A new insight into Serre s reduction problem

Transcription

1 A new insight into Serre s redution problem Thomas Cluzeau, Alban Quadrat RESEARCH REPORT N 8629 November 214 Projet-Team Diso ISSN ISRN INRIA/RR FR+ENG

2

3 A new insight into Serre s redution problem Thomas Cluzeau, Alban Quadrat Projet-Team Diso Researh Report n 8629 November pages Abstrat: The purpose of this paper is to study the onnetions existing between Serre s redution of linear funtional systems whih aims at finding an equivalent system defined by fewer equations and fewer unknowns and the deomposition problem whih aims at finding an equivalent system having a diagonal blok struture in whih one of the diagonal bloks is assumed to be the identity matrix. In order to do that, we further develop results on Serre s redution problem and on the deomposition problem obtained in Boudellioua and Quadrat (21); Cluzeau and Quadrat (28). Finally, we show how these tehniques an be used to analyze the deomposability problem of standard linear systems of partial differential equations studied in hydrodynamis suh as Stokes equations, Oseen equations and the movement of an inompressible fluid rotating with a small veloity around the vertial axis. Key-words: Mathematial systems theory, linear funtional systems, module theory, Serre s redution, the fatorization problem, the deomposition problem, appliations to hydrodynamis. Université de Limoges ; CNRS ; XLIM UMR 7252, DMI, 123 avenue Albert Thomas, 876 Limoges Cedex, Frane. thomas.luzeau@unilim.fr. INRIA Salay -Île-de-Frane, DISCO projet, Supéle, L2S, 3 rue Joliot Curie, Gif-sur-Yvette, Frane. alban.quadrat@ inria.fr. RESEARCH CENTRE SACLAY ÎLE-DE-FRANCE Par Orsay Université 4 rue Jaques Monod Orsay Cedex

4 Un nouveau point de vue sur le problème de rédution de Serre Résumé : Ce papier porte sur l étude des liens entre la rédution de Serre des systèmes fontionnels linéaires qui a pour but de trouver un système équivalent défini par moins d équations et moins d inonnues et le problème de déomposition qui a pour but de trouver un système diagonal par blos équivalent dans le as où l un des blos diagonaux est une matrie d identité. Pour ela, nous étendons des résultats obtenus dans Boudellioua and Quadrat (21); Cluzeau and Quadrat (28) sur la rédution de Serre et sur le problème de déomposition. Finalement, nous montrons omment es résultats peuvent être utilisés pour analyser le problème de la déomposabilité des systèmes linéaires d équations aux dérivées partielles lassiquement étudiés en hydrodynamique tels que les équations de Stokes, les équations d Oseen et le mouvement d un fluide inompressible en rotation à petite vitesse autour d un axe vertial. Mots-lés : Théorie mathématique des systèmes, systèmes fontionnels linéaires, théorie des modules, rédution de Serre, problème de fatorisation, problème de déomposition, appliations à l hydrodynamique.

5 A new insight into Serre s redution problem 3 Contents 1 Introdution 3 2 Algebrai analysis approah to linear funtional systems 5 3 Homomorphisms of finitely presented left D-modules 9 4 Fatorization problem 14 5 Deomposition problem General results The deomposition problem with an identity diagonal blok Serre s redution problem as a partiular deomposition problem Serre s redution From Serre s redution problem to the deomposition problem and vie versa Appliations to linear PD systems studied in hydrodynamis Oseen equations Impliit sheme for the Oseen equations and Stokes equations Fluid dynamis Appendix Wind tunnel model: deomposition Wind tunnel model: Serre s redution Oseen equations Impliit sheme for the Oseen equations Rotating fluid Introdution Mathematial systems theory aims at studying general systems defined by mathematial equations. These systems are usually defined by funtional equations, namely, systems whose unknowns are funtions, suh as ordinary differential (OD) or partial differential (PD) equations, differential time-delay equations, (partial) differene equations,... They an be linear, nonlinear, determined, overdetermined or underdetermined. A system an be studied by a broad spetrum of mathematial theories. For instane, mathematial models developed in natural sienes are usually studied by means of tehniques oming from mathematial physis, funtional analysis, probability and numerial analysis. There are at least two reasons for that. The first one is that it is generally diffiult to obtain purely analytial results for suh funtional systems. The seond one is the role that simulations play in nowadays life. Other funtional systems oming from mathematial physis, differential geometry, hamiltonian systems, algebrai geometry,... are usually studied by means of algebrai or differential geometry tehniques. More reently, the development of onstrutive versions of parts of pure mathematial theories (e.g., differential algebra, algebrai geometry, differential geometry, module theory, homologial algebra) and their implementations in effiient omputer algebra systems allow one to develop a more analyti study of ertain funtional systems studied, for instane, in ontrol theory and in mathematial physis. The questions raised in this approah are the intrinsi study of these systems, i.e., the study of their built-in properties, their symmetries and their solutions, the omputation of partiular forms for the systems (e.g., formal integrable forms, Gröbner or Janet bases, blok triangular forms, blok diagonal forms, equidimensional deomposition), of onservation laws,... This intrinsi study leads to important information on the system (e.g., dimension of the solution spae, invariants, asade integration, deoupling), the omputation of partiular solutions (e.g., exponential, hypergeometri, parametrizations),... RR n 8629

