· mm research preprints, 149{169 klmm, amss, academia sinica vol. 25, december 2006 149 on...

MM Research Preprints, 149–169KLMM, AMSS, Academia SinicaVol. 25, December 2006 149

On Solutions of Linear Functional Systems and

Factorization of Laurent-Ore Modules

Ziming LiKey Lab of Mathematics-Mechanization

Academy of Mathematics and System SciencesZhong Guan Cun, Beijing 100080, China

[email protected]

Min Wu 1)

Software Engineering InstituteEast China Normal University

North Zhangshan Road, Shanghai 200062, [email protected]

Abstract. We summarize some recent results on partial linear functional systems.By associating a finite-dimensional linear functional system to a Laurent-Ore module,Picard-Vessiot extensions are generalized from linear ordinary differential (difference)equations to finite-dimensional linear functional systems. A generalized Beke’s methodis also presented for factoring Laurent-Ore modules and it will allow us to find all “sub-systems”whose solution spaces are contained in that of a given linear functional system.

1. Introduction

This paper provides a survey of the work by M. Bronstein and the authors in a France-Sino Scientific Cooperation Project 2) from 2002 to 2005. Our project concerns finite-dimensional linear functional systems, and its outcome includes: a generalization of Picard-Vessiot extensions of linear ordinary differential (difference) equations, the notion of modulesof formal solutions, algorithms for computing the dimension of solution spaces and for reduc-ing linear functional systems to fully integrable ones, and generalizations of Beke’s factoriza-tion algorithm and of the eigenring method. The emphasis of this paper is on descriptionsof these results. Precise references are given for proofs and technical details.

A (partial) linear functional system consists of linear partial differential, shift, and q-shiftoperators, or any mixture thereof. By a finite-dimensional linear functional system, or a ∂-finite system for short, we mean a linear functional system whose module of formal solutionshas finite dimension (see Definition 4.5). Intuitively, a system is ∂-finite if and only if its

1) Partially supported by a KLMM Open Project Funding KLMM-0611.2)This joint project is supported in part by the French Government Scholarship (BGF no. 2002915), an

INRIA-Cafe Project Funding and two National Key Projects of China (no. 1998030600, 2004CB31830).

150 Z. Li and M. Wu

solution space has finite dimension. The following is an example:

P ′′(x, k)− 2x1−x2 P ′(x, k) + k(k+1)

1−x2 P (x, k) = 0

P (x, k + 2)− (2k+3)xk+2 P (x, k + 1) + k+1

k+2P (x, k) = 0 .(1)

The sequence of the Legendre polynomials {P (x, k)}∞k=1 is a solution of (1) with the initialconditions {P (0, 0) = 0, P ′(0, 0) = 0, P (0, 1) = 0, P ′(0, 1) = 1}.

Given a linear functional system L, we are interested in the following questions: (i) Does Lhave a nonzero solution? (ii) Is there a ring containing “all” the solutions of L? (iii) Howdoes one compute the dimension of the solution space of L? (iv) How does one find (if itexists) a “subsystem” whose solution space is properly contained in that of L? (v) Determinewhether the solution space of L can be written as a direct sum of those of its subsystems?

Our work is intended for answering these questions algorithmically for ∂-finite systems.In terms of modules of formal solutions (Definition 4.5) and Picard-Vessiot extensions (De-finition 4.8), the above questions translate respectively to: (i) Is a module M of formalsolutions trivial? (ii) Does there exist a Picard-Vessiot extension for a given system? (seeSection 4.) (iii) How does one compute the dimension of M? (see Section 5.) (iv) Howdoes one find a nontrivial submodule of M? (see Section 6.) (v) Is M decomposable? (seeSection 6.)

Many of the results in this paper are straightforward generalizations of their counterpartsof linear ordinary differential or difference equations. These generalizations are howevernecessary in view of their wider applicability and the complications caused by the appearanceof several differential and difference operators.

Throughout the paper, rings are not necessarily commutative and have arbitrary char-acteristic. Ideals, modules and vector spaces are all left ones. Fields are always assumed tobe commutative. Denote by R p×q the set of all p× q matrices with entries in a ring R, andby ein, for 1 ≤ i ≤ n, the unit vector in R1×n with 1 in the ith position and 0 elsewhere. Thenotation “∼=R” means “isomorphic as R-modules”. We use (·)τ to denote the transpose of avector or matrix, and 1n to denote the identity matrix of size n. Vectors are represented bythe boldfaced letters u,v,w etc. Vectors of unknowns are denoted x,y, z, etc. The symbol Cdenotes the field of complex numbers.

The paper is organized as follows. In Section 2., we present some preliminaries anddefine the notion of linear functional systems. In Section 3., we construct Picard-Vessiotextensions for fully integrable systems, which are a common special case of ∂-finite systems.In Section 4., modules of formal solutions are defined and Picard-Vessiot extensions aregeneralized for ∂-finite systems. In Section 5., we present some techniques for computinglinear dimension of a linear functional system. In Section 6., we generalize Beke’s algorithmand the eigenring approach to factor Laurent-Ore modules. Concluding remarks are madein Section 7.

2. Preliminaries

Let R be a ring and ∆ be a finite set of commuting maps from R to itself. A map in ∆is assumed to be either a derivation or an automorphism. Recall that a derivation δ is an

Solutions of Linear Functional Systems and Factorization of Laurent-Ore Modules 151

additive map satisfying the multiplicative rule δ(ab) = aδ(b) + δ(a)b for all a, b ∈ R. Thepair (R, ∆) is called a ∆-ring, and it is a ∆-field when R is a field.

For a derivation δ ∈ ∆, an element c of R is called a constant with respect to δ if δ(c) = 0.For an automorphism σ ∈ ∆, c is called a constant with respect to σ if σ(c) = c. An element cof R is called a constant if it is a constant with respect to all maps in ∆. The set of constantsof R, denoted by CR, is a subring. The ring CR is a subfield if R is a field.

Let (F, ∆) be a ∆-field. By reordering the indices, we can always assume that ∆ ={δ1, . . . , δ`, σ`+1, . . . , σm} for some ` ≥ 0, where the δi’s are derivation operators on F andthe σj ’s are automorphisms of F . The Ore algebra ([8]) over F is the polynomial ring S :=F [∂1, . . . , ∂m] in ∂i with the usual addition and a multiplication as follows:

∂i∂j = ∂j∂i, ∂sa = a∂s + δs(a), ∂ta = σt(a)∂t,

for any 1 ≤ i, j ≤ m, 1 ≤ s ≤ `, ` < t ≤ m and a ∈ F .Remark that ∂i(a), where a is an element of a ∆-ring, is meant to be δi(a) if ∂i is asso-

ciated to a derivation operator δi, and to be σi(a) if ∂i is associated to an automorphism σi;while ∂ia, where a is an element of the Ore algebra S, means the product of ∂i and a.

Definition 2.1 Let (F, ∆) be a ∆-field. A linear functional system over F is a system ofthe form A(z) = 0 where A is a p × q matrix with entries in the Ore algebra S and z is acolumn vector of q unknowns.

Example 2.2 The system (1), satisfied by the Legendre polynomials, can be rewritten asA(z) = 0 where A =

(∂2

x − 2x1−x2 ∂x + k(k+1)

1−x2 , ∂2k − (2k+3)x

k+2 ∂k + k+1k+2

)τ, with ∂x the differenti-

ation with respect to x and ∂k the shift operator with respect to k.

Let F be a ∆-field. A commutative ring R containing F is called a ∆-extension of Fif all the maps in ∆ can be extended to R in such a way that all derivations (resp. auto-morphisms) of F become derivations (resp. automorphisms) of R and the extended mapscommute pairwise.

