Desingularization of First Order Linear Difference Systems with Rational Function Coefficients
aa r X i v : . [ c s . S C ] F e b Desingularization of First Order Linear Difference Systems withRational Function Coefficients
Moulay A. Barkatou
XLIM UMR 7252 , DMI, University of Limoges; CNRS123, Avenue Albert Thomas, 87060 Limoges, [email protected]
Maximilian Jaroschek ∗ Technische Universit¨at Wien,Institut for Logic and ComputationFavoritenstraße 9-11, 1040 Vienna, AustriaJohannes Kepler Universit¨at Linz, Institute for AlgebraAltenberger Straße 69, 4040 Linz, [email protected]
ABSTRACT
It is well known that for a first order system of linear differenceequations with rational function coefficients, a solution that is holo-morphic in some left half plane can be analytically continued to ameromorphic solution in the whole complex plane. The poles stemfrom the singularities of the rational function coefficients of thesystem. Just as for differential equations, not all of these singular-ities necessarily lead to poles in solutions, as they might be whatis called removable. In our work, we show how to detect and re-move these singularities and further study the connection betweenpoles of solutions and removable singularities. We describe two al-gorithms to (partially) desingularize a given difference system andpresent a characterization of removable singularities in terms ofshifts of the original system.
KEYWORDS systems of linear difference equations, apparent singularities, desin-gularization, removable singularities.
ACM Reference format:
Moulay A. Barkatou and Maximilian Jaroschek. 2018. Desingularization ofFirst Order Linear Difference Systems with Rational Function Coefficients.In
Proceedings of ISSAC ’18, July 16-19, 2018, New York, USA, ,
First order linear difference systems are a class of pseudo-linearsystems [5, 9, 15] of the form ϕ ( Y ) = AY , where ϕ is the forward-or backward shift operator and A an invertible matrix with, in ourcase, rational function coefficients. To study properties of possi-ble solutions Y , it is not always necessary to explicitly computethe solution space, but one can rather obtain the information fromthe system itself. Properties that can be derived in this fashion ∗ The author is supported by the ERC Starting Grant 2014 SYMCAR 639270 and theAustrian research projects FWF Y464-N18 and FWF RiSE S11409-N23.Permission to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full cita-tion on the first page. Copyrights for components of this work owned by others thanACM must be honored. Abstracting with credit is permitted. To copy otherwise, or re-publish, to post on servers or to redistribute to lists, requires prior specific permissionand/or a fee. Request permissions from [email protected].
ISSAC ’18, July 16-19, 2018, New York, USA © 2018 ACM. 978-x-xxxx-xxxx-x/YY/MM...$15.00DOI: 10.1145/nnnnnnn.nnnnnnn comprise, among others, the asymptotic behavior [6, 7, 13], posi-tive/negative (semi-) definiteness [1, 22], holomorphicity, and clo-sure properties of (a class of) solutions [18].In the center of attention when analyzing difference and differ-ential systems lie the poles of the rational function coefficients. Itis well known that, like in the case of differential equations andsystems, not all poles of the coefficients of a difference systemlead to singularities for solutions. These apparent singularities cantherefore distort the properties of solutions and should be circum-vented in the analysis. One technique to do so is desingularization —transforming a given system (or operator) in a way that removes asmany poles of the system as possible to discard apparent singular-ities. In this paper we describe the first algorithm to desingularizefirst order linear difference systems with rational function coeffi-cients. Our main tool in the treatment of these systems are polyno-mial basis transformations. We show how to achieve desingular-ization by composing several basic and easy to compute transfor-mations, and our procedure results in the provably “smallest pos-sible” such desingularizing transformation in the sense that anyother desingularizing transformation can be obtained as a rightmultiple.The main contributions of this paper are:(1) The first algorithm to desingularize—partially, or, if possi-ble, completely—first order linear difference systems withrational function coefficients.(2) A non-trivial necessary and sufficient condition for a givensystem to be desingularizable at a given singularity.(3) With the help of (2), an analysis of the connection be-tween removable and apparent singularities of differencesystems and their meromorphic function solutions.(4) An algorithm for reducing the rank of the leading matrixat a singularity of a linear difference system.In the context of single linear difference equations [1, 2], lineardifferential equations [21] and, more general, Ore operators [10,11, 17], desingularization and the effects of removable singular-ities have been extensively studied in recent years. In [22], theauthor presents an extension of the idea of desingularization thatalso takes into account the leading number coefficients of Ore op-erators. For first order differential equations, a first algorithm fordesingularization was given in [3].It is possible to convert any first order linear difference systemto a difference operator of higher order and vice versa [4, 6–8].Desingularization of systems could therefore be done by comput-ing for a given system the corresponding operator, use existingechniques to desingularize the operator and then constructing thedesingularized system from the new operator. While this is possi-ble, the procedure comes with at least two caveats:(1) It can be observed that the coefficients grow very large inthe conversion process which has severe negative impacton the computation time.(2) Desingularization on operator level is done by finding asuitable left-multiple of the given operator. In general, thisleads to an increase in order, and thus to an increase in thedimension of the solution space.Both problems are avoided when dealing directly with systems in-stead of operators, making the results presented in this paper anessential tool for analyzing difference systems.The paper is organized as follows. In Section 2 we remind thereader of the formal definition of linear difference systems withrational function coefficients, well known results about meromor-phic function solutions and the notion of apparent singularities. InSection 3, we present an algorithm to remove poles of differencesystems and give a necessary and sufficient condition for a singu-larity to be removable. Lastly, the connection between removablepoles and apparent singularities is established in Section 4 beforeconcluding the paper in Section 5.