6 4 Cluzeau & Quadrat Following the latter approah, the purpose of this paper is to further develop ertain results obtained in Cluzeau and Quadrat (28) and Boudellioua and Quadrat (21) whih study the existene of fatorizations of a matrix of funtional operators defining a linear funtional system, the existene of equivalent blok diagonal forms for the system and the existene of equivalent forms defined by fewer unknowns and fewer equations than the original system. More preisely, if GL r (D) is the group formed by the r r matries with entries in a ring D whih are invertible and R D q p is a matrix whih defines the system equations R η =, where η F p is a vetor of unknown funtions whih belong to a funtional spae F having a left D-module struture, then the so-alled fatorization problem, deomposition problem and Serre s redution problem are respetively defined by: 1. Find R D r p and R D q r suh that R = R R. 2. Find V GL q (D) and W GL p (D) suh that ( R1 V R W = R 2 ) for ertain matries R 1 D s t and R 2 D (q s) (p t). 3. Find V GL q (D) and W GL p (D) suh that ( Is V R W = R 2 ) for a ertain matrix R 2 D (q s) (p s), where I s denotes the identity matrix of GL s (D). To do that, we study linear funtional systems within the algebrai analysis approah (also alled D-module theory) developed by Malgrange, Bernstein, Sato, Kashiwara... See Hotta et al. (28); Kashiwara (1995); Malgrange (1962); Quadrat (21) and the referenes therein. In this approah, a linear funtional system R η = is studied by means of the left D-module M finitely presented by the matrix R D q p whih defines the system equations and whose entries belong to a nonommutative polynomial ring D of funtional operators. Using the reent development of Gröbner or Janet basis tehniques for ertain lasses of nonommutative polynomial rings of funtional operators (Chyzak et al. (25)), results of algebrai analysis, using module theory and homologial algebra, were made algorithmi in Chyzak et al. (25); Cluzeau and Quadrat (28); Quadrat (21) and implemented in the OreModules and OreMorphisms pakages (Chyzak et al. (27); Cluzeau and Quadrat (29)). In this paper, we first omplete some of the main results obtained in Cluzeau and Quadrat (28). In partiular, we obtain a neessary and suffiient ondition for the existene of a non-trivial fatorization of the system matrix based on the onept of a non-generi solution developed in algebrai analysis. Even if this haraterization is not onstrutive, it generalizes a result obtained in Cluzeau and Quadrat (28) and gives another explanation to the well-known fat that, for a linear OD operator, the existene of a fatorization annot usually be deteted from the knowledge of the assoiated eigenring (see Barkatou (27); van der Put and Singer (23) and the referenes therein). We then onsider the deomposition problem and we obtain neessary and suffiient onditions for the existene of a diret deomposition of the module whih generalize a result obtained in Cluzeau and Quadrat (28). We study Serre s redution problem as a partiular ase of the deomposition problem, i.e., as the partiular ase where one of the diagonal blok is the identity matrix. We show how to use ertain homotopies of the trivial idempotents of the left D-module M, namely, of the and identity endomorphisms of M, and the solutions of an algebrai Riati equation (generalized inverses) to obtain neessary and suffiient onditions of Serre s redution. These onditions are then related to the ones obtained in Boudellioua and Quadrat (21) following Serre s ideas (see the referenes of Boudellioua and Quadrat (21)). In partiular, we state a orrespondene between these two approahes and show how to expliitly pass from one formulation to the other. Finally, we show how the above results an be used to prove that standard 2-dimensional linear PD systems studied in hydrodynamis (namely, Oseen equations and the movement of an inompressible fluid rotating with a small veloity around the vertial axis) are defined by indeomposable differential Inria

7 A new insight into Serre s redution problem 5 modules. These results give a mathematial proof that the matries of PD operators defining these systems are not equivalent to blok diagonal matries, and thus, that the equations of these systems annot be unoupled (whih would exhibit independent physial subphenomema). These results are obtained by proving that their endomorphism rings are yli differential modules that only admit the two trivial idempotents. The plan of the paper is the following. In Setion 2, we briefly review the main ideas of the algebrai analysis approah to linear systems theory. In Setion 3, we first present well-known results on the homomorphisms of finitely presented left modules and then study the multipliative struture of the endomorphism ring of a finitely presented module over a ommutative polynomial ring. In Setion 4, we omplete the results obtained in Cluzeau and Quadrat (28) on the fatorization problem. In Setion 5, we further develop the results of Cluzeau and Quadrat (28) on the deomposition problem. Serre s redution problem is first realled in Setion 6 and the main results are reviewed. We then study the deep onnetions existing between Serre s redution problem and the deomposition problem in the partiular ase when one of the diagonal bloks is the identity matrix. We exhibit a non-trivial orrespondene between the solutions of Serre s redution problem and those of the deomposition problem (whih are based on the solvability of an algebrai Riati equation). In Setion 7, we illustrate how the tehniques developed in this paper an be used to study the endomorphism ring of standard linear PD systems enountered in hydrodynamis and prove that some of these systems are indeomposable, i.e., that they annot be unoupled. Finally, the paper ends with Setion 8 whih is an appendix where the different omputations used in Setions 5, 6 and 7, and obtained by means of the OreMorphisms pakage, are given. Notation. In this artile, D will denote a left noetherian domain, namely a ring without zero divisors and whih is suh that every left ideal of D is finitely generated as a left D-module (see, e.g., Rotman (29)). When D is a (nonommutative) polynomial ring over a omputational field k, we shall further assume that Buhberger s algorithm terminates for any admissible term order and omputes a Gröbner basis (see Cluzeau and Quadrat (28) and referenes therein). Moreover, D q p denotes the D D- bimodule formed by the q p matries with entries in D. We simply note D p 1 by D p. The group of invertible matries of D p p is denoted by GL p (D). If M and N are two left D-modules, hom D (M, N) is the abelian group formed by the left D-homomorphisms (i.e., left D-linear maps) from M to N. If k is a field and D a k-algebra, then hom D (M, N) inherits a k-vetor spae struture. Two left D-modules M and N are said to be isomorphi, whih is denoted by M = N, if there exists an injetive and surjetive element of hom D (M, N). We denote by M N the diret sum of M and N (see, e.g., Rotman (29)). Finally, det(r) denotes the determinant of a square matrix R whose entries belong to a ommutative ring and diag(r 1, R 2 ) is the blok diagonal matrix formed by the matries R 1 and R 2. 2 Algebrai analysis approah to linear funtional systems In this paper, we study linear systems theory within the algebrai analysis framework (see Chyzak et al. (25); Cluzeau and Quadrat (28); Malgrange (1962); Quadrat (21) and the referenes therein). Let us briefly state again the main ideas of this approah. Let D be a left noetherian domain, F a left D-module and R D q p. A linear system, also alled behaviour in ontrol theory, is defined by the following abelian group: ker F (R.) := {η F p R η = }. If k is a field and D a k-algebra, then ker F (R.) inherits a k-vetor spae struture. To study the linear system ker F (R.), we first introdue the finitely presented left D-module M defined as the okernel of the following left D-homomorphism.R: D 1 q D 1 p λ λ R, i.e., with the notation im D (.R) := D 1 q R, defined by the following fator left D-module: M := D 1 p /(D 1 q R). RR n 8629