By a solution of a linear functional system A(z) = 0 over F , we mean a vector (s1, . . . , sq)τ

over some ∆-extension of F such that A(s1, . . . , sq)τ = 0, i.e., the application of the matrix Ato the vector is zero.

3. Fully integrable systems

A common special case of linear functional systems consists of fully integrable systems,which are of the form {∂i(z) = Aiz}1≤i≤m and correspond to the linear functional sys-tem A(z) = 0 where the matrix A is given by the stacking of blocks of the form (∂i ·1n−Ai).Fully integrable systems are of interest to our study, since to every ∂-finite system, we canassociate a fully integrable system whose solution space is isomorphic to that of the originalsystem (see Section 4.3.).

Definition 3.1 A system of the form

δi(z) = Aiz, 1 ≤ i ≤ ` , σi(z) = Aiz, ` + 1 ≤ i ≤ m, (2)

152 Z. Li and M. Wu

where Ai ∈ Fn×n and z is a column vector of n unknowns, is called an integrable system ofsize n over F if the following compatibility conditions are satisfied:

δi(Aj) = δj(Ai), 1 ≤ i < j ≤ `,

σi(Aj)Ai = σj(Ai)Aj , ` < i < j ≤ m,

σj(Ai)Aj = AiAj + δi(Aj), 1 ≤ i ≤ ` < j ≤ m.

(3)

The integrable system (2) is said to be fully integrable if the matrices A`+1, . . . , Am areinvertible.

Using Ore algebra notation, we write {∂i(z) = Aiz}1≤i≤m for the system (2) where theaction of ∂i is again meant to be δi for i ≤ ` and to be σi for i > `. Observe that theconditions (3) are derived from the condition ∂i(∂j(z)) = ∂j(∂i(z)) and are exactly thematrix-analogues of the compatibility conditions for first-order scalar equations in [11]. Fora linear ordinary difference equation, we often assume that its trailing coefficient is nonzero,while, for a first-order matrix difference equation, we assume that its matrix is invertible.These assumptions lead to the condition on invertibility of A`+1, . . . , Am in Definition 3.1.

Example 3.2 Let F = C(x, k), δx be the differentiation with respect to x and σk the shiftoperator with respect to k. Then A : { δx(z) = Axz, σk(z) = Akz } is a fully integrable systemwhere

Ax=

(x2−kx−k

x(x−k)(x−1)x2−kx+3k−2xkx(x−k)(x−1)

k(kx+x−x2−2k)(x−k)(x−1)

x3+x2−kx2−2x+2kx(x−k)(x−1)

)and Ak=

(k+1+kx2−xk2−x

(x−k)(x−1) −k+1+kx−k2−xk(x−k)(x−1)

x(k+1)(k+1+kx−k2−x)(x−k)(x−1)

(k+1)(x2−2kx−x+k2)k(x−k)(x−1)

).

In what follows, we generalize fundamental matrices and Picard-Vessiot extensions oflinear ordinary differential (difference) equations to fully integrable systems.

A square matrix with entries in a commutative ring is said to be invertible if its deter-minant is a unit in that ring.

Let F be a ∆-field and {∂i(z) = Aiz}1≤i≤m be a fully integrable system of size n over F .We define

Definition 3.3 An n × n matrix U with entries in a ∆-extension of F is a fundamentalmatrix for the system {∂i(z) = Aiz}1≤i≤m if U is invertible and ∂i(U) = AiU for each i,i.e., each column of U is a solution of the system.

A two-sided ideal I of a commutative ∆-ring R is said to be invariant if δi(I) ⊂ I for i ≤ `and σj(I) ⊂ I for j > `. The ring R is said to be simple if its only invariant ideals are (0)and R.

Definition 3.4 A Picard-Vessiot ring for a fully integrable system is a (commutative) ring Esuch that:

(i) E is a simple ∆-extension of F .

(ii) There exists some fundamental matrix U with entries in E for the system such that Eis generated by the entries of U and det(U)−1 over F .


Definitions 3.3 and 3.4 are natural generalizations of their analogues in the purely differ-ential case [20, (pages 12, 415)] and the ordinary difference case [19, (Errata)].

The existence of fundamental matrices and Picard-Vessiot extensions for fully integrablesystems is stated in the following

Theorem 3.5 Every fully integrable system over F has a Picard-Vessiot ring E. If F hascharacteristic 0 and CF is algebraically closed, then CE = CF . Furthermore, that extensionis minimal, meaning that no proper subring of E satisfies both conditions in Definition 3.4.

A detailed proof of the above theorem is found in [6].Consequently, if F has characteristic zero and an algebraically closed field of constants,

then all the solutions of a fully integrable system in its Picard-Vessiot ring form a CF -vectorspace whose dimension equals the size of the system.

We now present two examples for Picard-Vessiot extensions. The reader is referred to[25, §2.2] for detailed verifications.

Example 3.6 Consider the fully integrable system of size one:

∂i(z) = aiz where ai ∈ F and i = 1, . . . , m.

This is an extension of Example 1.19 in [20].Let E = F [T, T−1] be the ∆-extension such that δi(T ) = aiT and σj(T ) = ajT for i ≤ `

and j > `. Then E is a Picard-Vessiot ring of the given system if there does not exist aninteger k > 0 and a nonzero r ∈ F such that δi(r) = kair for i ≤ ` and σj(r) = ak

j r for j > `.Otherwise, assume that the integer k > 0 is minimal so that there exists a nonzero r ∈ Fsatisfying δi(r) = kair for i ≤ ` and σj(r) = ak

j r for j > `. Then E/(T k − r) is a Picard-Vessiot ring of the given system.

Example 3.7 Consider the system A in Example 3.2. Note that the change of variable3)

z = My where

M =(

x−kx x2

(x− k)k x2k

),

transforms A into another fully integrable system B : {δx(y) = Bxy, σk(y) = Bky} with

Bx =(

1 00 0

)and Bk =

(1 00 k

). It suffices to find a Picard-Vessiot ring of B. We get

that V =(

ex 00 Γ(k)

)is a fundamental matrix for B, and thus MV is for A. More-

over, F [ex,Γ(k), e−x,Γ(k)−1] is a Picard-Vessiot extension for A.

4. ∂-finite systems

In this section, we first discuss generic solutions of linear algebraic equations over ar-bitrary rings, then introduce the notions of Laurent-Ore algebras and modules of formalsolutions. These two notions allow us to generalize the results in Section 3. to ∂-finitesystems.

3)which can be found, for example, by computing the hyperexponential solutions of the system ([11, 25])

154 Z. Li and M. Wu

4.1. Generic solutions of linear algebraic equations over ringsLet R be an arbitrary ring. Denote by Z(R) the center of R, i.e. the set of all elements

that commute with every element in R. Then Z(R) is a subring of R. Consider a p × qmatrix A = (aij) with entries in R. For any R-module N , we can associate to A a Z(R)-linearmap λ : N q → Np given by

ξ := (ξ1, . . . , ξq)τ 7→ Aξ =

q∑

j=1

a1jξj , . . . ,

q∑

j=1

apjξj

τ

.

We therefore say that ξ ∈ N q is a solution “in N” of the system A(z) = 0 if λ(ξ) = 0,and write solN (A(z) = 0) for the set of all solutions in N . Clearly, solN (A(z) = 0) isa Z(R)-module. Note that λ is in general not R-linear since R is noncommutative.

