A Root-Free Splitting-Lemma for Systems of Linear Differential Equations
aa r X i v : . [ c s . S C ] N ov A Root-Free Splitting-Lemma forSystems of Linear Differential Equations
Eckhard Pfl¨ugelFaculty of Computing, Information Systems and MathematicsKingston UniversityPenrhyn RoadKingston upon ThamesSurrey KT1 2EEUnited KingdomE.Pfl[email protected]
Abstract
We consider the formal reduction of a system of linear differential equations andshow that, if the system can be block-diagonalised through transformation with aramified Shearing-transformation and following application of the Splitting Lemma[9], and if the spectra of the leading block matrices of the ramified system satisfya symmetry condition, this block-diagonalisation can also be achieved through anunramified transformation. Combined with classical results by Turritin [8] and Wasow[9] as well as work by Balser [1], this yields a constructive and simple proof of theexistence of an unramified block-diagonal form from which formal invariants such asthe Newton polygon can be read directly. Our result is particularly useful for designingefficient algorithms for the formal reduction of the system.
Mathematics Subject Classification:
Keywords:
Formal Reduction of Systems of Linear Differential Equations,Formal Solutions, Newton Polygons
When studying the formal reduction of a system of linear differential equations x dydx = A ( x ) y (1)where y is a vector with n ≥ A a square formal meromorphic powerseries matrix of dimension n of the form A ( x ) = x − r ∞ X j =0 A j x j ( A = 0)with pole order r >
0, the structure of the leading matrix A allows to reduce the problemto several problems of smaller size whenever A has several eigenvalues. The well-known1 plitting Lemma [9] states that if A is block-diagonal A = (cid:18) A A (cid:19) with the additional condition that A and A have no common eigenvalue, there existsa formal transformation matrix T ( x ) = ∞ X j =0 T j x j ( T = I ) (2)such that the change of variable y = T z transforms the system (1) into a new system x dzdx = B ( x ) z (3)where B = (cid:18) B B (cid:19) is of same pole order r and block-diagonal with the same block partition as in A . Thematrix B is computed by B = T [ A ] := T − AT − xT − dTdx . (4)Using the Splitting Lemma it is hence sufficient to study the case where the leading matrix A in (1) has only one eigenvalue. Using an exponential shift of the form y = exp( λ/x r ) z where λ is the unique eigenvalue of A one can (and we will throughout this paper) assumethat A is nilpotent.Several methods for finding transformation matrices which again lead to non-nilpotentleading matrices have been suggested [1, 2, 3, 7, 8, 9]. It can be shown that this, combinedwith the Splitting Lemma, gives rise to a recursive procedure which decomposes the initialsystem into new systems for which one has either n = 1 or r = 0. The structure of thematrix W of a formal fundamental matrix solution of the system Y ( x ) = F ( x ) x Λ e W ( x ) (5)can be determined uniquely through this method. Here F is an invertible formal mero-morphic matrix power series in a fractional power of x , Λ is a constant complex matrixcommuting with W and W is a diagonal matrix containing polynomials in the same frac-tional power of x without constant terms.For the purposes of this paper, it is useful to distinguish between the following twotypes of transformation matrices:1. Matrices containing formal meromorphic power series in the variable x , whose deter-minant is not the zero series. We will refer to this type of transformations as root-freetransformations . Two systems linked as in (4) by such a root-free transformationshall be called meromorphically equivalent or short equivalent .2. Matrices having coefficients which are formal meromorphic power series in a frac-tional power of x , whose determinant is not the zero series. We will call thesetransformations ramified transformations . If T is a ramified transformation, thesmallest integer q such that T ( x q ) is root-free is called the ramification index of T .We shall also say that T is a q -meromorphic transformation and takes a system intoa q -meromorphically