8 6 Cluzeau & Quadrat Let us explain why this module plays an important role in the algebrai analysis approah to linear systems theory. Let π hom D (D 1 p, M) be the left D-homomorphism sending λ D 1 p onto its residue lass π(λ) M (i.e., π(λ) = π(λ ) if and only if there exists µ D 1 q suh that λ = λ +µ R), {f j } j=1,...,p the standard basis of D 1 p, i.e., f j D 1 p is the vetor formed by 1 at the j th position and elsewhere, and y j := π(f j ) for j = 1,..., p. Then, every element m M is of the form m = π(λ) for a ertain λ = (λ 1... λ p ) D 1 p, whih yields m = π( p j=1 λ j f j ) = p j=1 λ i π(f j ) = p j=1 λ i y j and shows that {y j } j=1,...,p is a family of generators of M. These generators satisfy the following left D-linear relations: i = 1,..., q, p p p R ij y j = R ij π(f j ) = π(r ij f j ) = π((r i1... R ip )) =. j=1 j=1 j=1 If we note y := (y 1... y p ) T, then we have R y =. For more details, see Chyzak et al. (25); Cluzeau and Quadrat (28); Quadrat (21). Let M, M and M be left D-modules, f hom D (M, M) and g hom D (M, M ). If ker g = im f, f g then M M M is alled an exat sequene at M (see, e.g., Rotman (29)). If the above sequene is exat at M and if g =, then im f = M, i.e., f is surjetive, or if f =, then ker g =, i.e., g is injetive. By definition of M as the okernel of.r hom D (D 1 q, D 1 p ), we have the exat sequene D 1 q.r D 1 p π M, whih is alled a finite presentation of M. If we apply the ontravariant left exat funtor hom D (, F) (see, e.g., Rotman (29)) to the above exat sequene, we get the following exat sequene hom D (D 1 q (.R), F) hom D (D 1 p, F) π hom D (M, F), where (.R) (φ) = φ (.R) for all φ hom D (D 1 p, F) and π (ψ) = ψ π for all ψ hom D (M, F). For more details, see, e.g., Rotman (29). Using the isomorphism hom D (D 1 r, F) = F r defined by mapping the elements of the standard basis of D 1 r to elements of F, the above exat sequene yields the following exat sequene of abelian groups F q R. F p hom D (M, F), where (R.)(η) = R η for all η F p, whih finally shows that: ker F (R.) = {η F p R η = } = hom D (M, F). (1) More preisely, we an easily show that φ hom D (M, F) yields η := (φ(y 1 )... φ(y p )) T ker F (R.), where {y j } j=1,...,p is the family of generators of M defined as above, and if η ker F (R.), then φ η (π(λ)) := λ η for all λ D 1 p is a left D-homomorphism from M to F. See, e.g., Chyzak et al. (25). The isomorphism (1) shows that the linear system ker F (R.) an be studied by means of M and F (Malgrange (1962)). The finitely presented left D-module M enodes the algebrai side (i.e., the linear equations) of ker F (R.) and the left D-module F is the (funtional) spae in whih the solutions are sought. Example 1. A ommutative ring A is alled a differential ring if A is equipped with n ommuting derivations i, i = 1,..., n, i.e., maps i : A A satisfying i (a 1 + a 2 ) = i a 1 + i a 2, i (a 1 a 2 ) = ( i a 1 ) a 2 + a 1 i a 2 (Leibniz rule) for all a 1, a 2 A, and j i a = i j a for all a A. A differential field K is a field K endowed with a differential ring struture (whih yields i a 1 = a 2 i a). The ring D := A d 1,..., d n of PD operators with oeffiients in a differential ring (A, { i } i=1,...,n ) is defined as the nonommutative polynomial ring formed by elements of the form µ r a µ d µ, where a µ A, µ := (µ 1... µ n ) Z n, µ := µ µ n, d µ := d µ dµn n, and the d i s satisfy the relations: a A, i, j = 1,..., n, { di a = a d i + i a, d i d j = d j d i. Inria

9 A new insight into Serre s redution problem 7 If A := k[x 1,..., x n ] (resp., k(x 1,..., x n )), then the ring A d 1,..., d n is simply denoted by A n (k) (resp., B n (k)) and is alled the polynomial (resp., rational) Weyl algebra. Finally, if A := kx 1,..., x n (resp., A = k{x 1,..., x n }) is the integral domain of formal (resp., loally onvergent) power series in x 1,..., x n with oeffiients in the field k (resp., k = R or C), and Q(A) its quotient field, i.e., the ring of Laurent formal power series (resp., the ring of Laurent power series), then Q(A) d 1,..., d n is simply denoted by D n (k) (resp., D n (k)). If D = A d 1,..., d n is a ring of PD operators with oeffiients in a differential ring A, then R D q p is a q p matrix of PD operators. If F is a left D-module (e.g., F = A), then (1) shows that the solutions η F p of the PD system R η = are in a 1-1 orrespondene with the elements of hom D (M, F). See Chyzak et al. (25); MConnell and Robson (2) for other nonommutative polynomial algebras of funtional operators (e.g., time-delay or shift operators) suh as the Ore extensions and the Ore algebras. Let us briefly review a part of the lassifiation of finitely generated left D-modules, i.e., left D-modules whih an be defined by a finite number of generators. Definition 1 (Lam (1999); MConnell and Robson (2); Rotman (29)). Let D be a left noetherian domain and M a finitely generated left D-module. 1. M is free if there exists r Z suh that M = D 1 r. Then, r is alled the rank of the free left D-module M and is denoted by rank D (M). 2. M is stably free if there exist r, s Z suh that M D 1 s = D 1 r. Then, r s is alled the rank of the stably free left D-module M. 3. M is projetive if there exist r Z and a left D-module N suh that M N = D 1 r. 4. M is reflexive if the anonial left D-homomorphism ε : M M defined by ε(m)(f) = f(m) for all f M := hom D (M, D), where M = hom D (hom D (M, D), D), is an isomorphism (whih then yields M = M ). 5. M is torsion-free if the torsion left D-submodule of M, namely, t(m) := {m M d D \ {} : d m = }, is redued to, i.e., if t(m) =. The elements of t(m) are alled the torsion elements of M. 6. M is torsion if t(m) = M, i.e., if every element of M is a torsion element. 7. M is yli if there exists m M suh that M = D m := {d m d D}. 8. M is deomposable if there exist two proper left D-submodules M 1 and M 2 of M suh that: M = M 1 M 2. If M is not deomposable, then M is said to be indeomposable. 9. A non-zero left D-module M is alled simple if M has no non-zero proper left D-submodules. Similar definitions exist for finitely generated right D-modules. We refer to Chyzak et al. (25); Fabiańska and Quadrat (27); Quadrat and Robertz (27a) for algorithms whih test whether or not a finitely presented module M over some lasses of nonommutative polynomial rings admits a non-trivial torsion submodule, is torsion-free, projetive, stably free or free. These algorithms are implemented in the OreModules (Chyzak et al. (27)), QuillenSuslin (Fabiańska and Quadrat (27)) and Stafford (Quadrat and Robertz (27a)) pakages. A free module is learly stably free (take s = in 2 of Definition 1), a stably free module is projetive (take N = D 1 s in 3 of Definition 1) and a projetive module is torsion-free (sine it an be embedded into a free, and thus into a torsion-free module). More generally, we have the following results. RR n 8629

