x_{2} + 2) \\ - 4 & 2 (3 x_{1} + 2 x_{2} - 2) & 9 x_{1}^{3} - 12 x_{1}^{2} + 4 x_{1} + 4 \end{matrix}),

where $d_{2} = 4 x_{1} (3 x_{1} - 2) \notin M_{2}$ . We then have $det H_{2} = - 1 / (x_{1} (3 x_{1} - 2))$ , i.e., $H_{2} \in {GL}_{3} (E_{M_{2}} [x_{2}])$ and $R H_{2} = (1 0 0)$ . We can check that the ideal $(d_{1}, d_{2}) = E$ as we have the Bézout identity $c_{1} d_{1} + c_{2} d_{2} = 1$ , where $c_{1} = 1 / 4$ and $c_{2} = - (3 x_{1} - 2) / 16$ .

The matrix $Δ_{i} (x_{2}, - c_{1} d_{1} x_{2})$ is defined by:

\begin{matrix} (\begin{matrix} (9 x_{1}^{4} / 4 - 3 x_{1}^{3} + x_{1}^{2}) x_{2}^{2} + (3 x_{1}^{2} / 2 - x_{1}) x_{2} + 1 \\ - (18 x_{1}^{4} - 24 x_{1}^{3} + 8 x_{1}^{2}) x_{1} x_{2}^{3} / 8 + (27 x_{1}^{5} - 54 x_{1}^{4} + 36 x_{1}^{3} - 20 x_{1}^{2} + 8 x_{1}) x_{1} x_{2}^{2} / 8 - x_{1} x_{2} \\ 0 \end{matrix} \\ \begin{matrix} - x_{2} & - 2 x_{1} x_{2} \\ x_{1} x_{2}^{2} + (- 3 x_{1}^{2} / 2 + x_{1}) x_{2} + 1 & 2 x_{1}^{2} x_{2}^{2} - x_{1}^{2} (3 x_{1} - 2) x_{2} \\ 0 & 1 \end{matrix}) . \end{matrix}

We can easily check that we have $R (x_{1}, x_{2}) Δ_{i} (x_{2}, - c_{1} d_{1} x_{2}) = R (x_{1}, x_{2} - c_{1} d_{1} x_{2})$ as well as $Δ_{1} (x_{2}, - c_{1} d_{1} x_{2}) \in {GL}_{3} (D)$ . Moreover, the matrix $Δ_{2} (x_{2} - c_{1} d_{1} x_{2}, - c_{2} d_{2} x_{2})$ is defined by:

(\begin{matrix} 1 & 0 & 0 \\ 0 & (3 x_{1}^{2} / 2 - x_{1}) x_{2} + 1 & x_{1}^{2} (3 x_{1} - 2) x_{2} \\ (9 x_{1}^{2} - 12 x_{1} + 4) x_{1} x_{2} / 8 & (- 3 x_{1} + 2) x_{2} / 4 & (- 3 x_{1}^{2} / 2 + x_{1}) x_{2} + 1 \end{matrix}) .

We can easily check that we have $R (x_{1}, x_{2} - c_{1} d_{1} x_{2}) Δ_{2} (x_{2} - c_{1} d_{1} x_{2}, - c_{2} d_{2} x_{2}) = R (x_{1}, 0)$ and $Δ_{2} (x_{2} - c_{1} d_{1} x_{2}, - c_{2} d_{2} x_{2}) \in {GL}_{3} (D)$ . Defining the matrix

U_{1} = Δ_{1} (x_{2}, - c_{1} d_{1} x_{2}) Δ_{2} (x_{2} - c_{1} d_{1} x_{2}, - c_{2} d_{2} x_{2}) \in {GL}_{3} (D),

we then get $R (x_{1}, x_{2}) U_{1} (x_{1}, x_{2}) = R (x_{1}, 0) = (1 3 x_{1} / 2 - 1 0)$ .