As in the case of D-modules [16], we can associate to A ∈ Rp×q an R-module as follows: Ainduces the R-linear map ρ : R1×p → R1×q given by (r1, . . . , rp) 7→ (r1, . . . , rp)A. Let M =R1×q/(R1×pA), which is the quotient of R1×q by the image of the map ρ. We call M the R-cokernel of A and denote it by cokerR(A). Clearly, cokerR(A) is an R-module. Let e1p, . . . , epp

and e1q, . . . , eqq be the canonical bases of R1×p and R 1×q, respectively. Denote by π thecanonical map from R1×q to cokerR(A), and set ej = π(ejq) for 1 ≤ j ≤ q. Since π issurjective, M is generated by e1, . . . , eq over R. Note that ρ(eip) is the i-th row of A. Hence

0 = π(ρ(eip)) = π

q∑

j=1

aijejq

=

q∑

j=1

aijπ(ejq) =q∑

j=1

aijej , for 1 ≤ i ≤ p,

which implies that (e1, . . . , eq)τ is a solution of A(z) = 0 in M .Given two R-modules N1 and N2, denote by HomR(N1, N2) the set of all R-linear maps

from N1 to N2. Clearly, HomR(N1, N2) is a Z(R)-module.As illustrated by the following theorem, Proposition 1.1 of [16] remains true when D is

replaced by an arbitrary ring R.

Theorem 4.1 Let M = R1×q/(R1×pA

). Then HomR(M, N) and solN (A(z) = 0) are iso-

morphic as Z(R)-modules for any R-module N .

Remark 4.2 (i) The proof of Proposition 1.1 in [16] can be adapted to this theorem in astraightforward way (see [6]) and also, a slightly different but elementary proof is given in [25,Theorem 2.4.1].

(ii) The proof of Theorem 4.1 reveals that the vector e := (e1, . . . , eq)τ ∈ M q spec-ified above is a “generic” solution of the system A(z) = 0 in the sense that any solu-tion (s1, . . . , sq)τ of that system in N is the image of e under the map in HomR(M, N)sending ei to si.

4.2. Laurent-Ore algebrasLet F be a ∆-field and S = F [∂1, . . . , ∂m] be the corresponding Ore algebra. In the

differential case, an S-module is classically associated to a linear functional system [16, 20]. Inthe difference case, however, S-modules may not have appropriate dimensions, as illustratedby the following counterexample.


Example 4.3 Let σ 6= 1 be an automorphism of F and S = F [∂] be the corresponding Orealgebra. The equation ∂(y) = 0 cannot have a fundamental matrix (u) in any difference ringextension of F , for otherwise, 0 = ∂(u) = σ(u), thus u = 0. Therefore ∂(y) = 0 has onlytrivial solution. However, the S-module S/S∂ has dimension one as an F -vector space.

In [19, page 56], modules over Laurent algebras are used instead to avoid the aboveproblem. It is therefore natural to introduce the following extension of S: let θ`+1, . . . , θm

be indeterminates independent of the ∂i. Since the σ−1j are automorphisms of F , S =

F [∂1, . . . , ∂m, θ`+1, . . . , θm] is also an Ore algebra in which the θj are associated to the σ−1j .

Note that ∂jθj is in the center of S, since

(∂jθj)a = ∂jσ−1j (a)θj = σj(σ−1

j (a))∂jθj = a∂jθj , for all a ∈ F and j > `.

Therefore the left ideal T =∑m

j=`+1 S(∂jθj − 1) is a two-sided ideal of S, and we call thefactor ring L = S/T the Laurent-Ore algebra over F . Writing ∂−1

j for the image of θj in L, wecan write L (by convention) as L = F [∂1, . . . , ∂m, ∂−1

`+1, . . . , ∂−1m ] and view it as an extension

of S. For linear ordinary difference equations, L = F [σ, σ−1] is the algebra used in [19]. Forlinear partial difference equations with constant coefficients, L is the Laurent polynomialring used in [18, 26].

Except for the purely differential case in which ` = 0, a Laurent-Ore algebra L =F [∂1, . . . , ∂m, ∂−1

`+1, . . . , ∂−1m ] is not an Ore algebra since ∂j∂

−1j = ∂−1

j ∂j = 1.When revisiting Example 4.3 with Laurent-Ore algebras, we get that the left ideal gen-

erated by ∂ in L = F [∂, ∂−1] is L, therefore the dimension of L/(L∂) over F , which is zero,equals that of the solution space of ∂(y) = 0 in any difference ring extension.

In the sequel, a module over a Laurent-Ore algebra that is finite-dimensional over theground field is called a Laurent-Ore module for short.

4.3. Modules of formal solutionsLet F be a ∆-field, and S and L be the corresponding Ore and Laurent-Ore algebras.

Replacing R with L in Theorem 4.1 yields

Theorem 4.4 Let A ∈ Sp×q and M=cokerL(A). Then solN (A(z)=0) and HomL(M, N) areisomorphic as CF -vector spaces for any L-module N .

From Remark 4.2(ii) in which we replace arbitrary ring R with L, cokerL(A) describesthe properties of all the solutions of A(z) = 0 “anywhere”. This motivates us to define

Definition 4.5 Let A ∈ Sp×q. The L-module M = L1×q/(L1×pA) is called the module offormal solutions of the system A(z) = 0. The dimension of M as an F -vector space is calledthe linear dimension of the system. The system is said to be of finite linear dimension, orsimply, ∂-finite, if 0 < dimF M < +∞.

Note that we choose to exclude systems with dimF M = 0 in the above definition since suchsystem has only trivial solution in any L-module, particularly, in any ∆-extension of F .

Remark 4.6 For any A ∈ Sp×q, we can construct both its S-cokernel cokerS(A) and L-cokernel cokerL(A). Viewing L as a right S-module and cokerS(A) as a left S-module, wecan define the tensor product ([21]) L⊗S cokerS(A), which is a right S-module and a left L-module. Lemma 2.4.10 in [25] shows that cokerL(A) and L ⊗S cokerS(A) are isomorphicas L-modules. Thus dimF cokerL(A) does not exceed dimF cokerS(A).

156 Z. Li and M. Wu

SystemA(z) = 0

sol(A(z) = 0)Pξ

Module of formal solutions

M = Le1 + · · ·+ Leq

= Fb1 ⊕ · · · ⊕ Fbn

∂i(b1, . . . ,bn)τ = Bi(b1, . . . ,bn)τ , i = 1, 2,

(e1, . . . , eq)τ = P (b1, . . . ,bn)τ , P ∈ F q×n

Integrable connection(P, {∂i(y) = Biy})sol(∂i(y) = Biy)

ξ

¡¡

¡¡

¡¡

¡µ ZZ

ZZ

ZZ

ZZ~

1−1¾

Fig. 1. Relationships among Systems, Modules and Solutions

Let A(z) = 0 with A ∈ Sp×q be a system of linear dimension n and M be its module offormal solutions with an F -basis b1, . . . ,bn. Suppose that ∂i(b1, . . . ,bn)τ = Bi(b1, . . . ,bn)τ

where Bi ∈ Fn×n for 1 ≤ i ≤ m. By a straightforward verification, {∂i(x) = Bix}1≤i≤m is afully integrable system, which is called the integrable connection of A(z) = 0 with respect tothe basis b1, . . . ,bn of M .

∂-finite and fully integrable systems are connected by the next proposition whose proofis given in [6, Proposition 2] and [25, Proposition 2.4.12].

Proposition 4.7 Let A,b1, . . . ,bn, B1, . . . , Bm be as above, and B be the stacking of theblocks (∂i · 1n −Bi). Then

(i) cokerL(A) ∼=L cokerL(B).

(ii) Let {e1, . . . , eq} be the set of L-generators of M satisfying A(e1, . . . , eq)τ=0 and P∈F q×n

be given by (e1, . . . , eq)τ=P (b1, . . . ,bn)τ . Then, for any ∆-extension E of F , the corre-spondence ξ 7→ Pξ is an isomorphism of CE-modules between solE({∂i(x) = Bix}1≤i≤m)and solE(A(z) = 0).