Let C be a subfield of the field C of complex numbers, C( z ) thefield of rational functions over C and ϕ the C -automorphism of C( z ) defined by ϕ ( z ) = z +
1. A homogeneous system of first-orderlinear difference equations with rational function coefficients is asystem of the form ϕ ( Y ) = AY , (1)where Y is an unknown d -dimensional column vector, ϕ ( Y ) is de-fined component-wise, and A is an element of GL d (C( z )) , the groupof invertible matrices of size d × d with entries in C( z ) . We denotethe set of matrices of size d × d with entries in C[ z ] as Mat d (C[ z ]) . A(block) diagonal matrix with entries (respectively blocks) a , . . . , a d is denoted by diag ( a , . . . , a d ) . We will refer to system (1) as [ A ] ϕ .Given a matrix T ∈ GL d (C( z )) , we can apply a basis transfor-mation Y = T X , and substitute T X into system (1) to arrive at an equivalent system ϕ ( X ) = T [ A ] ϕ X , where T [ A ] ϕ is defined as T [ A ] ϕ : = ϕ ( T − ) AT . A difference system [ A ] ϕ can be rewritten as ϕ − ( Y ) = A ∗ Y , (2)where A ∗ : = ϕ − ( A − ) . We will refer to system (2) as [ A ∗ ] ϕ − . Atransformation Y = T X yields the equivalent system ϕ − ( X ) = T [ A ∗ ] ϕ − X , with T [ A ∗ ] ϕ − : = ϕ − ( T − ) A ∗ T . The set of meromorphic solutions of [ A ] ϕ form a vector space ofdimension d over the field of 1-periodic meromorphic functions. Itis well known [19] that any difference system [ A ] ϕ possesses a fun-damental matrix of meromorphic solutions. If F is a holomorphicsolution of (1) in some left half plane (Re z < λ for some λ ∈ R ),then it can be analytically continued to a meromorphic solution inthe whole complex plane C using the relations: F ( z ) = ϕ − ( A ) ϕ − ( A ) · · · ϕ − n ( A ) ϕ − n ( F )( z ) = A ( z − ) A ( z − ) · · · A ( z − n ) F ( z − n ) , which are valid everywhere except at the points of the form ζ + n where ζ is a pole of A and n is a positive integer ( n = , , . . . ). If F is a holomorphic solution of (1) in some right half plane (Re z > λ ),then it can be analytically continued to a meromorphic solution inthe whole complex plane C using the relations: F ( z ) = ϕ ( A ∗ ) ϕ ( A ∗ ) · · · ϕ n ( A ∗ ) ϕ n F ( z ) = A ∗ ( z + ) A ∗ ( z + ) · · · A ∗ ( z + n ) F ( z + n ) , which are valid everywhere except at the points of the form ζ − n where ζ is a pole of A ∗ and n is a positive integer ( n = , , . . . ).We will denote by P r ( A ) (respectively P l ( A ) ) the set of polesof A (respectively A ∗ ). The elements of P r ( A ) (respectively P l ( A ) )will be called the r- (respectively l-) singularities of the system (1).A point ζ ∈ C is said to be congruent to a given r- (respectively l-)singularity ζ of [ A ] ϕ if ζ = ζ + k (respectively ζ = ζ − k ) forsome positive integer k .The finite singularities of the solutions of [ A ] ϕ are among thepoints that are congruent to the singularities of the system. Definition 2.1.
Let ζ be a pole of A (respectively pole of A ∗ ). Itis called(1) a removable r- (respectively l-) singularity if any solutionof [ A ] ϕ which is holomorphic in some left (respectivelyright) half-plane can be analytically continued to a mero-morphic solution which is holomorphic at ζ + ζ − [ A ] ϕ which is holomorphic in some left (respectivelyright) half-plane can be analytically continued to a mero-morphic solution which is holomorphic at each point of ζ + N ∗ (respectively ζ − N ∗ ). Example 2.2.
A 2 × Y ( z + ) = AY = (cid:18) − ( z + ) z − ( z − ) z − (cid:19) Y ( z ) , A ∗ = (cid:18) ( z − ) z − z z (cid:19) . Here P r ( A ) = { } and the points that are congruent to ζ = , , , . . . . We have P l ( A ) = { } and the corresponding congruentpoints are − , − , − , . . . . It can be easily verified that a fundamen-tal matrix of solutions of this system is given by F ( z ) = (cid:18) z z + z + z + z + z + z + (cid:19) . We focus on studying r-singularities. L-singularities can be re-moved in the same way by considering A ∗ and ϕ − instead of A and ϕ .e give an algebraic characterization of removable singularities.Let q ∈ C[ z ] be an irreducible polynomial. For f ∈ C( z ) \ { } , wedefine ord q ( f ) to be the integer n such that f = q n ab , with a , b ∈C[ z ] \ { } , q ∤ a and q ∤ b . We put ord q ( ) = + ∞ . Let O q = { f ∈C( z ) : ord q ( f ) ≥ } be the local ring at q and O q / q O q the residuefield of C( z ) at q . Let π q denote the canonical homomorphism from C[ z ] onto C[ z ]/h q i . It can be extended to a ring-homomorphismfrom O q onto C[ z ]/h q i as follows: let f ∈ O q ; by definition of O q , f can be written f = a / b where a , b ∈ C[ z ] and q ∤ b . Wecan find u , v ∈ C[ z ] such that ub + vq = the value of f at q ,denoted by π q ( f ) , is then defined as π q ( ua ) . Sometimes we write f mod q for π q ( f ) . It is clear that π q is well-defined on O q andis a surjective ring-homomorphism. The kernel of π q is q O q , so O q / q O q and C[ z ]/h q i are isomorphic.If A = ( a i , j ) is a finite-dimensional matrix with entries in C( z ) ,we define the order at q of A by ord q ( A ) : = min i , j ( ord q ( a i , j )) . Wesay that A has a pole at q if ord q ( A ) <
0. We define the leadingmatrix of A at q (notation lc q ( A ) ) as the leading coefficient A , q inthe q -adic expansion of A : A = q ord q ( A ) ( A , q + qA , q + q A , q + . . . ) . Here the coefficients A i , q are matrices with entries in the field C[ z ]/h q i . Note that the matrix A , q is the value of the matrix q − ord q ( A ) A at q .For a rational function r = p / q with p monic and gcd ( p , q ) = ( r ) : = p and den ( r ) : = q . Similarly, for a matrix M ∈ Mat d (C( z )) we denote by den ( M ) the common denominator of allthe entries of M and denote by num ( M ) the polynomial matrixnum ( M ) : = den ( M ) M . Definition 2.3.