Finally, if we denote by

U_{2} = (\begin{matrix} 1 & - 3 x_{1} / 2 + 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) \in {GL}_{3} (D),

then, the matrix $R (x_{1}, 0)$ is then equivalent to $(1 0 0)$ , i.e., $R (x_{1}, 0) U_{2} = (1 0 0)$ . Hence, if we define the matrix $U = U_{1} U_{2} \in {GL}_{3} (D)$ , i.e.,

(\begin{matrix} (3 x_{1}^{2} / 2 - x_{1}) x_{2} + 1 & (- 9 x_{1}^{3} / 4 + 3 x_{1}^{2} - x_{1} - 1) x_{2} - 3 x_{1} / 2 + 1 & - 2 x_{1} x_{2} \\ (- 3 x_{1}^{3} / 2 + x_{1}^{2}) x_{2}^{2} - x_{1} x_{2} & (9 x_{1}^{4} / 4 - 3 x_{1}^{3} + x_{1}^{2} + x_{1}) x_{2}^{2} + (3 x_{1}^{2} / 2 - x_{1}) x_{2} + 1 & 2 x_{1}^{2} x_{2}^{2} \\ (9 x_{1}^{2} - 12 x_{1} + 4) x_{1} x_{2} / 8 & (- 27 x_{1}^{4} / 16 + 27 x_{1}^{3} / 8 - 9 x_{1}^{2} / 4 - x_{1} / 4 + 1 / 2) x_{2} & (- 3 x_{1}^{2} / 2 + x_{1}) x_{2} + 1 \end{matrix}),

we finally obtain $R U = (1 0 0)$ .

In the third point of Section 3.3, we saw that the case of a matrix

R \in D^{q \times p}

admitting a right-inverse over D can be solved by applying q times Theorem 8 on certain row vectors obtained during the process having smaller and smaller lengths. Hence, we obtain the following corollary.

We note that as the matrix

R (a_{1}, ..., a_{n})

has full row rank over a field k, there always exists a right-inverse

V \in k^{p \times q}

such that

R (a_{1}, ..., a_{n}) V = I_{q}

. Hence, we obtain that

R (U V) = (I_{q} 0)

, which also solves Problem 1. Another possibility is to first obtain a matrix

W \in {GL}_{p} (D)

such that

R (x_{1}, ..., x_{n}) W = R (x_{1}, a_{2}, ..., a_{n})

and then compute a Smith form of

R (x_{1}, a_{2}, ..., a_{n})

as we did for the row vector case.

3.4.4. Computation of bases of free modules

R \in D^{q \times p}

is a matrix which admits a right-inverse over D, then, in Section 3.2, we showed that a basis of the free D-module

M = D^{1 \times p} / (D^{1 \times q} R)

is defined by

{π (T_{i})}_{1 \leq i \leq (p - q)}

, where

π : D^{1 \times p} \to M

denotes the canonical projection on M and

T_{i}

is the i ^th row of the matrix

T \in D^{(p - q) \times p}

defined by:

Example 6

Let us consider again Example 5. If we consider $d_{i} = \partial / \partial x_{i}$ instead of $x_{i}$ , namely, $D = Q [d_{1}, d_{2}]$ , $R = (d_{1} d_{2}^{2} + 1 3 d_{2} / 2 + d_{1} - 1 2 d_{1} d_{2}) \in D^{1 \times 3}$ , denote by $x = (x_{1}, x_{2}, x_{3})$ and choose $F = C^{\infty} (R^{3})$ , we then obtain that the linear system of PDEs

{ker}_{F} (R .) = {y = (y_{1} y_{2} y_{3})^{T} \in F^{3} | d_{1} d_{2}^{2} y_{1} (x) + y_{1} (x) + \frac{3}{2} d_{2} y_{2} (x) + d_{1} y_{2} (x) - y_{2} (x) + 2 d_{1} d_{2} y_{3} (x) = 0}

admits the parametrization $(y_{1} (x) y_{2} (x) y_{3} (x))^{T} = Q (z_{1} (x) z_{2} (x))^{T}$ , where Q is the matrix of differential operators formed by the last two columns of the matrix U defined in Example 5 and $z = (z_{1} z_{2})^{T}$ is any arbitrary element of $F^{2}$ , i.e.:

\{\begin{matrix} y_{1} = (- \frac{9}{4} d_{1}^{3} + 3 d_{1}^{2} - d_{1} - 1) d_{2} z_{1} - \frac{3}{2} d_{1} z_{1} + z_{1} - 2 d_{1} d_{2} z_{2}, \\ y_{2} = (\frac{9}{4} d_{1}^{4} - 3 d_{1}^{3} + d_{1}^{2} + d_{1}) d_{2}^{2} z_{1} + (\frac{3}{2} d_{1}^{2} - d_{1}) d_{2} z_{1} + z_{1} + 2 d_{1}^{2} d_{2}^{2} z_{2}, \\ y_{3} = (- \frac{27}{16} d_{1}^{4} + \frac{27}{8} d_{1}^{3} - \frac{9}{4} d_{1}^{2} - \frac{1}{4} d_{1} + \frac{1}{2}) d_{2} z_{1} + (- \frac{3}{2} d_{1}^{2} + d_{1}) d_{2} z_{2} + z_{2} . \end{matrix}

Finally, if we denote by $T \in D^{2 \times 3}$ the matrix formed by the last two rows of the matrix $U^{- 1}$ , namely,

(\begin{matrix} d_{1} d_{2} & 1 & 0 \\ \frac{1}{4} (3 d_{1}^{2} - 2 d_{1}) d_{2}^{2} + \frac{1}{8} (- 9 d_{1}^{3} + 12 d_{1}^{2} - 4 d_{1}) d_{2} & \frac{1}{4} (3 d_{1} - 2) d_{2} & \frac{1}{2} (3 d_{1}^{2} - 2 d_{1}) d_{2} \end{matrix}),

we then have $T Q = I_{2}$ , i.e., the parametrization Q of ${ker}_{F} (R .)$ is injective.

Now, if

M = D^{1 \times p} / (D^{1 \times q} R)

is a projective D which is defined by a non full row rank matrix

R \in D^{q^{'} \times p^{'}}

, then, using Proposition 2, we first compute a full row rank matrix

R^{'} \in D^{q^{'} \times p^{'}}

satisfying

and we then apply the previous QS-algorithm to

R^{'} \in D^{q^{'} \times p^{'}}

to obtain

U \in {GL}_{p^{'}} (D)

such that

R^{'} U = (I_{q^{'}} 0)

. Let

S^{'} \in D^{p^{'} \times q^{'}}

Q^{'} \in D^{p^{'} \times (p^{'} - q^{'})}

T^{'} \in D^{(p^{'} - q^{'}) \times p^{'}}

be the matrices defined by:

We now need to precisely describe the isomorphism between M and

M^{'}

in order to get a basis of M from one of

M^{'}

. In order to do that, we take the same notations as the ones used at the end of Section 3.1, namely,

R_{1} = R

T_{1} = (R_{1}^{T} 0^{T})^{T}

R^{'} = (T_{1} S_{2}^{'})

p_{0} = p

p_{1} = q

q^{'} = p_{1} + p_{3}^{'}

p^{'} = p_{0} + p_{2}^{'}

. We first easily check that we have the following commutative exact diagram

where

X = (I_{q}^{T} 0^{T})^{T}

. Moreover, we also have the commutative exact diagram

where

Y = (I_{p_{0}}^{T} 0^{T})^{T}

Z = (I_{p_{1}}^{T} 0^{T})^{T}

and the isomorphism σ is defined by:

Hence, if we denote by

{f_{i}}_{1 \leq i \leq (p^{'} - q^{'})}

the standard basis of

D^{1 \times (p^{'} - q^{'})}

, using (15), we then obtain that

{σ (π^{'} (f_{i} T^{'})) = π (f_{i} (T^{'} Y))}_{1 \leq i \leq (p^{'} - q^{'})}

is a basis of M, i.e., a basis of M is defined by taking the residue classes of the rows of