Remark that the inverse of the correspondence in Proposition 4.3 (ii) is given by η 7→ Qη,where Q is a matrix in Ln×q such that (b1, . . . ,bn)τ = Q(e1, . . . , eq)τ . From Proposi-tion 4.7 (ii), all the solutions of the system A(z) = 0 can be obtained from those of itsintegrable connection {∂i(x) = Bix}1≤i≤m, and vice versa. Figure 1 illustrates such a re-lationship, and it also suggests reducing the problem of solving ∂-finite systems to that ofsolving fully integrable systems.

4.4. Fundamental matrices and Picard-Vessiot extensionsBased on the discussion in Section 4.3., we generalize the notions and results of funda-

mental matrices and Picard-Vessiot extensions for ∂-finite systems.

Definition 4.8 Let A(z) = 0 with A ∈ Sp×q be a ∂-finite system, M be its module of formalsolutions, {e1, . . . , eq} be a set of L-generators of M and b1, . . . ,bn be an F -basis of M suchthat A(e1, . . . , eq)τ = 0 and (e1, . . . , eq)τ = P (b1, . . . ,bn)τ where P ∈ F q×n.


A q × n matrix V with entries in a ∆-extension E of F is called a fundamental matrixfor A(z) = 0 if V = PU where U ∈ En×n is a fundamental matrix of the integrable connectionof A(z) = 0 with respect to b1, . . . ,bn.

A Picard-Vessiot ring for an integrable connection of A(z) = 0 is called a Picard-Vessiotring for A(z) = 0.

As a consequence of Theorem 3.5, we have

Theorem 4.9 Every ∂-finite system A(z) = 0 over F has a Picard-Vessiot ring E. If Fhas characteristic 0 and CF is algebraically closed, then CE = CF .

Assume that F has characteristic 0 with an algebraically closed field of constants. If Eis a Picard-Vessiot ring for the system A(z) = 0 then the dimension of solE(A(z) = 0) asa CF -vector space equals the linear dimension of A(z) = 0, whenever the latter is finite.

Example 4.10 Let F, δx, σk be as in Example 3.2, and A := {δx(z) = Axz, σk(z) = Akz}where

Ax =

x+1x

k(x+1−k)x2(k−1)

−k(x+1−k)x2(k−1)

x + 1 xk−k2+2x2+kx2+k−1x(k−1) −xk−k2+2x2+kx2

x(k−1)

x + 1 xk+2x2+kx2−2k2+kx(k−1) −xk+2x2+kx2−2k2+1

x(k−1)

,

Ak =

k+1k

k+1−xk−xx(k−1)

xk+x−k−1x(k−1)

x(k+1)k

1−2x+k−xk+x3

k−12x+xk−x3−k−1

k−1x(k+1)

k1−2xk−2x+k+x3

k−12xk+2x−k−x3−1

k−1

,

and z = (z1, z2, z3)τ . Note that Ak is singular, so A is integrable but not fully integrable. Wewill show in Example 5.4 that all solutions of A can be found by a change of variable z = Pywhere

P =

1 00 1

x(k−1)x2−1

x2−kx2−1

and y is a solution of the fully integrable system B : {δx(y) = Bxy, σk(y) = Bky} with

Bx =

( −x+x3−1+x2−xk−k+k2

x(x2−1)k(x+1−k)x2(x2−1)

−x−xk+x3−1−x2+k2−kx2

x2−1−k2+xk+kx2+3x2−1

x(x2−1)

)

and

Bk =

(xk+x+k2+2k+1

k(x+1) − k+1x(x+1)

− (kx2−x−k2−2k−1)xk(x+1)

x2+x−1−kx+1

).

So it suffices to compute a Picard-Vessiot extension of B. The same method to construct afundamental matrix for the system in Example 3.2 yields a fundamental matrix for B:

U =(

xkex −kxk

kx2ex (x2 − k − 1)xk+1

),

hence PU is for A. In addition, a Picard-Vessiot ring C(x, k)[ex, e−x, xk, x−k] for B is aPicard-Vessiot ring for A.

158 Z. Li and M. Wu

5. Computing linear dimension

We now describe how to compute linear dimension for a given linear functional sys-tem A(z) = 0.

Let N be a submodule of a free L-module Lq with a finite set of generators. One cancompute a Grobner basis of N over L (see [?] and [25, Chapter 3]), which gives rise to an F -basis of Lq/N . Thus, one can determine whether a linear functional system is ∂-finite, andconstruct an F -basis of its module of formal solutions.

The following proposition indicates that the same goal may be achieved by Grobner basiscomputation over Ore algebra S (see [8]) if cokerS(A) has finite dimension over F . Noticethat the linear dimension of A(z) = 0 never exceeds dimF cokerS(A) by Remark 4.6.

Proposition 5.1 Let N be a left submodule of S1×n such that S1×n/N has finite dimensionover F and N the submodule generated by N in L1×n. Then S1×n/(N ∩S1×n) and L1×n/Nare isomorphic as F -vector spaces.

The proof to Proposition 5.1 can be found in [25, Proposition 2.4.6].Another useful fact is described in the next proposition whose proof is found in [25,

Proposition 2.4.11(ii)].

Proposition 5.2 Suppose that cokerS(A) with A ∈ Sp×q has a finite F -basis f1, . . . , fdand ∂i(f1, . . . , fd)τ = Di(f1, . . . , fd)τ where Di ∈ F d×d. Let D be the stacking of the blocks(∂i · 1d −Di). Then cokerS(A) ∼=S cokerS(D) and cokerL(A) ∼=L cokerL(D).

The above proposition reveals that, to compute linear dimension of a system A(z) = 0such that cokerS(A) has finite dimension over F , it suffices to compute linear dimension ofthe (integrable) system {∂i(y) = Diy}1≤i≤m in which y = (y1, . . . , yd)τ .

In the situation described in Proposition 5.2, let H0 be the set of row vectors of D, N0

the submodule generated by H0 over S, and N the submodule generated by H0 over L. Weproceed as follows to compute a Grobner basis of N over L, which gives rise to an F -basisof Ld/N , the module of formal solutions of {∂i(y) = Diy}1≤i≤m.

Compute the rank of Dj for each j with ` + 1 ≤ j ≤ m. If each Dj has full rank,then {∂i(y) = Diy}1≤i≤m is fully integrable and so H0 is already a Grobner basis of Nover L (see [25, Example 3.2.4]). Otherwise, the row vectors of some Dj satisfy cer-tain linear relations over F , which, together with ∂j(y) = Djy, implies F -linear relationsamong ∂j(y1), . . . ∂j(yd). Assume that one of the relations is

f1∂j(y1) + · · ·+ fd∂j(yd) = 0

where j ∈ {` + 1, . . . , m} and f1, . . . fd ∈ F , not all zero. Applying ∂−1j to it yields

σ−1j (f1) y1 + · · ·+ σ−1

j (fd) yd = 0. (4)

Hence the vector (σ−1j (f1), . . . , σ−1

j (fd)) belongs to the L-submodule N , but it does notbelong to N0. Adding to H0 the new vectors obtained from the linear relations of theform (4), we have a new set H1 of generators for N . Now we compute a Grobner basis of Nover L using H1. By Lemmas 2.5.1 and 2.5.2 in [25], such a basis can be computed by merelyrank computation, Gaussian elimination and the “Reduce All”trick. It is unnecessary to form


any S-polynomials. This simplification is due to the integrability of {∂i(y) = Diy}1≤i≤m.A detailed description of this process is formulated as an algorithm named LinearReductionin [25, §2.5].

We now give some examples to compute linear dimensions.

Example 5.3 Let A1, . . . , Am be in Fn×n and

A =

∂1 · 1n −A1

...∂m · 1n −Am

∈ Smn×n .