10 8 Cluzeau & Quadrat Theorem 1 (Lam (1999); MConnell and Robson (2); Rotman (29)). Let D be a left noetherian domain. Then, we have the following impliations for finitely generated left/right D-modules: free stably free projetive reflexive torsion-free. The onverses of the above results are generally not true. Some of them hold for partiular domains playing partiular roles in linear systems theory. Theorem 2 (Lam (1999); MConnell and Robson (2); Quadrat and Robertz (214); Rotman (29)). We have the following results: 1. If D is a prinipal ideal domain, i.e., every left ideal I and every right ideal J of the domain D are prinipal, i.e., are of the form I = D d 1 and J = d 2 D for d 1, d 2 D (e.g., the ring A of OD operators with oeffiients in a differential field A suh as the ring B 1 (k), D 1 (k), D 1 (k)), then every finitely generated torsion-free left or right D-module is free. 2. If D = k[x 1,..., x n ] is a ommutative polynomial ring with oeffiients in a field k, then every finitely generated projetive D-module is free (Quillen-Suslin theorem). 3. If D is the Weyl algebra A n (k) or B n (k), where k is a field of harateristi (e.g., k = Q, R, C), then every finitely generated projetive left/right D-module is stably free and every finitely generated stably free left/right D-module of rank at least 2 is free (Stafford s theorem). 4. If D = D n (k), D n (k) or D = A, where A = kt and k is a field of harateristi, or A = k{t} and k = R or C, then every finitely generated projetive left/right D-module is stably free and every finitely generated stably free left/right D-module of rank at least 2 is free. A matrix R D q p is said to have full row rank if ker D (.R) := {µ D 1 q µ R = } =, i.e., if the rows of the matrix R are left D-linearly independent. If R D q p has full row rank, then we have D 1 q = D 1 q R D 1 p, whih yields q p. The next theorem haraterizes when a left D-module M, finitely presented by a full row rank matrix R, is a projetive or a free module. Theorem 3 (Fabiańska and Quadrat (27); Quadrat and Robertz (27a)). Let M = D 1 p /(D 1 q R) be a left D-module finitely presented by a full row rank matrix R D q p. Then, we have: 1. M is a projetive left D-module if and only if M is a stably free left D-module. 2. M is a stably free left D-module of rank p q if and only if R admits a right inverse, i.e., if and only if there exists a matrix S D p q suh that R S = I q. 3. M is a free left D-module of rank p q if and only if there exists U GL p (D) suh that: R U = (I q ). If U := (S Q), where S D p q and Q D p (p q), then we have the following isomorphisms ψ : M D 1 (p q) π(λ) λ Q, ψ 1 : D 1 (p q) M µ π(µ T ), where the matrix T D (p q) p is defined by: ( U 1 R := T ) D p p. In partiular, we have M = D 1 p Q = D 1 (p q). The matrix Q is alled an injetive parametrization of M. If T i denotes the i th row of T, then {π(t i )} i=1,...,p q is a basis of the free left D-module M of rank p q. The Quillen-Suslin theorem (resp., Stafford s theorem) is implemented in the QuillenSuslin pakage (Fabiańska and Quadrat (27)) (resp., Stafford pakage (Quadrat and Robertz (27a))). Hene, for D = k[x 1,..., x n ] and k = Q, A n (Q) or B n (Q), bases and injetive parametrizations of finitely generated free left D-modules an be omputed. Inria

11 A new insight into Serre s redution problem 9 3 Homomorphisms of finitely presented left D-modules In this setion, we first briefly review the haraterization of a left D-homomorphism of two finitely presented left D-modules. For more details, see Cluzeau and Quadrat (28); Rotman (29). Lemma 1 (Cluzeau and Quadrat (28)). Let M = D 1 p /(D 1 q R) and M = D 1 p /(D 1 q R ) be two finitely presented left D-modules and π : D 1 p M and π : D 1 p M the anonial projetions onto M and M. 1. The existene of f hom D (M, M ) is equivalent to the existene of a pair of matries P D p p and Q D q q satisfying the following relation: Hene, we have the following ommutative exat diagram R P = Q R. (2) D 1 q.r D 1 p π M.Q.P D 1 q.r D 1 p π M, i.e., (2) holds and f π = π (.P ), where f hom D (M, M ) is defined by: f λ D 1 p, f(π(λ)) = π (λ P ). (3) 2. Let P D p p and Q D q q satisfy (2) and R 2 D q 2 q be suh that ker D (.R ) = im D (.R 2). Then, the matries defined by { P := P + Z R, Q := Q + R Z + Z 2 R 2, for all Z D p q and Z 2 D q q 2, satisfy the identity R P = Q R and we have: λ D 1 p, f(π(λ)) = π (λ P ) = π (λ P ). For two finitely presented left D-modules M and M, the problem of haraterizing elements of hom D (M, M ) is onsidered in Cluzeau and Quadrat (28) for ertain lasses of nonommutative polynomial rings and algorithms are given (see Algorithms 2.1 and 2.2 of Cluzeau and Quadrat (28)). An implementation is available in the OreMorphisms pakage (Cluzeau and Quadrat (29)). If f hom D (M, M ), then we an define the following finitely generated left D-modules: ker f := {m M f(m) = }, im f := {m M m M : m = f(m)}, oim f := M/ ker f, oker f := M /im f. For two finitely presented left D-modules M and M, let us expliitly haraterize the kernel, image, oimage and okernel of f hom D (M, M ). Lemma 2 (Cluzeau and Quadrat (28)). Let M = D 1 p /(D 1 q R) (resp., M = D 1 p /(D 1 q R )) be a left D-module finitely presented by R D q p (resp., R D q p ) and f hom D (M, M ) defined by (3), where P D p p satisfies (2) for a ertain matrix Q D q q. 1. Let S D r p and T D r q be two matries suh that ker D (. ( P T R T ) T ) = im D (.(S T )), (4) RR n 8629

12 1 Cluzeau & Quadrat L D q r a matrix satisfying R = L S and Q = L T, and S 2 D r2 r a matrix suh that ker D (.S) = im D (.S 2 ). Then, we have: ker f = (D 1 r S)/(D 1 q R) ( = D 1 r / D 1 (q+r2) ( L T S2 T ) T ). (5) Hene, f is injetive if and only if the matrix (L T S2 T ) T admits a left inverse, i.e., if and only if there exists X = (X 1 X 2 ) D r (q+r2) suh that X 1 L + X 2 S 2 = I r. 2. With the above notations, we have: oim f = D 1 p /(D 1 r S) = im f = Moreover, we have the following ommutative exat diagram ( D 1 (p+q ) ( P T R T ) T ) /(D 1 q R ). ker f D 1 q.r D 1 p π M.L D 1 r.s D 1 p κ oim f ρ, where ρ : M oim f = M/ ker f is the anonial projetion. ( 3. We have oker f = D 1 p / D ( 1 (p+q ) P T R ) ) T T and the following long exat sequene D 1 r.(s T ) D 1 (p+q ).(P T R T ) T ɛ D 1 p oker f defining the beginning of a finite free resolution of oker f. Hene, f is surjetive if and only if the matrix (P T R T ) T admits a left inverse, i.e., if and only if there exists Y = (Y 1 Y 2 ) D p (p+q ) suh that Y 1 P + Y 2 R = I p. 4. We have the following ommutative exat diagram D 1 r.s D 1 p κ oim f.t.p.r D π 1 q D 1 p M f, oker f where f hom D (oim f, M ) is defined by f (κ(λ)) = π (λ P ) for all λ D 1 p. Inria