Let A ∈ GL d (C( z )) and q ∈ C[ z ] be an irreduciblepolynomial. We say that q is a ϕ -minimal pole of A if q | den ( A ) and for all j ∈ N ∗ , ϕ j ( q ) ∤ den ( A ) .We can now give an algebraic definition of desingularizabilityof difference systems in an inductive fashion. Definition 2.4.
Let A ∈ GL d (C( z )) and let q ∈ C[ z ] be an irre-ducible pole of A .(1) If q is ϕ -minimal, we say that the system [ A ] ϕ is partiallydesingularizable at q if there exists a polynomial transfor-mation T ∈ GL d (C( z ))∩ Mat d (C[ z ]) such that ord q ( T [ A ] ϕ ) > ord q ( A ) and ord p ( T [ A ] ϕ ) ≥ ord p ( A ) for any other irre-ducible polynomial p ∈ C[ z ] . If moreover, ord q ( T [ A ] ϕ ) ≥ [ A ] ϕ is desingularizable at q and wecall T a desingularizing transformation for [ A ] ϕ at q .(2) If q is not ϕ -minimal, then we call [ A ] ϕ (partially) desin-gularizable at q if there exists a desingularizing transfor-mation T for all poles of A of the form ϕ k ( q ) , k ≥
1, and T [ A ] ϕ is either (partially) desingularized at q or (partially)desingularizable at q .While it is immediate that, for a ϕ -minimal pole q , the algebraicnotion of desingularization implies that the roots of q are remov-able in the sense of Definition 2.1, the converse is not obvious andis proven later in Section 4. Consequently, the roots of q are appar-ent singularities if (and only if) the system A is desingularizableat all poles of A of the form ϕ − k ( q ) , k ≥
0. In practice, in order to desingularize a system at a non- ϕ -minimal pole q , one first re-moves the ϕ -minimal pole congruent to q . The resulting systemthen has a new ϕ -minimal pole ’closer’ to q . One can repeat thisprocess until q itself is ϕ -minimal and eventually removed. A desin-gularizing transformation for q is then given by the product of allthe transformations obtained during this process.Let us illustrate in the next example why we require removingall singularities left of a given pole, thus making it ϕ -minimal, be-fore considering it eligible for desingularization. Example 2.5.
The system [ A ] ϕ given by A = diag (cid:18) ( z + ) z , z + (cid:19) , can be transformed via T = diag ( z , ) to T [ A ] ϕ = diag ( z + , z + ) .The transformed systems still does not enable analytic continua-tion at 0 of solutions that are holomorphic in the left half-planewith Re ( z ) < ϕ -Minimal Desingularization We begin our discussion of removing r-singularities by deriving amethod for shifting a factor in the denominator of a given systemin a way that allows, if possible, cancellation with zeroes of thesystem. For this we bring the leading matrix of A into a specificform. Lemma 3.1.
Let A be a d × d matrix with entries in C( z ) and let q ∈ C[ z ] be an irreducible pole of A . Set n : = − ord q ( A ) and r : = rank ( lc q ( A )) , the rank of the leading matrix of A at q . There existsa unimodular polynomial transformation S such that S [ A ] ϕ is of theform (cid:16) q n A q n − A , (cid:17) , (3) where A , A are matrices with entries in O q of size d × r and d × d − r respectively with rank ( A ) = r . Proof.
The leading matrix lc q ( A ) of A at q is a matrix with en-tries in the residue field C[ z ]/h q i . There exists a non-singular ma-trix Q with entries in C[ z ]/h q i such that lc q ( A ) · Q is in a column-reduced form, i.e. the last d − r columns of lc q ( A )· Q are zero, and Q is of the form Q = C · ( I d + U ) , where C is a non-singular constantmatrix, I d the identity matrix of dimension d × d , and U is a strictlyupper triangular matrix. Taking S = Q as a matrix in Mat d (C[ z ]) will result in S [ A ] ϕ as desired. (cid:3) Example 3.2 (Example 2.2 continued).
If we set q = z −
2, thenthe leading matrix of the system in Example 2.2 at q islc q ( A ) = ( qA ) mod q = (cid:18) − (cid:19) . A suitable transformation to bring this matrix into a column-reducedform is S = (cid:18) (cid:19) . Applying S to A gives S [ A ] ϕ = (cid:18) z + z − − z − z − (cid:19) . emma 3.3. Let A ∈ GL d (C( z )) and q ∈ C[ z ] be a ϕ -minimalpole of A . Suppose that A is of the form (3) and let r = rank ( lc q ( A )) .If [ A ] ϕ is partially desingularizable at q then any desingularizingtransformation T for [ A ] ϕ can be written as T = D · ˜ T , where D = diag ( q , . . . , q | {z } r times , , . . . , | {z } d − r times ) and ˜ T ∈ GL d (C( z )) ∩ Mat d (C[ z ]) . Proof.