The system A(z) = 0 corresponds to the system {∂i(z) = Aiz}1≤i≤m. Let M be the moduleof formal solutions of A(z) = 0, and e1, . . . , en be the respective images of e1n, . . . , enn

in M . For e := (e1, . . . , en)τ ∈ Mn, we have A(e) = 0 or ∂i(e) = Aie for each i. Sincethe entries of Ai are in F , ∂i(ej) ∈

∑ns=1 Fes for all i, j, thus Lej ⊆

∑ns=1 Fes for all j.

So M =∑n

s=1 Les =∑n

s=1 Fes. In particular, dimF M ≤ n. Observe that dimF M = n ifand only if the system {∂i(z) = Aiz}1≤i≤m is fully integrable (see [25, Proposition 2.4.9]).

Example 5.4 Let F, δx, σk and the system A be given in Example 4.10. We follow theidea of the algorithm LinearReduction to compute linear dimension of A. Note that Ak

is singular. Solve the linear system (v1, v2, v3)Ak = 0 in v1, v2, v3. A nontrivial solutionof this system yields σk(z3) = xk

x2−1σk(z1) + x2−k−1

x2−1σk(z2). By an application of σ−1

k , we

get z3 = x(k−1)x2−1

z1 + x2−kx2−1

z2, thus,

z1

z2

z3

=

1 00 1

x(k−1)x2−1

x2−kx2−1

︸︷︷︸P

(z1

z2

).

Substitute this relation into A, we get δx(z1, z2)τ = Bx(z1, z2)τ and σk(z1, z2)τ = Bk(z1, z2)τ

where

Bx=

(−x+x3−1+x2−xk−k+k2

x(x2−1)k(x+1−k)x2(x2−1)

−x−xk+x3−1−x2+k2−kx2

x2−1−k2+xk+kx2+3x2−1

x(x2−1)

)and Bk=

(xk+x+k2+2k+1

k(x+1) − k+1x(x+1)

− (kx2−x−k2−2k−1)xk(x+1)

x2+x−1−kx+1

).

A straightforward calculation verifies that the first-order system B given by Bx and Bk isfully integrable, so B has linear dimension two by Example 5.3. According to the algorithmLinearReduction, all the solutions z of A can be obtained from the solutions y of B via achange of variable z = Py, and the modules of formal solutions for A and for B are thesame. Hence A has linear dimension two. 2

The following example shows that there are ∂-finite systems whose S-cokernels areinfinite-dimensional over F .

Example 5.5 Let A = (L1, L2)τ with L1 = ∂1∂2(∂1 + 1) and L2 = ∂1∂2(∂2 + 1), J theideal in S generated by L1 and L2, and M = cokerL(A). Since ∂1 and ∂2 are invertible in L,then M = L/(L(∂1+1)+L(∂2+1)), thus dimF M = 1. However, S/J is infinite-dimensionalover F .

160 Z. Li and M. Wu

We now conclude how to determine whether a linear functional system is ∂-finite. Asseen in Examples 5.3 and 5.4, when the system is given as an integrable system, we havea set of generators of M over F , so computing dimF M can be done by linear algebra. Inparticular, when A(z) = 0 is given by a finite-rank ideal in S, Proposition 5.1 shows thateither M = 0 (if the ideal contains a monomial in ∂`+1, . . . , ∂m) or an F -basis of M can becomputed via Grobner bases of ideals in S. There are algorithms and implementations forthis task [7, 8]. For a more general matrix A ∈ Sp×q, one can use the Grobner basis techniquedeveloped in [25, Chapter 3] for computing F -bases of L-modules. However, to compute thelinear dimension of A(z) = 0 for which cokerS(A) is finite-dimensional it suffices to computethe linear dimension of an integrable system according to Proposition 5.2. The algorithmLinearReduction supplies a tool for the latter task. Therefore, Grobner basis techniques in Lare necessary only when cokerS(A) is infinite-dimensional over F .

6. Factorization of Laurent-Ore modules

The work of this section is motivated by the algorithm FactorWithSpecifiedLeadersin [13, 14], where the idea of associated equations is extended to factor linear partial differ-ential equations with finite-dimensional solution spaces. In terms of modules over an Orealgebra S = F [∂1, . . . , ∂m] where ` = m, the problem solved by their algorithm can be formu-lated as follows: given a submodule N of Sn such that M = Sn/N is finite-dimensional overthe field F , finds all submodules of Sn that contain N . Such a submodule is called a factorof N since all its solutions are solutions of N . In their algorithm a factor is represented bya Grobner basis with respect to a pre-chosen monomial order. Observe that, for a (right)factor of a given order, there is only one possibility for its leading derivative in the ordinarycase, whereas, there are many possibilities in the partial case. Due to this complication, thealgorithm has to check every possibility to compute all the factors of a given order. In thisideal-theoretic approach the quotient module M does not come into play.

In the module-theoretic approach to be described in this section, we compute all sub-modules of the above quotient module M , and then recover the factors of N in the senseof [13, 14] via the canonical map from Sn to M . As all submodules of M are representedby linear bases over F , the problem of guessing leading derivatives goes away. The sameidea carries over to Laurent-Ore modules and results in a factorization algorithm for ∂-finitesystems.

6.1. Constructions with modules over Laurent-Ore algebrasGiven a ring R, we first review some notions of reducibility of R-modules defined in [22].An R-module M is reducible if M has a submodule other than 0 and M . Otherwise, M

is irreducible or simple. An R-module M is completely reducible or semisimple if for everysubmodule N1 there exists a submodule N2 such that M = N1⊕N2. Note that an irreduciblemodule is completely reducible as well. An R-module M is decomposable if M can bewritten as N1 ⊕ N2 where N1 and N2 are nontrivial submodules of M . Otherwise, Mis indecomposable. Clearly, an R-module M is reducible if it is decomposable, and M isirreducible when it is both indecomposable and completely reducible. By factoring an R-module, we mean finding its R-submodules.

As before, let F be a ∆-field with C the field of constants, S = F [∂1, . . . , ∂m] and


L = F [∂1, . . . , ∂m, ∂−1`+1, . . . , ∂

−1m ] be the corresponding Ore algebra and Laurent-Ore alge-

bra, respectively. In the sequel, unless otherwise specified, F has characteristic 0 and C isalgebraically closed.

Clearly, ordinary and partial differential modules in [20] are special cases of L-modules.The constructions in [20, §2.2] can be carried on L-modules in a similar way.

Let M be an L-module and N a submodule of M . The F -vector space M/N with theinduced actions: ∂i(w + N) = ∂i(w) + N for 1 ≤ i ≤ m and ∂−1

j (w + N) = ∂−1j (w) + N

for ` + 1 ≤ j ≤ m, is the quotient module.The direct sum of two L-modules M1 and M2 is M1 ⊕ M2 equipped with the actions:

∂i(w1 + w2) = ∂i(w1) + ∂i(w2) and ∂−1j (w1 + w2) = ∂−1

j (w1) + ∂−1j (w2) for 1 ≤ i ≤ m

and ` + 1 ≤ j ≤ m.The tensor product M1 ⊗M2 of two L-modules M1 and M2 is M1 ⊗F M2 equipped with

the actions: ∂i(w1 ⊗ w2) = ∂i(w1) ⊗ w2 + w1 ⊗ ∂i(w2) for i ≤ `, and ∂νj (w1 ⊗ w2) =

∂νj (w1)⊗ ∂ν

j (w2) for j > ` and ν ∈ {−1, 1}.The d-th exterior power ∧dM of an L-module M is the F -vector space ∧d

F M providedwith the actions given by the formulas ∂i(w1 ∧ · · · ∧wd) =

∑ds=1 w1 ∧ · · · ∧ (∂iws)∧ · · · ∧wd

for i ≤ ` and ∂νj (w1 ∧ · · · ∧wd) = ∂ν

j (w1) ∧ · · · ∧ ∂νj (wd) for j > ` and ν ∈ {−1, 1}.

