Regular Systems of Linear Functional Equations and Applications

Regular Systems of Linear Functional Equations and Applications Moulay A. Barkatou

Gary Broughton, Eckhard Pflugel ¨

Institut XLIM Universite´ de Limoges 123, Av. A. Thomas 87060 Limoges France

Faculty of CISM Kingston University Penrhyn Road Kingston upon Thames, Surrey KT1 2EE United Kingdom

[email protected]

ABSTRACT The algorithmic classification of singularities of linear differential systems via the computation of Moser- and superirreducible forms as introduced in [19] and [15] respectively has been widely studied in Computer Algebra ([11, 20, 6, 9]). Algorithms have subsequently been given for other forms of systems such as linear difference systems [4, 3] and the perturbed algebraic eigenvalue problem [17]. In this paper, we generalise these concepts to the general class of systems of linear functional equations. We derive a definition of regularity for these type of equations, and an algorithm for recognizing regular systems. When specialised to q-difference systems, our results lead to new algorithms for tasks computing polynomial solutions and regular formal solutions.

Categories and Subject Descriptors I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms

General Terms Algorithms

Keywords Computer Algebra, Regular Systems of Linear Functional Equations, Moser-reduction, Super-reduction

1.

INTRODUCTION

In this paper, we are concerned with the class of systems of linear functional equations which comprise common types of systems such as linear differential or linear (q−)difference systems. The theoretical framework for these types of systems is pseudoliinear algebra, which has been introduced by Jacobson in

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[email protected], [email protected] [16]. Bronstein and Petkoˆ vsek in [13] review the basic objects of Pseudo-Linear Algebra in the context of Computer algebra. It is in this spirit that the first author of the present paper treats in [2] the problem of (global) factorization of linear functional systems. In the present paper we are interested in the local analysis of these type of systems. For this we first introduce (see Sec. 3.1) a unifying framework that permits us to treat simultaneously, all types of linear functional systems. This can be done by using the language of pseudo-linear equations over a field of discrete valuation. Once this common framework is installed, we give a definition of regularity for pseudo-linear systems (see Sec. 3.2) and develop a method for recognizing regular systems inspired by Moser’s work on differential equations (see Sec. 3.3). The rest of the paper is organized as follows. In Section 2 we present the notion of regularity and Moser-reducibility of linear differential systems and outline current ways of implementing the Moser-reduction. Our presentation aims to help the reader understand what motivates the way in which similar concepts are introduced in the following sections. Similarly, Section3.4 deals with the notion of super-reduction. In Section 4 we give two algorithms for Moser- and superreduction for linear functional equations. These algorithms are obtained from the Moser- and super-reduction algorithms as reviewed in Sec. 4.1 and Sec. 4.3. The originality of our approach is to prove that replacing, at each reduction step, the differential transformation (5) by the pseudo-linear transformation (10) results in valid reduction algorithms. Let us mention here that the Moser- and super reduction of differential and difference systems has been dealt with, however to our knowledge, Moser-irreducible forms and its generalisation, the super-irreducible form, has not been considered with respect to q-difference systems. As an application we show, in Section 5.1, that the approach developed in [7, 1] for computing polynomial solution (and more generally formal series solutions at infinity) for differential and difference systems is also applicable for q−difference systems. Our algorithms are implemented in the Maple package ISOLDE [10] and two examples of computation using this package are given in Section 6.

2.

THE DIFFERENTIAL CASE

Let K be a subfield of C, and denote by K[[x]] the ring of formal power series in x over K and K((x)) its the quotient field. If a ∈ K((x)), we denote the x − adic valuation of a

by v(a) and similarly if A = (ai,j ) is a matrix with coefficients in K((x)), we define v(A) as v(A) = min v(aij ). Let d ϑ denotes the derivation x dx of K((x)). A linear differential system with a singularity at the origin is a system of the form ϑY = A(x)Y

(1)

where Y is a n−dimensional unknown vector and A(x) is an n × n matrix with entries in K((x)). The matrix A can be expanded as: A = x−r

∞ X

Aj xj

(A0 6= 0)

(2)

j=0

where Aj ∈ Mn (K) and r is a non-negative integer called the Poincar´e-rank of the system. Performing the change of variable Y = T (x)Z