Let n = − ord q ( A ) . Suppose we are given a desingular-izing transformation T ∈ GL d (C( z )) and let B = T [ A ] ϕ . Then wehave that ϕ ( T ) B = AT and hence ϕ ( T )( q n B ) = ( q n A ) T . Since ord q ( q n B ) > q of the other matricesinvolved in the equality are non-negative, we get π q ( q n A ) π q ( T ) = . By assumption, the matrix π q ( q n A ) is of the form π q ( q n A ) = (cid:0) A (cid:1) , where A is a d × r matrix with linearly independent columns. Thus,the first r rows of π q ( T ) must be zero, i.e the first r rows of T haveto be divisible by q . This yields the claim. (cid:3) Remark 1.
Note that the determinant of any desingularizing trans-formation T of A at q , not necessarily ϕ -minimal, is divisible by q ; infact q r | det ( T ) . It then follows that ϕ ( q ) divides det ( ϕ ( T )) ; in fact ϕ ( q ) r | det ( ϕ ( T )) . Lemma 3.4.
Let A ∈ GL d (C( z )) and let q ∈ C[ z ] with q | den ( A ) be an irreducible pole. If [ A ] ϕ is (partially) desingularizable at q thenthere exists a maximal positive integer ℓ such that ϕ ℓ ( q ) | num ( det ( A )) . Proof.
First, suppose q is ϕ -minimal. There are only finitelymany factors of num ( det ( A )) of positive degree because A is non-singular. Thus it suffices to show that there exists a positive integer ℓ such that ϕ ℓ ( q ) | num ( det ( A )) . Let T be a desingularizing trans-formation of A at q . Put B : = T [ A ] ϕ and denote det ( T ) by t . Then,due to the desingularization property, we have that ϕ ( T ) − num ( A ) T = den ( A ) den ( B ) num ( B ) ∈ Mat d (C[ z ]) . Hence det ( num ( A )) tϕ ( t ) ∈ C[ z ] . Let ℓ be the largest integer such that ϕ ℓ ( q ) | ϕ ( t ) . By Remark 1, ℓ is strictly positive. Since ϕ ℓ ( q ) ∤ t , it follows that ϕ ℓ ( q ) | det ( num ( A )) . Now from the relation det ( num ( A )) = den ( A ) d det ( A ) and since we assumed that den ( A ) has no factor of the form ϕ j ( q ) with j ∈ N ∗ we can conclude that ϕ ℓ ( q ) | num ( det ( A )) . To seethat the theorem holds for non- ϕ -minimal poles, let ˜ q be a non- ϕ -minimal pole congruent to q , i.e. there exists a positive integer k such that ϕ k ( ˜ q ) = q . Then ϕ k + ℓ ( ˜ q ) = ϕ ℓ ( q ) | num ( det ( A )) . (cid:3) Definition 3.5.
Let A ∈ GL d (C( z )) and q ∈ C[ z ] be an irreduciblepole of A . We define the ϕ -dispersion of A at q as : ϕ -dispersion ( A , q ) = max { ℓ ∈ N ∗ s.t. ϕ ℓ ( q ) | num ( det ( A ))} . When the latter set is empty we put ϕ -dispersion ( A , q ) =
0. Note that a necessary condition that [ A ] ϕ can be (partially) desin-gularized at q is that ϕ -dispersion ( A , q ) > Example 3.6 (Example 3.2 continued).
The determinant of S [ A ] ϕ in Example 3.2 is ( z + ) z − . Therefore the ϕ -dispersion of S [ A ] ϕ at q = z − [ A ] ϕ at a ϕ -minimal pole q . By repeatedly applying thealgorithm to [ A ] ϕ , it is then possible to desingularize the system atall removable singularities. It is sufficient to treat the case where q is a single and simple pole of A (i.e. qA has polynomial entries).This is stated in the following lemma. Lemma 3.7.
Let A ∈ GL d (C( z )) and let q ∈ C[ z ] be a ϕ -minimalpole of A . Set h = den ( A ) q so that the matrix hA = q − num ( A ) has asingle and simple pole at q . Then the system [ A ] ϕ is (partially) desin-gularizable at q if and only if the system [ hA ] ϕ is desingularizableat q . More precisely, a polynomial matrix T ∈ GL d (C( z )) is a desin-gularizing transformation for [ hA ] ϕ at q if and only if T (partially)desingularizes [ A ] ϕ at q . Proof.
It is a direct consequence of the following (trivial butinteresting) property: for all T ∈ GL d (C( z )) and h ∈ C[ z ] \ { } ,one has T [( hA )] ϕ = h · ( T [ A ] ϕ ) . (cid:3) Remark 2.
With the notation of the above lemma, the ϕ -dispersionof [ hA ] ϕ at q is greater than or equal to the ϕ -dispersion of [ A ] ϕ at q ,and equality holds if q is ϕ -minimal. It follows from the fact that det ( hA ) = h d · det ( A ) . Lemma 3.8.
Let A ∈ GL d (C( z )) . Suppose that A has a single,simple, irreducible pole at q . If [ A ] ϕ is desingularizable at q with ϕ -dispersion ℓ , then there exist a unimodular polynomial matrix S and a diagonal polynomial matrix D such that ( S · D )[ A ] ϕ is eitherdesingularized (with respect to q ) or desingularizable at ϕ ( q ) with ϕ -dispersion ℓ − . Proof.