Exterior powers of Laurent-Ore modules play an important role in the next section.

6.2. A module-theoretic approach to factorizationWe now describe an idea on factoring Laurent-Ore modules.Recall that a decomposable ([17]) element w ∈ ∧dM is an exterior product of d elements

in M , i.e., w = w1 ∧ · · · ∧wd.The following theorem generalizes Lemma 10 in [9] or the corresponding statement in

[20, page 111]:

Theorem 6.1 A Laurent-Ore module M has a d-dimensional submodule if and only if ∧dMhas a one-dimensional submodule generated by a decomposable element.

Remark that the operators ∂−1j are indispensable in the proof of Theorem 6.1 (see also [25,

Theorem 4.3.1]), and this proof yields a correspondence between d-dimensional submodulesand one-dimensional submodules generated by decomposable elements: if a d-dimensionalsubmodule of M has an F -basis v1, . . . ,vd, then the linear subspace generated by v1∧· · ·∧vd

in ∧dM is a one-dimensional submodule; conversely, if a one-dimensional submodule of ∧dMis generated by a decomposable element v1 ∧ · · · ∧ vd, then the F -linear subspace generatedby v1, . . . ,vd in M is a d-dimensional submodule.

Let M be a Laurent-Ore module with an F -basis {e1, . . . , en}. Set g =(nd

). Then the

module ∧dM has an F -basis {f1, . . . , fg}. Let e = (e1, . . . , en) and f = (f1, . . . , fg). ByTheorem 6.1, the problem of finding d-dimensional submodules of M is converted into thatof finding one-dimensional submodules of ∧dM whose generators are decomposable, andthus the factorization problem is reduced to two “subproblems”: finding one-dimensionalsubmodules and deciding the decomposability of their generators.

The first subproblem can be solved by a recursive method [15] for determining one-dimensional submodules of a Laurent-Ore module. Applying the method to ∧dM yieldsseveral finite subsets S1, . . . , St ⊂ F g with the following properties:

(a) The elements of each Sk are C-linearly independent.

162 Z. Li and M. Wu

(b) A one-dimensional F -space of ∧dM is a submodule if and only if it is an F -spacegenerated by the product of f and a nontrivial C-linear combination of elements ofsome Sk.

Now we deal with the second subproblem. Let S be one of the Sk with q elements, and wbe the product of f and a C-linear combination of elements of S, in which the coefficients areunspecified constants c1, . . . , cq. Consider the map φw : M → ∧d+1M given by v 7→ v ∧w.By Theorem 1.1 in [17, Chapter 4] and the proof of Theorem 6.1, w is decomposable ifand only if ker(φw) is of dimension d. The latter is equivalent to the condition that thematrix P of φw has rank (n−d). Hence, testing the decomposability of w amounts to a rankcomputation of P , i.e., identifying the unspecified constants c1, . . . , cq such that the rankof P is (n− d). This further amounts to solving several systems consisting of homogeneouspolynomial equations and inequations in c1, . . . , cq over F . Using a linear basis of F over C,we can translate every such system into finitely many subsystems over C. Each subsystemhas two portions: a set of polynomial equations and an inequation. If none of the subsystemshas a solution, then the product of f and any C-linear combination of elements of S is notdecomposable and thus does not lead to any d-dimensional submodule of M . Otherwise,substitute a solution into the matrix P , and compute a basis r1, . . . , rd of the rational kernelof P where rj ∈ Fn. Set uj = erj for j = 1, . . . , d. Then ⊕d

j=1Fuj is a d-dimensionalsubmodule of M .

A few words need to be said about those subsystems derived from the rank conditionfor P , since they may have infinitely many solutions after dehomogenization. We requirethat the substitution of any solution of a subsystem into P not only yields the required rankfor P , but also makes a fixed (n−d)×(n−d) minor nonzero. An (n−d)×(n−d) minor maycorrespond to several subsystems. This requirement can always be fulfilled, and will help usdescribe all d-submodules of M by a finite amount of information. We proceed as follows.Let T be such a subsystem. Using the nonzero minor corresponding to T and Cramer’srule, we may find a basis r1, . . . , rd of the rational kernel of P where the entries of the rj arein F (c1, . . . , cq) and their denominators divide the given minor. Set uj = erj for j = 1, . . . , d.Then ⊕d

j=1Fuj represents all d-dimensional submodules obtained by substituting solutionsof T for c1, . . . , cq into u1, . . . , ud. Note that we may check the set of solutions of T bytechniques from computational algebraic geometry. These considerations lead to a methodfor computing all submodules of M , which is described stepwise in [25, §4.4].

Remark 6.2 The above representation for d-dimensional submodules of M is rather naiveand has a lot of redundancy. To have more concise representations, one would partition d-dimensional submodules with respect to module isomorphism, and generalize the techniquesgiven in [20, page 112].

We now present two examples for factoring Laurent-Ore modules.

Example 6.3 [Legendre’s system] Let F, δx, σk be as given in Example 3.2 and L =F [∂x, ∂k, ∂

−1k ] be the Laurent-Ore algebra. A Grobner basis of the ideal generated by the

Legendre’s system (1) is

g1 = xk + x + (x2 − 1)∂x − (k + 1)∂k and g2 = k + 1 + (k + 2)∂2k − (2xk + 3x)∂k.


Let A = (g1, g2)τ ∈ L2×1, M = L/(Lg1 + Lg2) and e1, e2 be the images of 1 and ∂k in M ,respectively. Then e1, e2 form a basis of M over F and, in addition,

∂x

(e1

e2

)=

( −xk−xx2−1

k+1x2−1

−k−1x2−1

xk+xx2−1

)(e1

e2

), ∂k

(e1

e2

)=

(0 1

−k−1k+2

2xk+3xk+2

)(e1

e2

).

Apply the algorithm in [15], we find that M has no one-dimensional submodules, so Mis irreducible.

Example 6.4 Let F, δx, σk be as given in Example 3.2 and L = F [∂x, ∂k, ∂−1k ] the Laurent-

Ore algebra. Let M be an L-module with an F -basis {e1, e2, e3, e4} satisfying ∂x(e1, e2, e3, e4)τ

= Ax(e1, e2, e3, e4)τ and ∂k(e1, e2, e3, e4)τ = Ak(e1, e2, e3, e4)τ where

Ax=

0 1 0 0

−x3−x2k+2x2+xk+k2x+k2+k3

x2(−x+k)2(x2−x−k2)

(x−k)x0 0

0 0 0 1

0 0 −x3−x2k+x2+3xk+2x+k2x+4k2+5k+2+k3

x2(−x+k+1)− 2(−x2+x+k2+2k+1)

(−x+k+1)x

and

Ak =

0 0 1 00 0 0 1

− (x−k)x2

x−k−2 0 2x(x−k−1)x−k−2 0

−2x(x2−2xk−3x+k2+2k)(x−k−2)2

− (x−k)x2

x−k−22(x2−2xk−4x+k2+3k+2)

(x−k−2)22x(x−k−1)

x−k−2

.