(T^{'} Y) \in D^{(p^{'} - q^{'}) \times p_{0}}

where

S \in D^{p \times q}

is a generalized inverse of R, i.e., S satisfies

R S R = R

([44]). If we denote by

T^{'} = (T_{1}^{'} T_{2}^{'})

, where

T_{1}^{'} \in D^{(p^{'} - q^{'}) \times p}

and

T_{2}^{'} \in D^{(p^{'} - q^{'}) \times p_{2}}

and

Q^{'} = ((Q_{1}^{'})^{T} (Q_{2}^{'})^{T})^{T}

, where

Q_{1}^{'} \in D^{p \times (p^{'} - q^{'})}

and

Q_{2}^{'} \in D^{p_{2} \times (p^{'} - q^{'})}

, we then get

i.e., we need to select the first p columns of

T^{'}

and the first p rows of

Q^{'}

F

is a D-module, then applying the functor

{hom}_{D} (\cdot, F)

to the previous split exact sequence, by 2 of Proposition 1, we then obtain the following split exact sequence:

The system

{ker}_{F} (R .)

admits the injective parametrization

Q_{1}^{'}

, namely:

Remark 7

Let us consider $R \in D^{q \times p}$ and let us suppose that the D-modules ${im}_{D} (. R)$ , ${ker}_{D} (. R)$ and ${coim}_{D} (. R) ≜ D^{1 \times q} / {ker}_{D} (. R)$ are free. We now show how to use the previous results to compute a basis of these free D-modules:

A basis of ${im}_{D} (. R) = D^{1 \times q} R$ can be obtained as follows: we first compute the first syzygy D-module of ${im}_{D} (. R)$ and we obtain a matrix $R_{2} \in D^{r \times q}$ satisfying ${ker}_{D} (. R) = D^{1 \times r} R_{2}$ . Let us denote by $M_{2} = D^{1 \times q} / (D^{1 \times r} R_{2}) ≅ D^{1 \times q} R$ . Using the method previously described, we can compute a basis of the free D-module $M_{2}$ . We get $Q_{2} \in D^{q \times l}$ and $T_{2} \in D^{l \times q}$ such that we have the exact split sequence

$\begin{matrix} D^{1 \times r} & \overset{. R_{2}}{\to} & D^{1 \times q} & \overset{. Q_{2}}{\to} & D^{1 \times l} & \to 0, \\ \overset{. S_{2}}{\leftarrow} & \overset{. T_{2}}{\leftarrow} \end{matrix}$

where $S_{2} \in D^{q \times r}$ denotes a generalized inverse of $R_{2}$ . A basis of $D^{1 \times q} R$ is then given by the D-linearly independent rows of the matrix $T_{2} R \in D^{l \times p}$ and we have $D^{1 \times q} R = D^{1 \times l} (T_{2} R)$ .
Using the same notations as before, we have ${ker}_{D} (. R) = D^{1 \times r} R_{2}$ and a basis of the free D-module ${ker}_{D} (. R)$ can then be obtained by computing a basis of $D^{1 \times r} R_{2}$ as it was shown in the previous point.
Using again the same notations as in the first point, we get

${coim}_{D} (. R) = D^{1 \times q} / {ker}_{D} (. R) = D^{1 \times q} / (D^{1 \times r} R_{2}),$

and a basis of ${coim}_{D} (. R)$ can be computed using the general method previously described in this section.

To finish, all the algorithms presented in this section were implemented in the package QuillenSuslin ([13]). See the Appendix for more details and examples.

4. Flat multidimensional linear systems

4.1. Computation of flat outputs of flat multidimensional systems

Our first motivation to study and implement constructive versions of the Quillen-Suslin theorem was the computation of flat outputs and injective parametrizations of flat multidimensional linear systems and, particularly, differential time-delay systems. Let us first recall the main ideas of flat systems and their applications in control theory.