(3)

where T ∈ GL(n, K((x))) in (1) yields the following equivalent system

Further, a matrix polynomial T such that m(T [A]) = µ(A) can be computed in this way. Inspired by Moser’s work, Dietrich [14] appears to be the first author to provide an efficient algorithm for computing µ(A). Apparently unaware of his approach, Hilali and Wazner [15] published a different method which they extended to compute the so-called super-irreducible forms (see Section 3.4 for more details) of linear differential systems. The first author of this paper has given versions of these algorithms in [5, 8] for systems with rational function coefficients. Using a different approach, Levelt [18] has given an algorithm which can also be used to compute a system with minimal Poincar´e-rank. Inspired by all the previous works the first and third author of this paper have introduced new algorithms implementing Moser- and super-reduction [11] (see Sections 4.1 and 3.4 for more details) which seem more efficient than the previous implementations. The aim of the present paper is to show how those algorithms can be extended relatively easily to handling general systems of linear functional equations.

ϑZ = B(x)Z

(4)

3.

SYSTEMS OF PSEUDO-LINEAR EQUATIONS

B = Tϑ [A] := T −1 AT − T −1 ϑT.

(5)

3.1

where

With the idea of equivalent systems outlined, we can now define a regular system as a system which is equivalent to a system of Poincar´e rank r = 0. Given a system (1) with Poincar´e rank r > 0, we wish to compute an equivalent system for which r is minimal. This is useful as it allows a regular system to be detected – in this case, the minimal Poincar´e rank is r = 0. This problem can be solved by using the notion Moserreducibility as defined in [19]. Following Moser we introduce the two following rational numbers  0) if vA) < 0 −v(A) + rank(A n m(A) = 0 if v(A) ≥ 0 and µ(A) =

min

T ∈GL(n,(K((x)))

(m(Tϑ [A])).

The rational number µ(A) is called the Moser-invariant of (1). The system (1) is regular if and only if µ(A) = 0. The matrix A is called Moser-irreducible if m(A) = µ(A) otherwise it is called Moser- reducible. The reason for introducing m(A) lies in the fact that it is possible to give a necessary and sufficient condition for reducibility which only involves A0 and A1 , as stated in [19]: If r > 0 then the matrix A is reducible if and only if the associated Moser polynomial
θ(λ) := xrank A0 det x−1 A0 + A1 − λI (6) x=0

vanishes identically in λ. When A is reducible one can the construct a matrix T of the form T (x) = C diag(xd1 , · · · , xdn ) where C is an invertible constant matrix and where di ∈ {0, 1}, such that m(S[A]) < m(A). Applying the above procedure several times, if necessary, µ(A) can be determined.

Basic Definitions and Notation.

Discrete valuation fields - Basic Objects In this section, and throughout the remainder of the paper, we let F be a commutative discrete valuation field of characteristic zero and denote by v the valuation of F. Recall that v : F −→ Z∪{+∞} is a map with the following properties : for f, g ∈ F one has 1. v(f ) = +∞ ⇐⇒ f = 0, 2. v(f g) = v(f ) + v(g), 3. v(f +g) ≥ min (v(f ), v(g)), and equality holds if v(f ) 6= v(g). The valuation ring of F is O = {f ∈ F : v(f ) ≥ 0}. The set M = {f ∈ F : v(f ) > 0} coincides with the set of non-invertible elements of O. M is the (unique) maximal ¯ := O/M is a field. It is called the residue ideal of O, so F field of F. Let π denote the canonical homomorphism from ¯ We denote by U the group of units of O. One O onto F. has U = O \ M. Example 3.1. : F = C((x)) the field of formal Laurent power series, equipped with the x − adic valuation. Its valuation ring is O = C[[x]]. The residue field may be identified with C. An element t ∈ O is said to be a uniformizing element (or a local parameter) if v(t) = 1. Recall that the valuation ring O is a principal ideal ring, that every proper ideal of O can be written as tm O for some integer m > 0 and that the intersection of all proper ideals of O is the zero ideal. In particular, M = tO. As a consequence, each nonzero element f ∈ F can be uniquely written as f = utv(f ) for some unit u ∈ U.