Let us compute all two-dimensional submodules of M . Clearly,

f1 = e1 ∧ e2, f2 = e1 ∧ e3, f3 = e1 ∧ e4, f4 = e2 ∧ e3, f5 = e2 ∧ e4, f6 = e3 ∧ e4

form a basis of ∧2M over F . By the algorithm in [15], every one-dimensional submoduleof ∧2M has a generator of the form w = (f1, f2, f3, f4, f5, f6)

(∑6i=1 civi

)where

v1 =

−x2 + 2xk − k2

−2k3+k2−2x2−x3+x+k4−2x2k2+3xk+3k2x−4x2k+x4

x3

x+2x3+k3+2xk+k2−3x2k−3x2

x2

−2xk+k+k3+x2k+2k2−x−2k2xx2

−k+k2+x2−x−2xkx

−x2−2xk−2x+k2+2k+1x2

,

and the other expressions v2, . . . ,v6 are quite big and given in [25, pages 70–71].It remains to determine the decomposability of w. Consider the map M → ∧3M given

by v 7→ v ∧w, whose matrix is some P ∈ F 4×4 (we do not write down this matrix explicitlydue to its big size). The matrix P has rank 2 if and only if all its 3× 3 minors are zero and

164 Z. Li and M. Wu

there exists a nonzero 2× 2 minor. This yields four sets of solutions for the ci:

{ c1 = c1, c2 = c2, c3 = 0, c4 = c4, c5 = 0, c6 = 0 },{c1 = c1, c2 = c2, c3 = c3, c4 = −3c3

2 , c5 = 0, c6 = 0}

,{c1 = c3(2c4+3c3−4c5)

4c5, c2 = c2, c3 = c3, c4 = c4, c5 = c5, c6 = 0

},

{c1=c1, c2=−8c6c5−2c3c4−3c23−4c6c4−20c3c6−4c1c6+4c1c5+4c3c5−24c26

4c6, c3=c3, c4=c4, c5=c5, c6=c6

}.

Therefore M has two-dimensional submodules if and only if the ci in w satisfy one of thesefour relations.

Substitute these four relations into P respectively and compute the corresponding F -basesfor the rational kernel of P . Finally, we get all two-dimensional submodules:

Ni = {a1ui,1 + a2ui,2 | a1, a2 ∈ F} , i = 1, 2, 3, 4,

whereu1,1=(2c1x2k + c1k2x + c4x5 + c1x2 + 2c1x4 + c2x2k − 2x3c2k − 3x3c1k − 2x4c4k

+x2c2k2 + xk3c1 + x4c2 + c4x3k2 − c2x3 − 2c4x4 − 3c1x3 + c4x3 + 2c4x3k)/(x(2c1x− c1x2 + 2c1xk

−2c2k − c2k2 + k3c4 + 2c2x− c2x2 + 2k2c4 − 2c1k − c1k2 + 2c2xk − 2kc4x− 2k2c4x + kc4x2 + kc4 − c1 − c2))e1−(c4x3 + c1x2 − kc4x2 − c4x2 − c1xk − c1x)(x− k)/(2c1x− c1x2 + 2c1xk − 2c2k − c2k2 + k3c4 + 2c2x− c2x2

+2k2c4 − 2c1k − c1k2 + 2c2xk − 2kc4x− 2k2c4x + kc4x2 + kc4 − c1 − c2)e2 + e3,

u1,2 = (c1x4 − 2c1x2 + c4x5 + c1x− 2c4x3 − c1x3 − c4x4 + c2x3 − 2c2x2 + 2c4x2 + c1k2 + 2c1k3 + c1k4 − 2c2x2k

+3c1k2x− 4c1x2k + 5kc4x2 + k2c4x + c2xk + 3c1xk − 4c4x3k − 2c4x3k2 − 2c1x2k2 + 2c4xk3 + 3c4x2k2 + c4xk4

+c2xk2)/(x(2c1x− c1x2 + 2c1xk − 2c2k − c2k2 + k3c4 + 2c2x− c2x2 + 2k2c4 − 2c1k − c1k2 + 2c2xk − 2kc4x

−2k2c4x + kc4x2 + kc4 − c1 − c2))e1 − (c4x3 − c2x2 + kc4x2 + c2xk − 2k2c4x + c2x− 2xc4 + c1xk − 4kc4x− c1−c1k2 − 2c1k)(x− k)/((2c1x− c1x2 + 2c1xk − 2c2k − c2k2 + k3c4 + 2c2x− c2x2 + 2k2c4 − 2c1k − c1k2

+2c2xk − 2kc4x− 2k2c4x + kc4x2 + kc4 − c1 − c2))e2 + e4.

The submodules given by the last three solutions of the ci’s are more complicated (see [25,Example 4.4.1]).

6.3. Eigenrings and decomposition of Laurent-Ore modulesWe discuss another approach to factoring Laurent-Ore modules, which is not based on the

associated equations method. This method is first introduced in [23] to factor linear ordinarydifferential operators using eigenrings of the operators. Three algorithms are presentedthere for computing eigenrings. Significant improvements on these algorithms are describedin [4, 10]. Although the eigenring method does not always factor reducible operators, itoften yields factors quickly. This method has been generalized in [1, 5] for systems of lineardifference equations, and in [3] recently for systems of linear partial differential equations inpositive characteristic. We will generalize this method for factoring Laurent-Ore modules.

Let R be an arbitrary ring and M be an R-module. Recall that EndR(M) is the set ofall R-linear maps on M . Clearly, EndR(M) becomes a ring with the usual addition and thecomposition of maps adopted as the multiplication.

Definition 6.5 Let M be an R-module. A set of elements π1, . . . , πs of EndR(M) is calleda set of orthogonal idempotents if they satisfy

s∑

i=1

πi = 1 and πiπj = 0 whenever i 6= j, (5)

where 1 and 0 are the identity map and the zero map on M , respectively.


Although it is not stated in Definition 6.5, the maps πi are all idempotent. Indeed, thecondition (5) implies that π2

i =∑s

j=1 πiπj = πi

(∑sj=1 πj

)= πi for each i.

It is stated in Exercise 7 of [12, (Chapter 1, §1)] that

Proposition 6.6 Let M be an R-module. If EndR(M) has a set of orthogonal idempo-tents π1, . . . , πs then M = ⊕s

i=1πi(M). Conversely, if M can be written as a direct sumof submodules M = N1 ⊕ · · · ⊕ Ns then {π1, . . . , πs} is a set of orthogonal idempotentsof EndR(M) where πi is the projection from M to Ni.

For any R-module M , EndR(M) always has a set of orthogonal idempotents {0,1},which is called the trivial orthogonal idempotents of EndR(M). As a direct consequenceof Proposition 6.6, an R-module M is decomposable if and only if EndR(M) contains anontrivial set of orthogonal idempotents.

Let F be a ∆-field, C its field of constants and L = F [∂1, . . . , ∂m, ∂−1`+1, . . . , ∂

−1m ] the

Laurent-Ore algebra over F . Here we do not assume that C is algebraically closed.For an L-module M , the endomorphism ring EndL(M) is called the eigenring of M and

denoted E(M). Then a map φ ∈ EndF (M) belongs to E(M) if and only if φ commuteswith the ∂i and ∂−1

j for all i, j with 1 ≤ i ≤ m and ` + 1 ≤ j ≤ m. However, since M isan L-module on which the ∂−1

j act, the commutativity of φ with the ∂j for ` + 1 ≤ j ≤ m

implies ∂j ◦φ ◦ ∂−1j (w) = φ(w) and further φ ◦ ∂−1

j (w) = ∂−1j ◦φ(w) for w ∈ M . Hence, φ ∈

EndF (M) belongs to E(M) if and only if φ commutes with all the ∂i for 1 ≤ i ≤ m.Let M be a Laurent-Ore module with an F -basis {e1, . . . , en}. Suppose that

∂i(e1, . . . , en)τ = Bi(e1, . . . , en)τ , i = 1, . . . , m,

where Bi ∈ Fn×n for 1 ≤ i ≤ m and the Bj are invertible for j > `. In practice, the mapsin E(M) can be interpreted in terms of the Bi. Let φ ∈ EndF (M) and P ∈ Fn×n be itstransformation matrix given by (φ(e1), . . . , φ(en))τ = P (e1, . . . , en)τ . Let w =

∑ni=1 aiei∈M

where ai ∈ F . Then

φ(w) =n∑

i=1

aiφ(ei) = (a1, . . . , an)(φ(e1), . . . , φ(en))τ = (a1, . . . , an)P (e1, . . . , en)τ .