equivalent system.In this paper, we are interested in the situation where one cannot find transformations ofthe first type in order to obtain a system with a non-nilpotent leading matrix. In otherwords, the introducing of a ramification is necessary. This can be stated in terms of formalsolutions by saying that the dominant (negative) power of x in the matrix W , or alter-natively the biggest slope of the Newton polygon of the system, is a rational number [2, 5].In this case, the methods in [1, 3, 8, 9] apply a series of root-free, ramified and Shearing-transformations (in [2, 7] a different strategy is employed). A Shearing-transformation isa transformation of the form S ( x ) = x p /q x p /q . . . x p n /q where p j ∈ Z and q ∈ N . In [1] it is shown that it is always possible to achieve this byusing exponential shifts and a transformation of the form T ( x ) = R ( x ) S ( x ) (6)where R is a root-free transformation having a finite number of nonzero terms and S is aramified Shearing-transformation. The transformed system x dydx = ˆ A ( x ) y (7)has a coefficient matrix of the formˆ A ( x ) = x − r ∞ X j = p ˆ A j x j/q ( ˆ A p = 0 , p ≥
0) (8)where p is relatively prime to q and ˆ A p has several eigenvalues. Applying the SplittingLemma to (7) then yields a q -meromorphic transformation taking the system into a newsystem whose coefficient matrix is block-diagonal. Hence the remaining computations arecarried out on matrices containing ramified power series.One may ask under which conditions there exists also a root-free transformation whichachieves a block-diagonalisation of the original system (1) without introducing ramifica-tions, and how to compute such a transformation.We shall give a sufficient condition for the existence of such a root-free transformationand also provide a constructive method for computing its coefficients. Denote by spec( A )the set of eigenvalues of a complex square matrix A . We will prove the following theorem:3 heorem 1.1 (“Root-Free Splitting Lemma”) . Consider the system (1) and assume thereexists a Shearing-transformation S of ramification index q taking the system into one ofthe form (7) such that its leading matrix ˆ A p is similar to a block-diagonal matrix ˆ B p = ˆ B p
00 ˆ B p ! and suppose that for all λ ∈ spec( ˆ B p ) and λ ∈ spec( ˆ B p ) it holds λ = e πik/q λ for all k ∈ N . Then there exists a root-free transformation H with the following properties:1. H transforms the system (1) into an equivalent system with block-diagonal coefficientmatrix B = (cid:18) B B (cid:19) where the block sizes match those in the matrix ˆ B p .2. The matrix S [ B ] has the leading matrix ˆ B p and the same pole order as S [ A ] .3. A finite number of coefficients of the root-free transformation H can be computedfrom a finite number of the coefficients of the system (1). The classical Splitting Lemma can be seen as a particular case of this theorem byputting S as the identity matrix and q = 1.This paper is organised as following: in Section 2, we review the classical SplittingLemma. In the following section we introduce a special class of systems and give a variantof the Splitting Lemma, particular to this class. Using this, we will give the proof ofTheorem 1.1 in Section 4 and illustrate the benefits of our theorem concerning the formalreduction in Section 5 on an example. Notations : Throughout the paper, empty entries in matrices are supposed to be filledwith 0. We write diag( a , . . . , a n ) for a (block-)diagonal matrix whose diagonal entries arethe a i . The valuation of a polynomial or formal power series (with possibly negative orfractional exponents) is the smallest occurring power in the variable x . Other definitionsof notations are made as they appear in the text. As we have previously mentioned, the Splitting Lemma is a well-known result. Its proofis carried out in a constructive fashion and gives a method for computing the coefficients T j of the transformation matrix T as in (2), see for example [1, 2, 9]. We repeat ithere for reason of completeness. Also, we will formulate it for q -meromorphic systems inpreparation of the proof of Lemma 3.4. Lemma 2.1.