A non-linear ordinary differential control system defined by

\dot{x} = f (x, u)

is said to be flat if there exist some outputs y of the form

y = h (x, u, \dot{u}, ..., u^{(r)})

such that we have:

The outputs y is then called flat outputs of the control system

\dot{x} = f (x, u)

. See [16], [17] and the references therein for more details and references. We can prove that the trajectories of a flat system are in a one-to-one correspondence with those of a controllable linear ordinary differential system having an arbitrary state dimension but the same number of inputs, i.e., with those of a Brunovský canoncial system ([17]). We say that a flat non-linear system is Lie-Bäcklund equivalent to a controllable linear ordinary differential system ([17]). Controllable linear systems form the simplest class of systems studied in control theory and a large literature is developed for the analysis and the synthesis of this class of control systems. This result, as well as the fact that many classes of non-linear control systems commonly used in the literature were proved to be flat, has popularized this class of systems in the control theory community. The motion planning problem was shown to be easily tractable for flat systems and it was illustrated on several examples in the literature ([16], [17]). Finally, the fact that the trajectories of a flat non-linear systems are in a one-to-one correspondence with the ones of a linear controllable system can be used to construct feedback laws which stabilize a flat non-linear system around a given trajectory (tracking problem) ([16], [17]). See also [48] for applications to optimal control.

Unfortunately, no general algorithm is known for checking whether or not a non-linear control system is flat and for the computation of flat outputs despite many effort of the mathematical and control theory communities. We refer the reader to [67] for a historical account of the main developments of the underlying mathematical problem, the Monge problem, which was studied by Hadamard, Hilbert, Cartan and Goursat.

We illustrate these definitions on the model of a vertical take-off and landing aircraft considered in [17], namely,

where ε is a small parameter. It is proved in [17] that the smooth solutions of (16) can be parametrized by means of the following non-linear differential operator

where

y_{1}

and

y_{2}

are two arbitrary smooth functions satisfying the following condition:

Moreover, the arbitrary functions

y_{1}

and

y_{2}

can be expressed in terms of the system variables as follows:

Hence,

(y_{1}, y_{2})

is a flat output of the non-linear system (16) and its knowledge gives a way to generate the trajectories of (16). Finally, the flat ordinary differential system (16) is Lie-Bäcklund equivalent to the Brunovský linear system defined by

under the invertible transformation (

η_{1} = u_{1} - ε {\dot{θ}}^{2}

and

η_{2} = {\dot{η}}_{1}

The study of flat linear ordinary differential time-delay systems has recently been initiated in [18], [32]. As for non-linear ordinary differential systems, this class of systems shares some interesting mathematical properties which can be used to do motion planning and tracking as shown in [32] and the references therein on explicit examples. However, the theory of flat linear ordinary differential time-delay systems is still in its infancy and some concepts developed for non-linear ordinary differential systems seem to have no counterparts for this second class of systems. In particular, for flat linear differential time-delay systems, we can wonder which kind of linear systems could play a similar role as the one played by the linear controllable systems (or Brunovský systems) for flat non-linear systems. To answer this question, we first need to understand which kind of equivalence plays a similar role for differential time-delay linear systems as the one played by the Lie-Bäcklund equivalence for non-linear differential systems. To our knowledge, these important questions have not be tackled in the literature till now. This section aims at constructively answer these two questions.

As the differential time-delay systems is a particular class of multidimensional systems, we can define the concept of a flat multidimensional linear system in terms of the existence of an injective parametrization of the trajectories of the system ([5], [44], [65]).

In terms of the module-theoretic/behaviour approach recently developed for multidimensional linear systems ([5], [41], [34], [65], [66]), it means that the module M intrinsically associated with the multidimensional linear system is free over the commutative polynomial ring D of functional operators ([5], [16], [17], [32], [44]).