13 A new insight into Serre s redution problem 11 To study the deomposition problem, namely the problem of reognizing whether or not a finitely presented left D-module M is deomposable (see of 8 of Definition 1), we shall fous on the ase M = M, i.e., on the study of the endomorphism ring end D (M) := hom D (M, M) of M. In many standard examples oming from linear systems theory and mathematial physis (see, e.g., the examples onsidered in Setion 7), D is a ommutative polynomial ring. In this partiular ase, hom D (M, M ) inherits a D-module struture (whih is usually not the ase for a nonommutative ring D) and an expliit desription of the D-module hom D (M, M ) in terms of generators and relations an be given. For more details and expliit algorithms, we refer to Cluzeau and Quadrat (28). Till the end of this setion, we assume that D is a ommutative ring. From Lemma 1, it follows that the ring end D (M) an be written as the fator of two D-modules, i.e., we have end D (M) = B := A/(D p q R), where A := {P D p p Q D q q : R P = Q R} is a ring alled eigenring. Indeed, we learly have A, I p A and if P 1, P 2 A, i.e., R P 1 = Q 1 R and R P 2 = Q 2 R for some matries Q 1, Q 2 D q q, then we have R (P 1 + P 2 ) = (Q 1 + Q 2 ) R and R (P 1 P 2 ) = (Q 1 Q 2 ) R so that P 1 + P 2 A and P 1 P 2 A. The other properties of a ring an easily be heked. The ring A is a nonommutative ring sine P 1 P 2 is usually different from P 2 P 1. Moreover, D p q R is a two-sided ideal of A. Indeed, if P 1, P 2 A and Z 1 R, Z 2 R D p q R, where Z i D p q for i = 1, 2, then we have: { P1 (Z 1 R) + P 2 (Z 2 R) = (P 1 Z 1 + P 2 Z 2 ) R, (Z 1 R) P 1 + (Z 2 R) P 2 = (Z 1 Q 1 + Z 2 Q 2 ) R. Thus, B = A/(D p q R) is a nonommutative ring and κ := id p π : A B is the anonial projetion onto B. In partiular, the produt of B is defined by: P 1, P 2 A, κ(p 1 ) κ(p 2 ) = κ(p 1 P 2 ). We all opposite ring of B, denoted by B op, the ring defined by B as an abelian group but equipped with the opposite multipliation defined by: b 1, b 2 B, b 1 b 2 := b 2 b 1. If φ : B end D (M) is the abelian group isomorphism mapping κ(p ) to φ(κ(p )) defined by then we have λ D 1 p, φ(κ(p ))(π(λ)) = π(λ P ), λ D 1 p, (φ(κ(p 2 )) φ(κ(p 1 )))(π(λ)) = π(λ P 1 P 2 ) = φ(κ(p 1 P 2 ))(π(λ)) = φ(κ(p 1 ) κ(p 2 ))(π(λ)), i.e., using the opposite ring B op, we obtain: φ(κ(p 2 ) κ(p 1 )) = φ(κ(p 1 ) κ(p 2 )) = φ(κ(p 2 )) φ(κ(p 1 )). Sine φ(κ(i p )) = id M, φ is a ring isomorphism, i.e.: end D (M) = B op. Algorithm 2.1 in Cluzeau and Quadrat (28) omputes a family of generators {f i } i=1,...,s of the finitely generated D-module end D (M). The f i s are given by means of two matries P i D p p and Q i D q q satisfying R P i = Q i R, i.e., f i (π(λ)) = π(λ P i ) for all λ D 1 p and i = 1,..., s (see Lemma 1). Let us now explain how to obtain a finite family of D-linear relations among these generators, i.e., X F =, where F = (f 1... f s ) T and X D t s. A D-linear relation s j=1 d j f j = between the f i s is equivalent to the existene of Z D p q satisfying: RR n 8629 s d j P j = Z R. (6) j=1

14 12 Cluzeau & Quadrat To solve (6), let us introdue a few definitions and a standard result whih holds for matries with entries in a ommutative ring D. If F D q p, then row(f ) D 1 q p denotes the row vetor obtained by onatenating the rows of F. If F D q p and F D q p, then K := F F stands for the Kroneker produt of F and F, namely, the matrix K D q q p p defined by (F ij F ) 1 i q, 1 j p. If F D q p, G D r q and H D s r, then a standard result on Kroneker produts states that we have: row(h G F ) = row(g) (H T F ). Applying the above identity to (6), we get: s d j row(p j ) row(z) (I p R) = (d 1... d s row(z)) j=1 row(p 1 ). row(p s ) I p R =. If we introdue the matries U := ( row(p 1 ) T... row(p s ) T ) T D s p 2, V := I p R D p q p2, W := (U T V T ) T D (s+p q) p2, then there exist X D t s and Y D t p q satisfying ker D (.W ) = D 1 t (X i j entry of the matrix Y and for i = 1,..., t, Y i,1... Y i,q Y i,(q+1)... Y i,2 q Z i =.. Dp q, (7) Y ). If Y i,j denotes the Y i,(p 1) q+1... Y i,p q then s j=1 X ij P j = Z i R, and thus the f i s satisfy the following D-linear relations: i, = 1,..., t, s X ij f j =. (8) j=1 Hene, we get end D (M) = D 1 s /(D 1 t X), i.e., end D (M) is finitely presented by the matrix X D t s. Now, the ring struture of end D (M) is haraterized by the expressions of the f i f j s in terms of the generators f k s of the D-module end D (M), i.e.: i, j = 1,..., s, f i f j = s γ ijk f k, γ ijk D. (9) The γ ijk s look like the struture onstants appearing in the theory of finite-dimensional algebras. The matrix Γ formed by the γ ijk satisfies F F = Γ F. Γ is alled a multipliation table in group theory. If D f 1,... f s denotes the free assoiative D-algebra generated by the f i s and s s J = X ij f j, i = 1,..., t, f i f j γ ijk f k, i, j = 1,..., s j=1 is the two-sided ideal of D f 1,... f s generated by the relations (8) and (9), then the nonommutative ring end D (M) is defined by end D (M) = D f 1,... f s /J, whih shows that end D (M) an be defined as the quotient of a free assoiative algebra by a two-sided ideal generated by linear and quadrati relations over D. k=1 k=1 Inria