Consider the system (7) and assume that ˆ A p is block-diagonal ˆ A p = ˆ A p
00 ˆ A p ! uch that spec( ˆ A p ) ∩ spec( ˆ A p ) = ∅ . Then there exists a formal q -meromorphic transformation of the form ˆ T ( x ) = ∞ X j =0 ˆ T j x j/q ( ˆ T = I ) such that the transformed system is block-diagonal with the same block partition as in ˆ A p . Proof
We use a transformation of the special formˆ T ( x ) = (cid:18) I ˆ U ( x )ˆ V ( x ) I (cid:19) with ˆ U = ˆ V = 0. Denote by ˆ B the matrix ˆ T [ ˆ A ]. Inserting the series expansion for ˆ A, ˆ B and ˆ T and comparing coefficients gives the recursion formulaˆ A p ˆ T h − ˆ T h ˆ A p = h X j =1 ( ˆ T h − j ˆ B j + p − ˆ A j + p ˆ T h − j ) + (( p + h ) /q − r ) ˆ T p + h − qr , h > T j = 0 for j <
0. Equation (9) is of the formˆ A p ˆ T h − ˆ T h ˆ A p = ˆ B h + p − ˆ A h + p + ˆ R h (10)where ˆ R h = h − X j =1 ( ˆ T h − j ˆ B j + p − ˆ A j + p ˆ T h − j ) + (( p + h ) /q − r ) ˆ T p + h − qr depends only on ˆ B j with j < h + p and ˆ T j with j < h . Using the special form ofˆ T h = (cid:18) U h ˆ V h (cid:19) , ˆ B h = (cid:18) ˆ B h
00 ˆ B h (cid:19) and decomposing ˆ R h into block-structure accordingly gives the following system of equa-tions: ˆ B p + h + ˆ R h = 0 , (11)ˆ B p + h + ˆ R h = 0 (12)where ˆ B p + h and ˆ B p + h are unknown, andˆ A p ˆ U h − ˆ U h ˆ A p = ˆ R h , (13)ˆ A p ˆ V h − ˆ V h ˆ A p = ˆ R h with unknowns ˆ U h and ˆ V h . Given ˆ R h , the first two equations (11) and (12) can be solvedby setting ˆ B p + h = − ˆ R h and ˆ B p + h = − ˆ R h . The remaining equations can be solveduniquely for ˆ U h and ˆ V h because the matrices ˆ A p and ˆ A p have no eigenvalues in common,see e.g. [4]. (cid:3) On ( ω, P )-Commutative Systems
In this section, we study a particular class of q -meromorphic systems. Starting point ofour considerations was [1, Lemma 5, Section 3.3] observing that a system transformedby a Shearing-transformation has a special structure. We will state this more generallyand give conditions under which this special structure is preserved by the Splitting-Lemma. Definition 3.1.
Let q > be a positive integer, ω = e πi/q and P ∈ C n × n . We call aformal q -meromorphic matrix ˆ A as in (8) ( ω, P )-commutative if ˆ A j P = ω j P ˆ A j ( j ≥ p ) . A system of the form (7) is called ( ω, P ) -commutative if its coefficient matrix is ( ω, P ) -commutative. Remark 3.1.
The considerations in [1] correspond, in our notation, to the case of a ( ω, P )-commutative system where P is an invertible diagonal matrix. However, this restrictionis not necessary in this section and we will develop our theory first for arbitrary matrices P . Remark 3.2.
Two complex matrices A and B satisfying AB = ωBA are called ω -commutative in [6]. In terms of their notation, the j th coefficient of a ( ω, P )-commutativematrix and the matrix P are ω j -commutative.The following lemma gives an alternative characterisation for ( ω, P )-commutative sys-tems which will be useful later. Note that similar concepts are used in [1]. Lemma 3.1.
Consider a system of the form (7). Then the following two statements areequivalent:i) ˆ A is ( ω, P ) -commutative.ii) ˆ A ( x ) P = P ˆ A ( e πi x ) . Proof
A direct calculation shows: ˆ A j P = ω j P ˆ A j ∀ j ≥ p ⇐⇒ x − r ∞ X j = p ˆ A j x j/q P = x − r ∞ X j = p e πij/q P ˆ A j x j/q ⇐⇒ ˆ A ( x ) P = P ˆ A ( e πi x ) . (cid:3) It is also straightforward to see that we have
Lemma 3.2.
Consider a system of the form (7) and suppose ˆ A is ( ω, I ) -commutativewhere I denotes the n × n identity matrix. Then ˆ A is an unramified formal meromorphicpower series matrix.
6e make the following definition: for two eigenvalues λ and λ of ˆ A p we define anequivalence relation ∼ l by λ ∼ l λ ⇐⇒ ∃ k ∈ { , . . . , q − } : λ = ω lk λ and denote by ω l -spec( A p ) the set { [ λ ] ∼ l | λ ∈ spec( ˆ A p ) } where we will, slightly abusing notation, identify λ with [ λ ] ∼ l .Given C ∈ GL( n, C ), it is clear that the matrix C − ˆ AC is ( ω, C − P C )-commutative.