One can verify that the conditions ∂i(φ(w)) = φ(∂i(w)) hold for w ∈ M and 1 ≤ i ≤ mif and only if δi(P ) = BiP − PBi for i ≤ ` and σj(P ) = BjPB−1

j for j > `. Hence theeigenring E(M) can be defined equivalently to be

E(M) = {P ∈ Fn×n | δi(P ) = BiP − PBi for i ≤ ` and σj(P ) = BjPB−1j for j > `}. (6)

Clearly, the identity matrix 1n ∈ E(M) and E(M) is a C-subalgebra of Fn×n of dimensionnot greater than n2. Moreover, C · 1n ⊆ E(M) where C · 1n denotes the set of all matricesof the form c · 1n where c ∈ C.

As a natural generalization of the results in [2], [20, Proposition 2.13] or [23] for the caseof linear ordinary differential equations, we have

Theorem 6.7 Let M be an L-module of dimension n. Then

166 Z. Li and M. Wu

(i) If E(M) 6= C · 1n then M is reducible.

(ii) If M is decomposable then E(M) 6= C · 1n.

(iii) If M is completely reducible, then M is irreducible if and only if E(M) = C · 1n.

Given a Laurent-Ore module M of dimension n, we now use the formula (6) to com-pute E(M). Let P ∈ E(M) be a matrix of n2 indeterminates zij . From (6), we get asystem ∂i(z) = Aiz where Ai ∈ Fn2×n2

and z = (z11, . . . , z1n, . . . , zn1, . . . , znn)τ . This sys-tem is clearly ∂-finite, so its rational solutions can be found by a specialized version of themethod in [15]. A C-basis of all rational solutions of this system yields a C-basis {P1, . . . , Pr}of all rational solutions of E(M). Without loss of generality, we assume that P1 = 1n.Therefore E(M) = ⊕r

i=1C · Pi. If r = 1, then E(M) is trivial and M is indecomposableby Theorem 6.7 (ii). Otherwise, each eigenvalue λ of a nontrivial P ∈ E(M) will producea submodule {w ∈ M | Pw = λw} of M . If E(M) has a set of nontrivial orthogonalidempotents π1, . . . , πs, then we derive a decomposition M = π1(M) ⊕ · · · ⊕ πs(M). If Mis furthermore completely reducible, a maximal decomposition of M can be obtained byapplying the eigenring method recursively on the submodules in the above decomposition.

Example 6.8 Let F, δx, σk be as given in Example 3.2, L = F [∂x, ∂k, ∂−1k ] the Laurent-

Ore algebra and M an L-module of dimension two. Suppose that {e1, e2} is a basis of Msatisfying ∂x(e1, e2)τ = Bx(e1, e2)τ and ∂k(e1, e2)τ = Bk(e1, e2)τ where Bx =

(1 00 0

)and

Bk =(

1 00 k

).

We now compute the eigenring of M . Let P ∈ E(M) be a 2×2 matrix with indeterminateentries z11, z12, z21 and z22. The conditions δx(P ) = BxP−PBx and σk(P ) = BkPB−1

k yieldsa system A : {δx(z) = Axz, σk(z) = Akz} where z = (z11, z12, z21, z22)τ ,

Ax =

0 k(−x+x2−kx+2k)(x−k)(x−1)

x2−kx+3k−2xx(x−k)k(x−1) 0

−x2−kx+3k−2xx(x−k)k(x−1) −x3−kx2−2x+3k+kx

x(x−k)(x−1) 0 x2−kx+3k−2xx(x−k)k(x−1)

−k(−x+x2−kx+2k)(x−k)(x−1) 0 x3−kx2−2x+3k+kx

x(x−k)(x−1)k(−x+x2−kx+2k)

(x−k)(x−1)

0 −k(−x+x2−kx+2k)(x−k)(x−1) −x2−kx+3k−2x

x(x−k)k(x−1) 0

and

Ak =1γ

x2−2kx−x+k2

k α −xαβ −x2−2kx−x+k2

k2 β xk β2

1k(k+1)αβ 1

k+1α2 − 1k2(k+1)β

2 − 1k(k+1)αβ

x(k+1)(x2−2kx−x+k2)k β −(k + 1)x2β2 (k+1)(x2−2kx−x+k2)2

k2x(k+1)(2kx+x−x2−k2)

k β

xk β2 xαβ x2−2kx−x+k2

k2 β x2−2kx−x+k2

k α,

,

with α = k + 1 + kx2− k2x− x, β = k + 1 + kx− k2− x and γ = (x− k)(x− k− 1)(x− 1)2.All rational solutions of A are of the form

c1

(1

x− 1, − 1

k(x− 1),

xk

x− 1, − x

x− 1

)+ c2

(− x

x− 1,

1k(x− 1)

, − xk

x− 1,

1x− 1

),


for c1, c2 ∈ C. So

E(M) =

{(c1−c2x

x−1c2−c1k(x−1)

(c1−c2)xkx−1

c2−c1xx−1

), for any c1, c2 ∈ C

}

= C

(1

x−1 − 1k(x−1)

kxx−1 − x

x−1

)⊕ C

(− x

x−11

k(x−1)

− kxx−1

1x−1

).

Recall that the necessary condition for {P1, . . . , Ps} ⊂ E(M) being a set of orthogonal idem-potents is that P 2

i = Pi for each i. Substitute

P =

(c1−c2x

x−1c2−c1k(x−1)

(c1−c2)xkx−1

c2−c1xx−1

)

into the relation P 2 = P , we obtain three solutions:

P0 = 12, P1 =

( − 1x−1

1k(x−1)

− kxx−1

xx−1

), P2 =

(x

x−1 − 1k(x−1)

kxx−1 − 1

x−1

).

Among which, we find P1P2 = 0 and P1 + P2 = 12. So {P1, P2} is a set of nontrivialorthogonal idempotents of E(M). We have

P1(M) = {P1(w) | w ∈ M} = {(a1, a2)P1(e1, e2)τ | a1, a2 ∈ F} = F ·(e1 − 1

ke2

)

and

P2(M) = {P2(w) | w ∈ M} = {(a1, a2)P2(e1, e2)τ | a1, a2 ∈ F} = F ·(e1 − 1

kxe2

).

Therefore, P1(M)⊕ P2(M) is a decomposition of M into two nontrivial submodules. 2

The eigenring method, however, may fail to find any factor of a Laurent-Ore module eventhis module is reducible. This happens when the eigenring of that module is trivial.

7. Concluding remarks

In this paper we have discussed how to solve and factor ∂-finite systems. A key tech-nique described here is to use the notion of modules of formal solutions to connect ∂-finitesystems with fully integrable systems, while the latter systems are very similar to linearordinary differential (difference) equations. This technique naturally gives rise to Picard-Vessiot extensions for ∂-finite systems. Since Picard-Vessiot extensions are a stepping-stoneto introduce Galois groups, it would be interesting to extend (part of) the Galois theoryfor linear ordinary (difference) equations to ∂-finite systems. We presented some methodsfor determining linear dimension of a linear functional system. We also generalized Beke’smethod and the eigenring approach to factor Laurent-Ore modules. The work on factoringLaurent-Ore modules is however preliminary, because efficiency and applications of thesetwo methods have not yet been considered.

168 Z. Li and M. Wu

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