15 A new insight into Serre s redution problem 13 Using (7), the struture onstants γ ijk s an be omputed as follows. The omputation of the normal form of the rows row(p i P j ) with respet to a Gröbner basis of the D-module D 1 (s+p q) W for i, j = 1,..., s yields a matrix (Γ 1 Γ 2 ) D s2 (s+p q), where Γ 1 D s2 s and Γ 2 D s2 p q. Then, the matrix Γ 1 defines the multipliation table of the family of generators {f i } i=1,...,s of end D (M). The omputation of the endomorphism ring end D (M) (i.e., generators, relations and multipliation table) for a finitely presented module over a ommutative polynomial ring D is implemented in the OreMorphisms pakage (Cluzeau and Quadrat (29)). Example 2. Let us onsider the motion of a fluid in a one-dimensional tank studied in Dubois et al. (1999) and defined by the following linear system of OD time-delay equations { y1 (t 2 h) + y 2 (t) 2 u(t h) =, (1) y 1 (t) + y 2 (t 2 h) 2 u(t h) =, where h a positive real number. Let D = Q(α)[, δ] be the ommutative polynomial ring of OD time-delay operators with rational onstant oeffiients, i.e., y(t) = ẏ(t), δ y(t) = y(t h) and δ = δ, ( ) δ δ R = 1 δ 2 D 2 3 (11) 2 δ the matrix defining (1), and the D-module M = D 1 3 /(D 1 2 R) finitely presented by R. Applying Algorithm 2.1 of Cluzeau and Quadrat (28) to R, the D-module struture of end D (M) is generated by f e1, f e2, f e3, f e4 end D (M) defined by f α (π(λ)) = π(λ P α ) for all λ D 1 3, where α = (α 1 α 2 α 3 α 4 ) D 1 4, {e i } i=1,...,4 is the standard basis of D 1 4 and: α 1 α 2 2 α 3 δ P α = α α 4 α 1 2 α 4 2 α 3 δ, α 4 δ α 4 δ α 1 + α 2 + α 3 (δ 2 + 1) ( ) α1 2 α Q α = 4 α α 4. α 2 α 1 Let us simply set f i := f ei. We an hek that the generators {f i } i=1,...,4 of the D-module struture of end D (M) satisfy the following D-linear relations: (δ 2 1) f 4 =, δ 2 f 1 + f 2 f 3 =, f 1 + δ 2 f 2 f 3 =. (12) A omplete desription of the nonommutative ring end D (M) is given by the knowledge of the expressions of the ompositions f i f j in the family of generators {f k } k=1,...,4 for i, j = 1,..., 4: f 1 f i = f i f 1 = f i, i = 1,..., 4, f 3 f 3 = (δ 2 + 1) f 3, f 2 f 2 = f 1, f 3 f 4 = 2 f 1 2 f f 4, f 2 f 3 = f 3 f 2 = f 3, (13) f 4 f 3 =, f 2 f 4 = 2 f 1 2 f 2 + f 4, f 4 f 4 = 2 f 4. f 4 f 2 = f 4, Denoting by f f r the omposition of an element f in the first olumn by an element f r in the first row of the table below, we an write (13) in the form of the following multipliation table: f f r f 1 f 2 f 3 f 4 f 1 f 1 f 2 f 3 f 4 f 2 f 2 f 1 f 3 2 f 1 2 f 2 + f 4 f 3 f 3 f 3 (δ 2 + 1) f 3 2 f 1 2 f f 4 f 4 f 4 f 4 2 f 4 RR n 8629

16 14 Cluzeau & Quadrat We finally obtain end D (M) = D f 1, f 2, f 3, f 4 /J, where J = (δ 2 1) f 4, δ 2 f 1 + f 2 f 3, f 1 + δ 2 f 2 f 3, f 1 f 1 f 1,..., f 4 f f 4 is the two-sided ideal of the free D-algebra D f 1, f 2, f 3, f 4 generated by the polynomials defined by the identities (12) and (13). 4 Fatorization problem In this setion, we omplete results of Cluzeau and Quadrat (28) to obtain a neessary and suffiient ondition for the existene of a strit fatorization of a linear funtional system. Let us first give a neessary and suffiient ondition for the existene of a fatorization of R D q p. Lemma 3. If R D q p, then the following assertions are equivalent: 1. There exist two matries L D q r and S D r p suh that: R = L S. (14) 2. There exist a finitely presented left D-module M and f hom D (M, M ), where M = D 1 p /(D 1 q R), suh that: oim f = D 1 p /(D 1 r S), ker f = (D 1 r S)/(D 1 q R). (15) Proof Let M := D 1 p /(D 1 r S) be the left D-module finitely presented by S. The relation R = L S indues the ommutative exat diagram D 1 q.r D 1 p π M.L D 1 r.s D 1 p κ M, whih defines f hom D (M, M ) by f(π(λ)) = κ(λ) for all λ D 1 p. Indeed, if π(λ) = π(λ ) for some λ D 1 p, then there exists µ D 1 q suh that λ = λ + µ R, whih yields: f(π(λ)) = κ(λ) = κ(λ ) + κ(µ R) = κ(λ ) + κ((µ L) S) = κ(λ ) = f(π(λ )). Using ker D (.(I T p S T ) T ) = D 1 r (S I r ), 1 and 2 of Lemma 2 yield (15). 2 1 is proved in Theorem 3.1 of Cluzeau and Quadrat (28). For the sake of ompleteness, we repeat the proof here. Let M := D 1 p /(D 1 q R ) and f hom D (M, M ) satisfy (15). From Lemma 1, f is defined by (3) where P D p p satisfies (2) for a ertain matrix Q D q q. Using (2) and (4) of Lemma 2, we get im D (.(R Q)) ker D (. ( P T R T ) T ) = im D (.(S T )), whih shows that there exists a matrix L D q r suh that R = L S and Q = L T. Using Gröbner basis tehniques for a polynomial ring D, the fatorization (14) an be omputed (see, e.g., Chyzak et al. (25, 27)). Definition 2. A fatorization R = L S, where R D q p, L D q s and S D s p, is alled strit if: im D (.R) im D (.S). If F is a left D-module and R = L S, then ker F (S.) ker F (R.), i.e., every F-solution of S η = is a F-solution of R η =. Hene, finding solutions of a linear funtional system is an appliation of the problem of fatoring matries of funtional operators. Proposition 1. If R = L S is not a strit fatorization, then we have ker F (R.) = ker F (S.) for all left D-modules F. Inria