Lemma 3.3.
Let ˆ A as in (8) be ( ω, P ) -commutative and let λ and λ be two eigenvaluesof ˆ A p with λ p λ . Then there exists C ∈ GL( n, C ) such that ˆ B = C − ˆ AC is ( ω, ˜ P ) -commutative and ˆ B p = ˆ B p
00 ˆ B p ! , ˜ P = (cid:18) ˜ P
00 ˜ P (cid:19) with ˜ P = C − P C , λ ∈ spec( ˆ B p ) , λ ∈ spec( ˆ B p ) and ω p - spec( ˆ B p ) ∩ ω p - spec( ˆ B p ) = ∅ . Proof
The existence of a matrix C so that ˆ B p = C − ˆ A p C satisfies the conditions of thelemma can be seen easily from elementary properties of matrix decomposition. We willuse techniques similar as in [6] in order to show that ˜ P has the required block diagonalstructure. Let ˜ P = (cid:18) ˜ P ˜ P ˜ P ˜ P (cid:19) where the block partition matches that in the matrix ˆ B p . Inserting into the equationˆ B p ˜ P = ω p ˜ P ˆ B p yields in particular the two conditionsˆ B p ˜ P − ω p ˜ P ˆ B p = 0and ˆ B p ˜ P − ω p ˜ P ˆ B p = 0 . The first of these two equations is of the formˆ B p X − Xω p ˆ B p = 0 . The assumption ω p -spec( ˆ B p ) ∩ ω p -spec( ˆ B p ) = ∅ implies that the matrices ˆ B p and ω p ˆ B p have no eigenvalue in common. The above equation therefore has the unique solution˜ P = 0. A very similar argument applies to the second equation. This proves the lemma. (cid:3) Remark 3.3.
The matrices C , ˜ B and ˜ P are in general not uniquely determined.We now show that for a ( ω, P )-commutative system which has block-diagonal structureas in Lemma 3.3, application of the Splitting Lemma preserves the property of being( ω, P )-commutative. 7 emma 3.4 ( “Splitting Lemma for ( ω, P )-Commutative Systems”). Consider thesystem (7) and assume that ˆ A is ( ω, P ) -commutative with ˆ A p and P block-diagonal withblocks of same dimension ˆ A p = ˆ A p
00 ˆ A p ! , P = (cid:18) P P (cid:19) such that ω p - spec( ˆ A p ) ∩ ω p - spec( ˆ A p ) = ∅ . Then there exists a ( ω, P ) -commutative q -meromorphic transformation of the form ˆ T ( x ) = ∞ X j =0 T j x j/q ( T = I ) (14) such that the transformed system is ( ω, P ) -commutative and block-diagonal with the sameblock partition as in ˆ A p and P . Proof
The existence of the transformation ˆ T is given by Lemma 2.1, the classical Split-ting Lemma. What remains to show is that ˆ T and the transformed system are ( ω, P )-commutative. Denote by ˆ B the coefficient matrix of the transformed system. Using thenotations as in (9) and (10), we will show that the following relations hold:ˆ R k P = ω p + k P ˆ R k , (15)ˆ T k P = ω k P ˆ T k , (16)ˆ B p + k P = ω p + k P ˆ B p + k (17)for k ∈ N . The case k = 0 holds trivially by putting ˆ R = 0 since ˆ T = I and ˆ B p = ˆ A p .Let h be an arbitrary positive integer. We will see that if the above relations hold for k = 0 , . . . , h − h . The claim follows then by induction.We computeˆ R h P = h − X j =1 ( ˆ T h − j ˆ B j + p − ˆ A j + p ˆ T h − j ) P + (( p + h ) /q − r ) ˆ T p + h − qr P = h − X j =1 ω p + h P ( ˆ T h − j ˆ B j + p − ˆ A j + p ˆ T h − j ) + (( p + h ) /q − r ) ω p + h P ˆ T p + h − qr = ω p + h P ˆ R h where we have used (16) and (17) for k = 0 , . . . , h − A is( ω, P )-commutative. This proves (15) for k = h .We decompose ˆ R h into blocks accordingly to the block structure of ˆ B h and P and findusing (11) ˆ B p + k P = − ˆ R k P = − ω p + k P ˆ R k = ω p + k ˆ B p + k P . We can show an analogous relationship for ˆ B p + k and P using (12). Hence we can seethat (17) holds for k = h . 