Using Proposition 3 and the Quillen-Suslin theorem (see 4 of Theorem 2), we then get the following important corollary.

When R has a full row rank, then, using Theorem 3, a constructive test for flatness of multidimensional linear systems with constant coefficients consists in checking if the

q \times q

minors of R do not simultaneously vanish on complex common zeros ([24], [62]). This last result can algorithmically be checked by computing a Gröbner or Janet basis of the ideal I of D generated by the

q \times q

minors of R and check whether or not

1 \in I

. We can also check whether or not R admits a right-inverse over D ([4], [5], [44]).

In the general case, using Theorem 3, the projectiveness of M can constructively be obtained by verifying the vanishing of

{ext}_{D}^{i} (N, D)

, for

i = 1, ..., n

, where N is the transposed D-module

N = D^{1 \times q} / (D^{1 \times p} R^{T})

. Other possibilities are to compute the so-called global dimension of M ([57]) by means of Proposition 2 and Corollary 2 as it was shown in [53], check whether or not R admits a generalized inverse S over D, i.e., check for the existence of a matrix

S \in D^{p \times q}

satisfying

R S R = R

([44]) or check some straightforward conditions on the so-called Fitting ideals of M as it is explained in [11].

However, we point out that, till now, there has been no easy way for obtaining the flat outputs of the system, i.e., the bases of the free D-module M. Hence, we are led to use constructive versions of the Quillen-Suslin theorem developed in the symbolic algebra community ([19], [27], [29], [37]) for computing a basis of the free D-module M. It was our first main purpose for developing the package QuillenSuslin ([13]). See the Appendix for more details and examples.

Example 7

Let us consider the following differential time-delay linear system ([32]):

\{\begin{matrix} {\dot{y}}_{1} (t) - y_{1} (t - h) + 2 y_{1} (t) + 2 y_{2} (t) - 2 u (t - h) = 0, \\ {\dot{y}}_{1} (t) + {\dot{y}}_{2} (t) - \dot{u} (t - h) - u (t) = 0 . \end{matrix}

(20)

Let us denote by $D = Q [\frac{d}{d t}, δ]$ the commutative ring of differential time-delay operators with rational coefficients, where $(d / d t) y (t) = \dot{y} (t)$ and $(δ y) (t) = y (t - h)$ , $h \in R_{+}$ . Let us also denote the matrix of functional operators defining (20) by:

R = (\begin{matrix} \frac{d}{d t} - δ + 2 & 2 & - 2 δ \\ \frac{d}{d t} & \frac{d}{d t} & - \frac{d}{d t} δ - 1 \end{matrix}) \in D^{2 \times 3} .

Using the algorithms developed in [5], [44] and implemented in the package OreModules ([4]), we obtain that R admits a right-inverse over D defined by

S = \frac{1}{2} (\begin{matrix} 0 & 0 \\ \frac{d}{d t} δ + 2 & - 2 δ \\ \frac{d}{d t} & - 2 \end{matrix}),

a fact proving that $M = D^{1 \times 3} / (D^{1 \times 2} R)$ is a projective, and thus, a free D-module by the Quillen-Suslin theorem (see 4 of Theorem 2).

Using a constructive version of the Quillen-Suslin theorem (see also the heuristic methods developed in [5], [44]), we obtain the following split exact sequence of D-modules

\begin{matrix} 0 \to & D^{1 \times 2} & \overset{. R}{\to} & D^{1 \times 3} & \overset{. Q}{\to} & D \to 0, \\ \overset{. S}{\leftarrow} & \overset{. T}{\leftarrow} \end{matrix}

(21)

where $T = (1 0 0)$ and $\begin{matrix} Q = \frac{1}{2} (\begin{matrix} 2 \\ - \frac{d^{2}}{d t^{2}} δ + \frac{d}{d t} δ^{2} - \frac{d}{d t} + δ - 2 \\ \frac{d}{d t} δ - \frac{d^{2}}{d t^{2}} \end{matrix}) \end{matrix}$ .