We first take S as in Lemma 3.1 so that S [ A ] ϕ has theform S [ A ] ϕ = © « ˜ A , q ˜ A , A , q ˜ A , ª®¬ , where the ˜ A i , j are blocks with polynomial entries, the diagonalblocks are of size r = rank ( lc q ( A )) and d − r respectively. Take D = diag ( q I r , I d − r ) as in Lemma 3.3. Then the matrix B : = ( S · D )[ A ] ϕ has the form B = ˜ A , ϕ ( q ) ˜ A , ϕ ( q ) ˜ A , ˜ A , ! . The resulting system [ B ] ϕ has at worst a simple and single pole at ϕ ( q ) with ϕ -dispersion ℓ − (cid:3) Example 3.9 (Example 3.6 continued).
The rank of the leadingmatrix in Example 3.2 is 1. We apply the transformation D = (cid:18) z − (cid:19) , o S [ A ] ϕ of Example 3.6 and arrive at the system ( S · D )[ A ] ϕ = (cid:18) z + z − − z − (cid:19) . The determinant of ( S · D )[ A ] ϕ is ( z + ) z − . The new ϕ -dispersionis 2. Theorem 3.10.
Let A be desingularizable at a single, simple, irre-ducible pole q . Then there exists an integer n , unimodular polynomialmatrices S , . . . , S n and diagonal polynomial matrices D , . . . , D n such that T = S · D · · · S n · D n , is a desingularizing transformation for A at q . Furthermore, anyother desingularizing transformation T ′ for A at q can be writtenas T ′ = T · ˜ T with ˜ T ∈ GL d (C( z )) ∩ Mat d (C[ z ]) . (4) Proof.
By Lemma 3.4, a desingularizable system [ A ] ϕ has strictlypositive ϕ -dispersion ℓ . Applying the transformation S · D as inLemma 3.8 gives a system equivalent to [ A ] ϕ having at worst apole at ϕ ( q ) (instead of q ) but with reduced ϕ -dispersion. Afterat most ℓ such transformations, the resulting matrix T [ A ] ϕ has tobe desingularized at q . This shows that T can be chosen as in thestatement of the theorem. To see that any other desingularizingtransformation T ′ of [ A ] ϕ at q can be written as in (4), we firstnote that since S is unimodular, for any such T ′ we have T ′ = S · ( S − · T ′ ) | {z } = : T ′′ ∈ GL d (C( z ))∩ Mat d (C[ z ]) , and therefore we can assume that A is of the form (3). Then, as wasshown in Lemma 3.3, we can write T ′′ = D · ˜ T , with ˜ T ∈ GL d (C( z )) ∩ Mat d (C[ z ])) . Again, we can repeat thisreasoning n times until we arrive at the desired form. (cid:3) Example 3.11 (Example 3.9 continued).
The leading matrix of ( S · D )[ A ] ϕ as in Example 3.9 at ϕ ( q ) = z − D = diag ( z − , ) , and get ( S · D · D )[ A ] ϕ = (cid:18) z + z − z + (cid:19) . Again, the leading matrix of this system at ϕ ( q ) = z is column-reduced and of rank 1. Finally, after applying the transformation D = diag ( z , ) , we get the desingularized system ( S · D · D · D )[ A ] ϕ = (cid:18) − z + z (cid:19) . Collecting all the transformations, we see that a desingularizingtransformation for A at q = z − T = S · D · D · D = (cid:18) z − z + z (cid:19) . As was already shown in Lemma 3.4, a positive ϕ -dispersionis a necessary condition for a removable singularity. For a givensystem [ A ] ϕ and an irreducible polynomial q , the ϕ -dispersion canbe obtained by computing the largest integer root of the resultant res z ( q ( z + k ) , num ( det ( A ))) . This, together with Theorem 3.10 andits proof gives rise to Algorithm 1. Algorithm 1: desingularize A( A , q ) Input: A with entries in C( z ) and a single, simple, ir-reducible pole q ∈ C[ z ] . Output: ( T , T [ A ] ϕ ) s.t. T [ A ] ϕ is desingularized at q , or ( I d , A ) if desingularization is not possible.1 T ← I d WHILE ( ϕ -dispersion ( A , q ) > AND den ( A ) = mod q ) DO .1 A ← lc q ( A ) .2 S ← as in the proof of Lemma 3.1.2 .3 D ← diag ( q , . . . , q , , . . . , ) with rank ( A ) many elements equal to q .2 .4 A ← ϕ ( S · D ) − · A · ( S · D ) .5 T ← T · S · D .6 q ← ϕ ( q ) IF (den ( A ) = mod q ) RETURN ( I d , A ) ELSE RETURN ( T , A ) We can give a necessary and sufficient condition for a pole to bedesingularizable. It can be seen as the shift analogue of the nilpo-tency of the leading matrix at the considered pole of the system,which is a necessary condition for an apparent singularity in thedifferential setting [3].
Proposition 3.12.
Let q ∈ C[ z ] be a ϕ -minimal pole of the sys-tem [ A ] ϕ . Let ˜ A = q n A , so that ord q ( ˜ A ) = and π q ( ˜ A ) = lc q ( A ) .If A is (partially) desingularizable at q then there exists a positiveinteger k such that π q ( ˜ Aϕ − ( ˜ A ) . . . ϕ − k ( ˜ A )) = . (5) Proof.