8t remains to show (16), which is equivalent to showingˆ U h P = ω h P ˆ U h , (18)ˆ V h P = ω h P ˆ V h . (19)We will only show that the first of these two equations holds, the second can be dealtwith similarly. Multiplying (13) with ω p P on the left and with P on the right andcombining the two equations yieldsˆ A p ( ˆ U h P − ω h P ˆ U h ) − ( ˆ U h P − ω h P ˆ U h ) ω p ˆ A p = 0 . This equation is of the form ˆ A p X − Xω p ˆ A p = 0 . The assumption ω p -spec( ˆ A p ) ∩ ω p -spec( ˆ A p ) = ∅ implies that the matrices ˆ A p and ω p ˆ A p have no eigenvalue in common. The above equation therefore has the unique solution X = 0, from which we conclude (18). This completes the proof of the lemma. (cid:3) Remark 3.4.
We observe that the two block matrices in the transformed system are( ω, P )-commutative and ( ω, P )-commutative respectively. We define a generalised Shearing-transformation as a transformation of the form SC where S is a Shearing-transformation and C ∈ GL( n, C ). Proposition 4.1.
Consider a system as in (7) with leading matrix ˆ A p and let q ≥ . Thefollowing statements are equivalent:i) There exists a system as in (1) and a generalised Shearing-transformation ˜ S of ram-ifications index q such that ˜ S [ A ] = ˆ A .ii) The system (7) is ( ω, ˜ P ) -commutative, the matrix ˜ P is similar to a diagonal matrixand spec( ˜ P ) ⊆ { , ω, ω , . . . , ω q − } . Furthermore, if λ is an eigenvalue of ˆ A p withmultiplicity s , the numbers ωλ, . . . , ω ( q − λ are all eigenvalues of the same multiplic-ity s . Proof
We proof i ) ⇒ ii ): let ˜ S = SC be the generalised Shearing-transformation. Since˜ S ( e πi x ) = ˜ S ( x ) ˜ P where ˜ P = C − P C with P = S ( e πi ), we find with ˆ A = ˜ S [ A ]˜ P ˆ A ( e πi x ) = ˜ P ˜ S − ( e πi x ) A ( e πi x ) ˜ S ( e πi x ) − ˜ P x ˜ S − ( e πi x ) ˜ S ′ ( e πi x )= ˆ A ( x ) ˜ P showing that ˆ A is ( ω, ˜ P )-commutative where ˜ P satisfies the stated properties. The claimedsymmetry in the spectrum of ˆ A p can be shown as in the proofs of [1, Lemma 5, Section3.3] and [6, Theorem 5] since we have ˜ P − ˆ A p ˜ P = ω p ˆ A p and ω p is a primitive q th root ofunity. 9n order to prove the converse direction, we first assume that ˜ P = diag( ω α , . . . , ω α n )with α j ∈ { , . . . , q − } and define the Shearing-transformation S ( x ) = diag( x − α /q , x − α /q , . . . , x − α n /q ) . We observe that S ( e πi x ) = ˜ P − S ( x ). Transform the given system using this transforma-tion S and denote the coefficient matrix of the transformed system by B . We compute B ( e πi x ) = S − ( e πi x ) ˆ A ( e πi x ) S ( e πi x ) − xS − ( e πi x ) S ′ ( e πi x )= S − ( x ) ˜ P ˆ A ( e πi x ) ˜ P − S ( x ) − xS − ( x ) S ′ ( x )= B ( x )showing that B is ( ω, I )-commutative and hence (Lemma 3.2) B must be a unramifiedformal meromorphic power series matrix. The case of the general matrix ˜ P follows by firstapplying a constant similarity transformation which diagonalises P . (cid:3) We can now give the proof of our main theorem.