Using the split exact sequence (21), we can check that we have

M = D^{1 \times 3} / (D^{1 \times 2} R) ≅ (D^{1 \times 3} Q) = D,

i.e., we find again that M is a free D-module of rank 1.

Now, if $F$ is a D-module (e.g., $F = C^{\infty} (R)$ ), by applying the functor ${hom}_{D} (\cdot, F)$ to the split exact sequence (21), we then obtain the following split exact sequence of D-modules (see 2 of Proposition 1):

\begin{matrix} 0 \leftarrow & F^{2} & \overset{R .}{\leftarrow} & F^{3} & \overset{Q .}{\leftarrow} & F \leftarrow 0 . \\ \overset{S .}{\to} & \overset{T .}{\to} \end{matrix}

Hence, for any D-module $F$ , we get that the system ${ker}_{F} (R .)$ defined by (20) is parametrized by the following injective parametrization:

\forall x_{1} \in F, \{\begin{matrix} y_{1} (t) = x_{1} (t), \\ y_{2} (t) = \frac{1}{2} (- {\ddot{x}}_{1} (t - h) + {\dot{x}}_{1} (t - 2 h) - {\dot{x}}_{1} (t) + x_{1} (t - h) - 2 x_{1} (t)), \\ u (t) = \frac{1}{2} ({\dot{x}}_{1} (t - h) - {\ddot{x}}_{1} (t)) . \end{matrix}

(22)

We refer the reader to [53], [55] for a constructive algorithm for the computation of bases, and thus, of flat outputs of a class of linear systems defined by partial differential equations with polynomial or rational coefficients. See [54], [53] for an implementation of this algorithm in the package Stafford of the library OreModules.

Finally, we say that the

D = k [x_{1}, ..., x_{n}]

-module

M = D^{1 \times p} / (D^{1 \times q} R)

is π-free, where

π \in D

, if the

D_{π}

-module

D_{π} \otimes_{D} M

is free, where

D_{π}

denotes the localization

of the ring D with respect to the multiplicatively closed subset

S_{π} = {1, π, π^{2}, ...}

of D ([57]). By extension, we can define the concept of a π-flat system. See [5], [32], [33] for more details. Given a finitely presented

D = k [x_{1}, ..., x_{n}]

-module

M = D^{1 \times p} / (D^{1 \times q} R)

, constructive algorithms computing the corresponding polynomials π and basis of the free

D_{π}

-module

D_{π} \otimes_{D} M

were given in [5] and implemented in the OreModules package ([4], [6]). However, we can also use Remark 5 to compute the corresponding basis in the case where

π = x_{i}

. We can also follow the simple idea developed in Section 9.4.3.

4.2. Equivalences of flat multidimensional systems

Using a QS-algorithm, the purpose of this section is to prove that a flat multidimensional linear system with constant coefficients is algebraically equivalent to a linear controllable 1-D system obtained by setting all but one functional operator to 0 in the system matrix. In particular, the algebraic equivalence we use is the natural equivalence developed in module theory, namely, two multidimensional linear systems are said to be algebraically equivalent if their canonical associated modules are isomorphic over the underlying commutative polynomial ring of functional operators D. This equivalence is nothing else than the natural substitute to the Lie-Bäcklund equivalence for multidimensional linear systems. In the case of ordinary differential linear systems, we already know that Lie-Bäcklund transformations correspond to isomorphisms of the underlying modules (see e.g. [17] and the references therein). Finally, we prove that a flat differential time-delay linear system is algebraically equivalent to the controllable ordinary differential system without delays, namely, the system obtained by setting all the delay amplitudes to 0. This last system plays a similar role as the one played by the Brunovský canoncial form in the non-linear case.