17 A new insight into Serre s redution problem 15 Proof. Sine, by definition of S, we have D 1 q R D 1 r S, D 1 r S = D 1 q R is equivalent to the existene of F D r q suh that S = F R. Combining this identity with R = L S, we get (I q L F ) R = and (I r F L) S =, and thus there exist two matries X D r q2 and Y D r r2 suh that { L F = Iq + X R 2, (16) F L = I r + Y S 2, where R 2 D q2 q (resp., S 2 D r2 r ) satisfies ker D (.R) = im D (.R 2 ) (resp., ker D (.S) = im D (.S 2 )). Then, using (16), we an easily hek that we have R η = L (S η) = { L θ =, S η = θ, S η =, sine θ F r satisfies S 2 θ =, and thus θ = F (L θ) Y (S 2 θ) = by (16), and onversely S η = F (R η) = { F ζ =, R η = ζ, R η =, sine ζ F q satisfies R 2 ζ =, and thus ζ = L (F ζ) X (R 2 ζ) = by (16), i.e., ker F (R.) = ker F (S.). Remark 1. Homologial algebra tehniques an be used to give another proof of Proposition 1. Indeed, by 1 2 of Lemma 3, the fatorization R = L S defines f hom D (M, M ), where M = D 1 p /(D 1 r S), suh that we have (15). Then, applying the ontravariant left exat funtor hom D (, F) to the anonial short exat sequene ker f M oim f and using (1) and (15), we get the following long exat sequene ker F (S.) ker F (R.) hom D (ker f, F) ext 1 D (oim f, F) ext 1 D (M, F) ext 1 D (ker f, F) ext 2 D (oim f, F) ext 2 D (M, F)..., where the ext i D (M, F) s are the so-alled extension abelian groups (see, e.g., Rotman (29)). Hene, if R = L S is not a strit fatorization, i.e., D 1 r S = D 1 q R, or equivalently ker f =, then hom D (ker f, F) =, and thus ker F (S.) = ker F (R.). Now, if F is a so-alled injetive left D-module, i.e. if we have ext i D (P, F) = for all left D-modules P and for i 1 (see, e.g., Rotman (29)), then the above long exat sequene redues to the following short exat sequene: ker F (S.) ker F (R.) hom D (ker f, F). We then get ker F (R.)/ ker F (S.) = hom D (ker f, F) = ker F ((L T S has full row rank, i.e., S 2 =, then we finally obtain: S T 2 ) T.) by (1) and (5). In partiular, if ker F (R.)/ ker F (S.) = ker F (L.). In partiular, this result holds for F = C (R n ) and D = R d 1,..., d n = R[d 1,..., d n ]. Let us now introdue the onept of a generi solution of the linear system ker F (R.). Definition 3. Let F be a left D-module, M = D 1 p /(D 1 q R) a finitely presented left D-module, and π : D 1 p M the anonial projetion onto M. Then, η ker F (R.) is alled a generi solution if φ η hom D (M, F), defined by φ η (π(λ)) = λ η for all λ D 1 p, is injetive. For instane, with the notations of Setion 2, y = (y 1... y p ) T is a generi solution of ker M (R.) = end D (M) orresponding to id M. The next result is a reformulation of the onept of a strit fatorization in terms of homomorphisms. RR n 8629

18 16 Cluzeau & Quadrat Theorem 4. If R D q p, then the following assertions are equivalent: 1. The matrix R admits a strit fatorization, i.e., there exist L D q r and S D r q suh that R = L S with im D (.R) im D (.S). 2. There exist a finitely presented left D-module F and f hom D (M, F) suh that ker f. 3. There exists a finitely presented left D-module F suh that the linear system ker F (R.) admits a non-generi solution in the sense of Definition 3. Proof. By Lemma 3, the existene of a fatorization R = L S is equivalent to the existene of a finitely presented left D-module F and f hom D (M, F) suh that oim f = D 1 p /(D 1 r S) and ker f = (D 1 r S)/(D 1 q R). Moreover, the fatorization is strit if and only if ker f, i.e., if and only if the linear system ker F (R.) admits a non-generi solution. Theorem 4 shows that the fatorization problem annot simply be solved by studying the ring end D (M) sine the fatorizations of R orrespond to finitely presented left D-modules F whih are usually not equal to M. Example 3. We illustrate the known fat that an operator R D = B 1 (Q) an admit a strit fatorization R = L S even if end D (M) is redued to k id M (see van der Put and Singer (23); Barkatou (27)). Let us onsider the OD operator R = d 2 + t d D. An element of end D (M) an be defined by P = a d + b, where a, b Q(t), whih satisfies R P = Q R for a ertain Q D. We have: R P = (d 2 + t d) (a d + b) = a d 3 + (2 ȧ + t a + b) d 2 + (ä + t (ȧ + b) + 2 ḃ) d + b + t ḃ. Hene, Q has the form Q = a d +, where Q(t), whih yields Q R = (a d + ) (d 2 + t d) = a d 3 + (t a + ) d 2 + (a + t ) d, and thus R P = Q R is equivalent to the following linear OD system: 2 ȧ + b =, ä + t (ȧ + b ) + 2 ḃ a =, b + t ḃ =. (17) If we note u := ḃ, then the last equation of (17) gives u + t u =, i.e., u = 1 e t2 /2, and thus we have t b = /2 1 ds + e s2 2, where 1 and 2 are two arbitrary onstants, i.e., 1, 2 Q. Sine b Q(t), we get 1 = and b = 2 and the above system beomes: ä t ȧ a = d dt (ȧ t a) =, b = 2, = 2 ȧ + 2. The integration of the first equation gives ȧ t a = 3 so that we get a = ( t e s2 /2 ds) e t2 /2, where 3 and 4 are two arbitrary onstants, i.e., 3, 4 Q. Sine a Q(t), we must have 3 = 4 =, i.e., a = and b = = 2. Hene, we obtain P = Q = 2, i.e., every element of end D (M) has the form of f = 2 id M, where 2 Q, and thus ker f = if 2. An algorithm for omputing rational solutions of linear OD systems an be found in Barkatou (1999). See also Barkatou (27); van der Put and Singer (23) and referenes therein for the omputation of the eigenring of a linear OD operator and a first order linear OD system. Theorem 4 asserts that R admits a strit fatorization if and only if there exists a finitely presented left D-module F and f hom D (M, F) suh that ker f. If we take F = D/(D d) = Q(t) and f hom D (M, F) defined by f(π(λ)) = κ(λ) for all λ D, where κ : D F is the anonial projetion onto F, then we get ker f = (D d)/(d R), whih shows that the OD equation η + t η = admits the non-generi solution η = 1 and yields the strit fatorization R = L S, where L = d + t and S = d. Inria