Proof of Theorem 1.1
Let C ∈ GL( n, C ) such that ˆ B = C − ˆ AC has a leading matrixˆ B p as in the assumptions of the Theorem. In a similar way as in the proof of Lemma3.3, we can see that C is block-diagonal, matching the block structure of ˆ B p . Using thisand Proposition 4.1, we obtain that ˆ B is ( ω, ˜ P )-commutative where ˜ P = C − S ( e πi x ) C is similarly block-diagonal. Note that p and q being relatively prime, the two conditions ω p -spec( ˆ B p ) ∩ ω p -spec( ˆ B p ) = ∅ and ω -spec( ˆ B p ) ∩ ω -spec( ˆ B p ) = ∅ are equivalent. Wecan therefore apply Lemma 3.4 to ˆ B in order to obtain a ( ω, ˜ P )-commutative transfor-mation matrix ˆ T such that ˆ T [ ˆ B ] is ( ω, ˜ P )-commutative and block-diagonal with matchingblock structure.We claim that the transformation matrix H ( x ) = S ( x ) C ˆ T ( x ) C − S − ( x )is root-free satisfying the desired properties of the theorem. It is clear that H [ A ] is block-diagonal. But one verifies that H is ( ω, I )-commutative and hence is root-free. Theremaining properties follow immediately. (cid:3) Consider the situation where the system (1) is q -meromorphically equivalent to a systemas in (7) whose leading matrix ˆ A p has several eigenvalues but is not invertible. Theexponential matrix polynomial W in a formal fundamental matrix solution (5) is then W ( x ) = diag( w ( x ) , w ( x ) , . . . , w n ( x ))where the leading terms of diagonal entries of the form w k ( x ) = λ k x − r + pq + · · · ( λ k = 0)10re given by nonzero eigenvalues λ k of ˆ A p . The diagonal entries having valuation greaterthan − r + pq correspond to eigenvalues zero. In particular, these entries might involve ram-ifications different to q or no ramifications at all. Algorithms using the classical SplittingLemma will not be able to compute these entries without first introducing the ramification q . In order to see how we can remedy this situation, we use the fact that ˆ A p is similar toa matrix of the form ˆ B p
00 ˆ B p ! where ˆ B p is invertible and ˆ B p is nilpotent. Hence the conditions for Theorem 1.1 aresatisfied and we will obtain a root-free transformation H which splits the system into x dydx = (cid:18) B B (cid:19) y. (20)This makes it possible to work independently on the two matrices B and B : for thefirst matrix we can use a Shearing-transformation introducing the (necessary) ramification q . For the second matrix however we now recursively apply the formal reduction process.In order to illustrate this approach, consider the following example with n = 5, r = 2and x − A ( x ) = x − − x − x − − x − x − − x − x − x − − x − x − x − x − − x − − . The Shearing-transformation S ( x ) = diag( S ( x ) , S ( x )) with S ( x ) = (cid:18) √ x (cid:19) , S ( x )) = √ x x transforms the system into a system of the form (7) with ramification index q = 2, p = 1and block-diagonal leading matrix with the two blocksˆ B = (cid:18) − (cid:19) , ˆ B = and the condition of Theorem 1.1 is satisfied since the first block matrix is invertible andthe second is nilpotent. We obtain a root-free transformation which is of the form H ( x ) = (cid:18) I U ( x ) V ( x ) I (cid:19) , U ( x ) = (cid:18) − x + 49 x − x + 18 x x − x x − x − x + 46 x − x + 16 x (cid:19) and V ( x ) = x − x + 290 x − x + 89 x − x + 8 x − x x − x + 274 x − x + 28 x − x − x + 9 x − x . Applying this transformation to the original system yields the following block-diagonalmatrix: B ( x ) = x − − x − − x − x − x − x − x − − x − + O (1) . Applying the Shearing-transformation S to the first block matrix will result in a ramifiedsystem of smaller size and invertible leading matrix equalling the first block of ˆ B . Theformal reduction can now be applied to the second block. In this example, it is foundthat another Shearing-transformation of ramification index q = 3 results in a system withinvertible leading matrix. This decomposition can be interpreted as separation of the dif-ferent slopes of the Newton-polygon, see Theorem 5.1 below.We conclude that if the algorithm employed for computing the transformation (4)keeps the introduced ramification minimal (as for example the algorithm in [1]), using theroot-free Splitting Lemma allows the recursive computation of W using minimal ramifi-cations. This approach leads to a root-free transformation taking a system of the form(1) into a system from which all the leading terms of the matrix W (or, alternatively theNewton polygon) can be determined directly. We state this as Theorem 5.1.