Proof

Using Proposition 3, we obtain that $M = D^{1 \times p} / (D^{1 \times q} R)$ is a free D-module. Using the fact that R has full row rank, by Theorem 8, there exists a matrix $U \in {GL}_{p} (D)$ such that $R U = \bar{R}$ , where $\bar{R} = R (x_{1}, 0, ..., 0)$ . Therefore, we have the following commutative exact diagram

\begin{matrix} 0 & 0 & 0 \\ ↓ & ↓ & ↓ \\ 0 \to & D^{1 \times q} & \overset{. R}{\to} & D^{1 \times p} & \overset{π}{\to} & M & \to 0 \\ ∥ & ↓ . U & ↓ f \\ 0 \to & D^{1 \times q} & \overset{. \bar{R}}{\to} & D^{1 \times p} & \overset{κ}{\to} & M^{'} & \to 0, \\ ↓ & ↓ & ↓ \\ 0 & 0 & 0 \end{matrix}

where $κ : D^{1 \times p} \to M$ denotes the canonical projection onto M and the D-isomorphism $f : M \to M^{'}$ is defined by:

\forall m = π (λ), λ \in D^{1 \times p}, f (m) = κ (λ U) .

Applying the functor ${hom}_{D} (\cdot, F)$ to the previous commutative exact diagram and using the fact that horizontal exact sequences split because $M ≅ M^{'}$ is a free D-module, we then obtain the following commutative exact diagram:

\begin{matrix} 0 & 0 & 0 \\ ↑ & ↑ & ↑ \\ 0 \leftarrow & F^{q} & \overset{R .}{\leftarrow} & F^{p} & \overset{π^{☆}}{\leftarrow} & {ker}_{F} (R .) & \leftarrow 0 \\ ∥ & ↑ U . & ↑ f^{☆} \\ 0 \leftarrow & F^{q} & \overset{\bar{R} .}{\leftarrow} & F^{p} & \overset{κ^{☆}}{\leftarrow} & {ker}_{F} (\bar{R} .) & \leftarrow 0 . \\ ↑ & ↑ & ↑ \\ 0 & 0 & 0 \end{matrix}

The D-isomorphism $f^{☆} : {ker}_{F} (\bar{R} .) \to {ker}_{F} (R .)$ defined by:

\forall η \in {ker}_{F} (R .), f^{☆} (


Module M	${ext}_{D}^{i} (N, D)$	$\dim_{D} (N)$	Primeness


With torsion	$t (M) ≅ {ext}_{D}^{1} (N, D)$	$n - 1$	∅


Torsion-free	${ext}_{D}^{1} (N, D) = 0$	$n - 2$	Minor left-prime


Reflexive	${ext}_{D}^{i} (N, D) = 0,$	$n - 3$
	$i = 1, 2$

...	...	...	...

...	${ext}_{D}^{i} (N, D) = 0,$	0	Weakly zero
	$1 \leq i \leq n - 1$		left-prime


Projective	${ext}_{D}^{i} (N, D) = 0,$	-1	Zero left-prime
	$1 \leq i \leq n$

Applications of the Quillen-Suslin theorem to multidimensional systems theory

Abstract

French Abstract

Short Table of Contents

1. Introduction

1.1. Notation.

2. A module-theoretic approach to systems theory

3. The Quillen-Suslin theorem

3.1. Projective and stably free modules

3.2. Stably free and free modules

3.3. Solution of Problem 2 in some special cases

3.3.1. Matrices over a principal ideal domain D

3.3.2. (p–1)×p matrices over an arbitrary commutative ring D

3.3.3. 1×p rows over an arbitrary commutative ring D

3.4. A QS-algorithm for commutative polynomial rings

3.4.1. Normalization Step

3.4.2. Local Loop

3.4.3. Patching

3.4.4. Computation of bases of free modules

4. Flat multidimensional linear systems

4.1. Computation of flat outputs of flat multidimensional systems

4.2. Equivalences of flat multidimensional systems

3.3.2. $(p - 1) \times p$ matrices over an arbitrary commutative ring D

3.3.3. $1 \times p$ rows over an arbitrary commutative ring D