Let T be a desingularizing transformation for [ A ] ϕ at q and B = T [ A ] ϕ . Then for all non-negative integers k one has ϕ ( T ) Bϕ − ( B ) . . . ϕ − k ( B ) = Aϕ − ( A ) . . . ϕ − k ( A ) ϕ − k ( T ) , and hence ϕ ( T )( q n B ) ϕ − ( q n B ) . . . ϕ − k ( q n B ) = ˜ Aϕ − ( ˜ A ) . . . ϕ − k ( ˜ A ) ϕ − k ( T ) . As ord q ( q n B ) > q ( ϕ − j ( q n B )) = ord ϕ j ( q ) ( B ) ≥ ord ϕ j ( q ) ( A ) ≥ , for all j ∈ N ∗ , we get that π q ( ˜ Aϕ − ( ˜ A ) . . . ϕ − k ( ˜ A ) ϕ − k ( T )) = . Now we conclude by remarking that for k large enough π q ( T ( z − k )) is invertible. (cid:3) We will now show that the factorial relation (5) is a sufficientcondition for a matrix A to be partially desingularizable at q . Proposition 3.13.
Let q ∈ C[ z ] be a ϕ -minimal pole of [ A ] ϕ . Let ˜ A = q n A , so that ord q ( ˜ A ) = and π q ( ˜ A ) = lc q ( A ) . If [ A ] ϕ is suchthat the factorial relation (5) holds for some integer k ≥ then [ A ] ϕ is (partially) desingularizable at q . roof. Let k be minimal so that (5) holds. Put M : = π q ( ˜ Aϕ − ( ˜ A ) · · · ϕ − k + ( ˜ A )) and N : = π q ( ϕ − k ( ˜ A )) . By definition of k , the matrix M is nonzero (but singular) and wehave M · N = . With d : = dim ( A ) it follows that0 < rank ( M ) ≤ s : = d − rank ( N ) < d . Let P ∈ GL d (C[ z ]/h q i) such that P · N has its last ( d − s ) rowslinearly independent over C[ z ]/h q i while its s first rows are zero.Consider the matrix : U = ϕ k − ( P − ) as an element of Mat d (C[ z ]) then by applying the unimodular transformation Y = U X , we canassume that the matrix N has the following form: N = (cid:18) O s O s , d − s N , N , (cid:19) , where N , and N , are matrices with entries in C[ z ]/h q i of size ( d − s )× s and ( d − s )×( d − s ) respectively, so that the last d − s rowsof N are linearly independent over C[ z ]/h q i . As M · N = d − s last columns of M are zero. Let ˜ A = ( ˜ A i , j ) ≤ i , j ≤ bepartitioned in four blocks as N . Then π q ( ϕ − k ( ˜ A , j )) = j = , s first rows of ˜ A are divisible by ϕ k ( q ) . Usingthe substitution Y = DX where D = diag ( ϕ k − ( q ) I s , I d − s ) , we geta new system which still has a pole at q of multiplicity at most n .Indeed, we have B : = ϕ ( D ) − AD = q − n ϕ k − ( q ) ˜ A , ϕ k ( q ) ˜ A , ϕ k ( q ) ϕ k − ( q ) ˜ A , ˜ A , ! = q − n (cid:18) ϕ k − ( q ) ˜ A ′ , ˜ A ′ , ϕ k − ( q ) ˜ A , ˜ A , (cid:19) , (6)for some matrices ˜ A ′ , , ˜ A ′ , with entries in O q . It is clear thatden ( B ) | den ( A ) and that ord q ( B ) ≥ ord q ( A ) . Now we will provethat the factorial relation (5) holds for ˜ B : = q n B with k − k . For this we remark first that ϕ ( D ) ˜ Bϕ − ( ˜ B ) · · · ϕ − k + ( ˜ B ) = ˜ Aϕ − ( ˜ A ) · · · ϕ − k + ( ˜ A ) ϕ − k + ( D ) . It then follows that π q ( ϕ ( D )) π q ( ˜ Bϕ − ( ˜ B ) · · · ϕ − k + ( ˜ B )) = M · π q ( ϕ − k + ( D )) . We have that π q ( ϕ − k + ( D )) = π q ( diag ( q I s , I d − s )) = diag ( O s , I d − s ) , hence M · π q ( ϕ − k + ( D )) = d − s last columns of M arezero). Now π q ( ϕ ( D )) = π q ( diag ( ϕ k ( q ) I s , I d − s )) is invertible (since q and ϕ k ( q ) are co-prime), it then follows that π q ( ˜ Bϕ − ( ˜ B ) · · · ϕ − k + ( ˜ B )) = . If k − B and the polynomial q until we arrive at k =
1. When k = π q ( ˜ B ) = q ( ˜ B ) > q ( B ) ≥ − n + (cid:3) This proof motivates the following alternative desingularizationalgorithm. In contrast to Algorithm 1, instead of shifting a singu-larity towards a zero of the system, it performs the analogous taskof moving a zero towards the singularity until they cancel eachother
Algorithm 2: desingularize B( A , q ) Input: A with entries in C( z ) and a single, simple, ir-reducible pole q ∈ C[ z ] . Output: ( T , T [ A ] ϕ ) s.t. T [ A ] ϕ is desingularized at q .1 T ← I d WHILE (den ( A ) = mod q ) DO .1 ℓ ← ϕ − dispersion ( A , q ) .2 IF ( ℓ ≤ THEN RETURN ( T , A )2 .3 n ← ord q ( A ) ; ˜ A ← q n A .4 k ← M ← I d ; N ← π q ( ˜ A ) .5 WHILE ( M · N , AND k ≤ ℓ ) DO .5.1 M ← M · N ; k ← k + N ← π q ( ϕ − k ( ˜ A )) .6 U ← as in the proof of Proposition 3.12.2 .7 D ← diag ( ϕ k − ( q ) I s , I d − s ) with s = d − rank ( N ) .2 .8 A ← ϕ ( U · D ) − · A · ( U · D ) .9 T ← T · U · D RETURN ( T , A Remark 3.