The given system (1) is meromorphically equivalent to a system x dydx = B ( x ) y where the matrix B is block-diagonal B ( x ) = diag( B (1) ( x ) , B (2) ( x ) , . . . , B ( ν ) ( x )) with ν ≤ n and there exist a diagonal transformation with blocks of same sizes S ( x ) = diag( S (1) ( x ) , S (2) ( x ) , . . . , S ( ν ) ( x )) with S ( k ) Shearing-transformations of ramification index q k ( k = 1 , . . . , ν ) such that eachof the matrices ˆ B ( k ) = S ( k ) [ B ( k ) ] , k = 1 , . . . , ν has either no pole at x = 0 or an invertibleleading matrix. In the latter case, let r k = r − p k q k (gcd( p k , q k ) = 1) be the pole order and ˆ B ( k ) p k the leading matrix of ˆ B ( k ) . Then | ω - spec( ˆ B ( k ) p k ) | = 1 and if λ k is an eigenvalue of ˆ B ( k ) p k with multiplicity s k , the eigenvalues ωλ k , . . . , ω q k − λ k are all of the same multiplicity s k . There are q k s k diagonal entries in the matrix W of the form w k,j ( x ) = ω j λ k x − r k + · · · ( j = 0 , . . . , q k − here the dots denote terms with higher powers of x . The Newton polygon of the systemcorresponding to this block admits a single slope r k of length q k s k . Proof
The proof follows from the procedure we have outlined in this section, Proposition4.1, and additional application of the root-free Splitting Lemma to B if B p is similarto a block-diagonal matrix such that, for each block, all eigenvalues are congruent modulo ∼ p . (cid:3) The issue of how to implement the root-free Splitting Lemma in a computer algebra sys-tem deserves additional attention. In particular, the question arises whether there exists amore direct way of obtaining the root-free transformation matrix. Furthermore, it seemslikely that a combination of the results of this paper together with our method in [7], wherewe have given a different generalisation of the Splitting Lemma, will lead to a significantimprovement of the algorithmic formal reduction of systems of linear differential equations.
References [1] W. Balser.
Formal Power Series and Linear Systems of Meromorphic Ordinary Dif-ferential Equations . Springer New York, 2000.[2] M.A. Barkatou. An algorithm to compute the exponential part of a formal fundamentalmatrix solution of a linear differential system.
Journal of App. Alg. in Eng. Comm. andComp. , 8(1):1–23, 1997.[3] G. Chen. An algorithm for computing the formal solutions of differential systems inthe neighbourghood of an irregular singular point. In
Proceedings of ISSAC ’90 , pages231–235, Tokyo, Japan, 1990. ACM Press.[4] F.R. Gantmacher.
The Theory of Matrices . Volumes 1 and 2, Chelsea, New York,1959.[5] A. Hilali and A. Wazner. Un algorithme de calcul de l’invariant de Katz d’un syst`emediff´erentiel lin´eaire.
Ann. Inst. Fourier , 36(3):67–81, 1986.[6] O. Holtz, V. Mehrmann, and H. Schneider. Potter, Wielandt, and Drazin on thematrix equation AB = ǫBA : new answers to old questions. Amer. Math. Month. ,111:655–667, 2004.[7] E. Pfl¨ugel. Effective formal reduction of linear differential systems.
Appl. Alg. Eng.Comm. Comp. , 10(2):153–187, 2000.[8] H.L. Turritin. Convergent solutions of ordinary linear homogeneous differential equa-tions in the neighborhood of an irregular singular point.
Acta Math. , 93:27–66, 1955.[9] W. Wasow.