19 A new insight into Serre s redution problem 17 We refer the reader to the AlgebraiAnalysis pakage (Cluzeau et al. (213)) whih omputes general homomorphisms of two finitely presented differential modules by integrating linear PD systems in the unknown oeffiients of a fixed order ansatz for P. For instane, for the above example, the AlgebraiAnalysis pakage integrates (17) to get that the general endomorphism of M is defined by: t P = ( ) e s2 /2 ds t e t2 /2 d + 1 e s2 /2 ds + 2, 1,..., 4 Q. If M is a simple left D-module (see 9 of Definition 1) and f hom D (M, F) \ {}, then we have ker f =, whih shows that f is injetive. If M is a left D-module finitely presented by R D q p, then R does not admit a strit fatorization by Theorem 4. Moreover, if F = M, then im f = M sine im f is a non-trivial left D-submodule of M, whih shows that a non-trivial f end D (M) is an automorphism, i.e., f aut D (M). This result is the so-alled Shur s lemma stating that the endomorphism ring end D (M) of a simple left D-module M is a division ring (see, e.g., MConnell and Robson (2)). Example 4. Let us show that M = D/(D d 1 + D d 2 ) = k[x 1, x 2 ] is a simple left D = A 2 (Q)-module. If L is a non-trivial left D-submodule of M and z := d y L, where d D \ {}, y = π(1) is the generator of M and π : D M the anonial projetion onto M, then we an assume without loss of generality that d k[x 1, x 2 ] sine y satisfies the following relations: { d1 y =, (18) d 2 y =. Using (18), we get d i z = d i (d y) = d d i y + d x i y = d x i y = for i = 1, 2. Thus, there exists d D suh that y = d z L for a ertain d D \ {}, i.e., L = M, whih proves that M is a simple left D-module. Using Proposition 2.5 of Cluzeau and Quadrat (28), we an easily prove that aut D (M) = k \ {}. 5 Deomposition problem 5.1 General results The existene of a non-trivial deomposition M = M 1 M 2 of a left D-module M is known to be equivalent to the existene of a non-trivial idempotent element f end D (M), i.e., f 2 = f, where f is neither id M nor. See, e.g., MConnell and Robson (2); Cluzeau and Quadrat (28). Let us state a haraterization of an idempotent element of end D (M). Lemma 4 (Cluzeau and Quadrat (28)). Let M = D 1 p /(D 1 q R) be the left D-module finitely presented by R D q p and R 2 D r q a matrix suh that ker D (.R) = im D (.R 2 ). Then, f end D (M), defined by a matrix P D p p satisfying R P = Q R for a ertain Q D q q, is an idempotent element of end D (M), i.e., f 2 = f, if and only if there exists Z D p q suh that: Then, there exists a matrix Z D q r suh that: P 2 = P + Z R. (19) Q 2 = Q + R Z + Z R 2. In partiular, if R D q p has full row rank, then we have Q 2 = Q + R Z. An algorithm for the omputation of idempotents of end D (M) is given in Algorithm 4.1 of Cluzeau and Quadrat (28). If f 2 = f end D (M), then we have M = ker f im f. Indeed, we have m = f(m) + (m f(m)) for all m M, where m f(m) ker f. Let us now generalize Lemma 4.4 of Cluzeau and Quadrat (28). RR n 8629

20 18 Cluzeau & Quadrat Lemma 5. Let R D q p, ker D (.R) = im D (.R 2 ), ker D (.R 2 ) = im D (.R 3 ), M = D 1 p /(D 1 q R) and f end D (M) an idempotent defined by P D p p satisfying P 2 = P + Z R and R P = Q R for a ertain matrix Z D p q and a matrix Q neessarily of the form Q 2 = Q + R Z + Z R 2 for a ertain matrix Z D q r. Moreover, let S D r r be suh that R 2 Q = S R 2. If there exist D p q, 2 D q r, U D p r and V D q s suh that { R + (P Ip ) + Q + Z = U R 2, 2 R (Q I q + R ) S + R U + Z (2) = V R 3, then the matries defined by P := P + R and Q := Q + R + 2 R 2 satisfy R P = Q R, P 2 = P, Q 2 = Q and f(π(λ)) = π(λ P ) for all λ D 1 p. If R has full row rank, then (2) redues to the following algebrai Riati equation: Proof. Considering P := P + R, we an hek that R + (P I p ) + Q + Z =. (21) P 2 P = ( R + (P I p ) + Q + Z) R, whih shows that P 2 = P if and only if the first equation of (2) holds for a ertain U D p r. Now, using the first equation of (2), we an hek that the matrix Q := Q + R + 2 R 2 satisfies Q 2 Q = R ( R + (P I p ) + Q + Z) + ( 2 R (Q I q + R ) S + Z ) R 2 = ( 2 R (Q I q + R ) S + R U + Z ) R 2, and thus Q 2 = Q if and only the seond equation of (2) holds for a ertain V D q s. Finally, (2) redues to (21) when R has full row rank. Remark 2. If D is a polynomial ring over a omputational field k, then a solution D p q of the first equation of (2) an be obtained by onsidering an ansatz for for a fixed total degree and by solving the quadrati equations in the parameters of the ansatz so that all the normal forms of the rows of R + (P I p ) + Q + Z with respet of a Gröbner basis of the D-module D 1 r R 2 redue to zero. In this way, we an obtain a solution of the first equation of (2) for a ertain U D p r. Then, the seond equation of (2) an be solved by onsidering an ansatz for 2 for a fixed total degree and by solving the quadrati equations in the parameters of the ansatz so that all the normal forms of the rows of 2 R (Q I q + R ) S + R U + Z with respet of a Gröbner basis of the D-module D 1 s R 3 redue to zero. We an get a solution 2 of the seond equation of (2) for a ertain V D q s. The interest of defining an idempotent f of end D (M) by two idempotent matries P D p p and Q D q q (i.e., two projetors) is that the left D-modules ker D (.P ), im D (.P ), ker D (.Q) and im D (.Q) then satisfy { D 1 p = ker D (.P ) im D (.P ), D 1 q = ker D (.Q) im D (.Q), (22) whih shows that ker D (.P ), im D (.P ), ker D (.Q) and im D (.Q) are finitely generated projetive left D- modules (see 3 of Definition 1). In this ase, we also have ker D (.P ) = im D (.(I p P )) and im D (.P ) = ker D (.(I p P )) and similarly with Q. Let us state again two standard results of homologial algebra that will be used in what follows. Proposition 2 (Rotman (29)). Let M f Then, the following assertions are equivalent: M g M be a short exat sequene. 1. There exists u hom D (M, M) suh that g u = id M. 2. There exists v hom D (M, M ) suh that v f = id M. Inria