All systems that are desingularizable via Algorithm 1are also desingularizable via Algorithm 2 and vice versa.Example 3.14.
For A as in Example 2.2 and q = z − A = ( z − ) A = (cid:18) z − − ( z + ) ( z − ) (cid:19) , M = π q ( ˜ A ( z ) ˜ A ( z − ) ˜ A ( z − )) = ˜ A ( ) ˜ A ( ) ˜ A ( ) = (cid:18) , −
12 6 (cid:19) , N = π q ( ϕ − ( ˜ A ) = ˜ A (− ) = (cid:18) − − (cid:19) , π q ( ˜ A ( z ) ˜ A ( z − ) ˜ A ( z − ) ˜ A ( z − )) = ˜ A ( ) ˜ A ( ) ˜ A ( ) ˜ A (− ) = , so k =
3. If we chose U = (cid:18)
12 12 (cid:19) , then ϕ ( U ) − AU = U − AU = ( z − ) (cid:18) ( z + ) −( z + ) ( z − ) (cid:19) . We have s =
1, so with D = diag ( ϕ ( q ) , ) = diag ( z , ) we get B = ϕ ( D ) − ( ϕ ( U ) − AU ) D = ( z − ) (cid:18) z − z ( z + ) ( z − ) (cid:19) . Note that, as expected, we have that˜ B ( ) ˜ B ( ) ˜ B ( ) = . Here we can repeat the above process on B to desingularize asmuch as possible the matrix A at q = z −
2. In this particularexample q is removable by the transformation T = U · diag ( z ( z − )( z − ) , ) . Indeed, one can see that T [ A ] ϕ = ϕ ( T ) − AT = (cid:18) − z ( z − ) (cid:19) , has polynomial entries. The transformation T is the same as inExample 3.11 up to a right factor diag ( , ) . .3 Rank Reduction Consider a system [ A ] ϕ and let q be a ϕ − minimal factor of den ( A ) with multiplicity n ≥
1, such that [ A ] ϕ the is not partially desingu-larizable at q . This implies that there’s no positive integer k suchthat relation (5) holds. As the quantity n cannot be reduced, it’snatural to ask if it is possible to reduce the rank of the leadingmatrix lc q ( A ) by applying a polynomial transformation T to [ A ] ϕ .We shall give a criterion for the existence of a polynomial trans-formation T such ord q ( T [ A ] ϕ ) = ord q ( A ) and rank ( lc q ( T [ A ] ϕ )) < rank ( lc q ( A )) Proposition 3.15.
Let q ∈ C[ z ] be a ϕ -minimal pole of [ A ] ϕ . Let ˜ A ( z ) = q n A ( z ) , so that ord q ( ˜ A ) = and π q ( ˜ A ) = lc q ( A ) . Then a nec-essary and sufficient condition for the existence of a polynomial trans-formation T such that ord q ( T [ A ]) = ord q ( A ) and rank ( lc q ( T [ A ])) < rank ( lc q ( A )) is that there exists a positive integer k such that rank ( π q ( ˜ Aϕ − ( ˜ A ) . . . ϕ − k ( ˜ A ))) < rank ( lc q ( A )) . (7) Proof.
Necessary condition : Suppose first that there exists apolynomial matrix T with the desired properties and let B = T [ A ] ϕ .Similarly to the proof of Proposition 3.12, one gets for all non-negative integers k : ϕ ( T )( q n B ) ϕ − ( q n B ) . . . ϕ − k ( q n B ) = ˜ Aϕ − ( ˜ A ) . . . ϕ − k ( ˜ A ) ϕ − k ( T ) . Since ord q ( q n B ) = = ord q ( ˜ A ) and all the other factors in bothsides of this equality have non-negative orders at q we get that π q ( ϕ ( T )) π q (( q n B )) π q ( ϕ − ( q n B ) . . . ϕ − k ( q n B )) = π q ( ˜ Aϕ − ( ˜ A ) . . . ϕ − k ( ˜ A )) π q ( ϕ − k ( T )) . By using the fact that the rank of a product of matrices is less orequal to the rank of each factor we get that the rank of the prod-uct in the right hand side of the previous equality is bounded byrank ( π q (( q n B ))) = rank ( lc q ( B )) and hencerank ( π q ( ˜ Aϕ − ( ˜ A ) . . . ϕ − k ( ˜ A )) π q ( ϕ − k ( T ))) ≤ rank ( lc q ( B )) < rank ( lc q ( A )) . Now let k be the smallest positive integer such that the matrix π q ( T ( z − k )) is of full rank. Thenrank ( π q ( ˜ Aϕ − ( ˜ A ) . . . ϕ − k ( ˜ A ))) = rank ( π q ( ˜ Aϕ − ( ˜ A ) . . . ϕ − k ( ˜ A )) π q ( ϕ − k ( T ))) < rank ( lc q ( A )) . Sufficient condition : Let r = rank ( lc ( A )) and let k be minimal sothat (7) holds. Put M : = π q ( ˜ Aϕ − ( ˜ A ) · · · ϕ − k + ( ˜ A )) and N : = π q ( ϕ − k ( ˜ A )) . By definition of k , the matrix M is nonzero, has the same rank r aslc q ( A ) and we have the strict inequalityrank ( M · N ) < r = rank ( M ) . This implies in particular that rank ( N ) < d = dim ( A ) . Let s : = d − rank ( N ) . As in the proof of Proposition 3.13, we can assumethat N has the following form: N = (cid:18) O s O s , d − s N , N , (cid:19) , where N , and N , are matrices with entries in C[ z ]/h q i of size ( d − s )× s and ( d − s )×( d − s ) respectively, so that the last d − s rows of N are linearly independent over C[ z ]/h q i . Let M = ( M i , j ) ≤ i , j ≤ be partitioned in four blocks as N . Then we have M · N = (cid:18) M , M , (cid:19) · (cid:0) N , N , (cid:1) . As the matrix ( N , N , ) is of full rank, we get thatrank (cid:18) M , M , (cid:19) = rank ( M · N ) < r . Let ˜ A = ( ˜ A i , j ) ≤ i , j ≤ be partitioned in four blocks as N . Then π q ( ϕ − k ( ˜ A , j )) = j = ,
2. Using the substitution Y = DX where D = diag ( ϕ k − ( q ) I s , I d − s ) , we get a system [ B ] ϕ of theform (6). with den ( B ) | den ( A ) and ord q ( B ) ≥ ord q ( A ) . Note that π q ( q n B ) = π q ( ϕ k ( q ) ) I s O s , d − s O d − s , s I d − s ! · (cid:18) π q ( ˜ A , ) π q ( ˜ A , ) π q ( ˜ A , ) π q ( ˜ A , ) (cid:19) · (cid:18) π q ( ϕ k − ( q )) I s O s , d − s O d − s , s I d − s (cid:19) .It follows that if k ≥
2, then rank ( π q ( q n B )) = rank ( lc q ( A )) , but wewill prove that the factorial relation (7) holds for ˜ B : = q n B with k − k . As in the proof of Proposition 3.13, we have that π q ( ϕ − k + ( D )) = π q ( diag ( q I s , I d − s )) = diag ( O s , I d − s ) , hence M · π q ( ϕ − k + ( D )) = (cid:18) O s M , O d − s M , (cid:19) , whose rank is less than r . Now π q ( ϕ ( D )) = diag ( π q ( ϕ k ( q )) I s , I d − s ) is invertible (since q and ϕ k ( q ) are co-prime), it then follows thatrank ( π q ( ˜ Bϕ − ( ˜ B ) · · · ϕ − k + ( ˜ B ))) = rank ( M · π q ( ϕ − k + ( D )) < r = rank ( lc q ( B )) . If k − B and the polynomial q until we arrive at k =
1. Then wehave that π q ( q n B ) = π q ( ϕ ( q ) ) I s O s , d − s O d − s , s I d − s ! · (cid:18) O s M , O d − s M , (cid:19) , whose rank is less than r . (cid:3) The proof of Proposition 3.15 suggests that Algorithm 2 canbe easily adapted to minimize the rank of the leading matrix ofa ϕ -minimal pole. In particular, a T can be computed such thatord p ( T [ A ] ϕ ) ≥ ord p ( A ) for p ∈ C[ z ] . It is to note that rank reduc-tion for a pole in A via Algorithm 2 comes at the potential cost ofan increase in order of a pole of A ∗ , as the next example shows. Example 3.16.
Consider the system with A = © « z ( z + ) z + z
00 0 z ª®¬ , A ∗ = © « z ( z − ) z − z
00 0 z − ª®®¬ . We have rank ( lc z ( A )) = z − ( A ∗ ) = −
1, and computing arank reducing transformation for [ A ] ϕ via Algorithm 2 gives T = iag ( z , z , ) , which results in T [ A ] ϕ = © « z z ª®¬ , T [ A ] ∗ ϕ = © « ( z − ) z − ª®®¬ , with rank ( lc z ( T [ A ] ϕ )) = z − ( A ∗ ) = −
2. We note thatwe merely shifted an already present pole in A ∗ to the right, asopposed to adding a new factor to the system. In this section we establish the connection between the analyti-cal notion of apparent and removable singularities of meromor-phic solutions and the algebraic concept desingularization of dif-ference systems. The key observation is the fact that the factorialrelation (5) provides a sufficient condition for a singularity to beremovable..
Proposition 4.1.
Let ζ ∈ P r ( A ) be a pole of A of order ν ≥ such that ζ − j < P r ( A ) for all positive integers j . Let ˜ A = ( z − ζ ) ν A ,so that ˜ A ( ζ ) , . If ζ is a removable r-singularity of [ A ] ϕ , then thereexists a positive integer k such that ˜ A ( ζ ) A ( ζ − ) · · · A ( ζ − k ) = . In particular, the matrix A ( ζ − j ) is singular for some non-negativeinteger j . Proof.
Using a result due to Ramis [4, 14, 20], one can easilyprove that for any complex number η with − Re η large enough,there exist a meromorphic fundamental matrix solution F ( z ) whichis holomorphic for − Re z large enough and satisfies F ( η ) = I d .Choose a positive integer k such that − Re ( ζ − k ) is large enoughand take a fundamental matrix solution F ( z ) as above with F ( ζ − k ) = I d . Then one can write F ( z + ) = A ( z ) A ( z − ) A ( z − ) · · · A ( z − k ) F ( z − k ) , and hence ( z − ζ ) ν F ( z + ) = ( z − ζ ) ν A ( z ) A ( z − ) A ( z − ) · · · A ( z − k ) F ( z − k ) . Taking the limit as z goes to ζ , we get that0 = ˜ A ( ζ ) A ( ζ − ) A ( ζ − ) · · · A ( ζ − k ) . (cid:3) Corollary 4.2.
Let ζ ∈ P r ( A ) such that there is a ϕ -minimal q with q ( ζ ) = . If ζ is a removable singularity of [ A ] ϕ , then [ A ] ϕ isdesingularizable at q . Proof.
Let n : = − ord q ( A ) . We can apply Proposition 3.13 toreduce the multiplicity of q in den ( A ) from n to n −
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