In the sequel we suppose that F is complete with respect to v and fix a uniformizing element t of F. Let R be a set of ¯ that is a subset R of O such that representatives of O in F, ¯ is bijective. 0 ∈ R and the map: a ∈ R 7→ π(a) ∈ O/M = F ThenPevery element f ∈ F can be uniquely expanded as i f = +∞ i=n fi t , where n = v(f ) and fi ∈ R with fn 6= 0. For A = (ai,j ) ∈ Mn (F) we define the valuation of A, by v(A) = min (v(ai,j )). Every nonzero matrix A with entries i,j

+∞ X

δ(f ) = δ(tm−1 z) = δ(tm−1 )φ(z) + tm−1 δ(z). and f −1 δ(f ) = t−(m−1) δ(tm−1 )z −1 φ(z) + z −1 δ(z). It then follows that ω(δ) = v(f −1 δ(f )) ≥ inf (v(t−(m−1) δ(tm−1 )), v(z −1 δ(z)))

in F can be uniquely written as A = t−r

take τ = f , otherwise put z = t1−m f . The element z is a local parameter, for v(z) = v(t) = 1. Now compute

Ai ti

(7)

and by using Lemma 3.1 ω(δ) ≥ inf (v(t−1 δ(t)), v(z −1 δ(z)))

i=0

where r = −v(A) and where the Ai are matrices with entries in R with A0 6= 0.

Now if v(t−1 δ(t)) ≥ v(z −1 δ(z)) then take τ = z, otherwise take τ = t. 2

The degree of a φ-derivation Let φ be an isometry of F, that is an automorphism of F such that v(φf ) = v(f ) for all f ∈ F. Let δ be a φ−derivation, that is a map δ : F → F satisfying, for all a, b ∈ F δ(a+b) = δa+δb and δ(ab) = φa δb+δa b (Leibniz rule) We let C be the field of constants of F, C = {f ∈ F : φf = f and δf = 0}. Note that when φ = 1F the identity map of F then δ is just a derivation in the standard way. Otherwise, one can easily show that, δ has to be of the form γ(1F − φ) for some γ ∈ F (see for example [13]). Lemma 3.1. For all integer k, one has v(t−k δ(tk )) ≥ v(t−1 δ(t)) Proof Let us notice first that it suffices to prove the lemma for k ≥ 1 because v(t−k δ(tk )) = v(tk δ(t−k )). Indeed, for any nonzero f ∈ F one has f δ(f −1 ) = −δ(f )φ(f −1 ) and hence v(f δ(f −1 )) = v(φ(f −1 )δ(f )) = v(f −1 δ(f )). Now using the Leibniz rule, one can prove, by induction on k ≥ 1, that t−k δ(tk ) = t−1 δ(t)

k−1 X

ui

i=0 −1

where u = φ(t)t . The result follows by noticing that v(u) = 1. Finally, the lemma is true for k = 0. 2 We define the degree of δ as ω(δ) =

inf

f ∈F,f 6=0

v(f

−1

δ(f )).

In the case φ 6= 1F and δ 6= 0, one has δ = γ(1F − φ) for some γ ∈ F. Since v(f − φ(f )) ≥ v(f ) for all nonzero f ∈ F one has ω(1F − φ) ≥ 0 and ω(δ) = v(γ) + ω(1F − φ) ≥ v(γ). Lemma 3.2. Suppose δ 6= 0 and is continuous. Then ω(δ) = v(τ −1 δ(τ ))

Example 3.2. d 1. F = C((x)), φ = 1F and δ = dx . In this case ω(δ) = −1 d −1 = v(x δx) and ω(x dx ) = 0.

2. F = C((x−1 )) equipped with the valuation: if f = P+∞ −i with fi0 6= 0, v(f ) = i0 . Here t = x−1 i=i0 fi x is a uniformizing element of F. For φ = 1F and d δ = dx one has ω(δ) = +1 = v(t−1 δ(t)) = v(x −1 )) x2 d while ω(x dx ) = 0. 3. F = C((x−1 )), v as above and t = x−1 . Let φ the C−automorphism defined by φx = x − 1 and δ = 1 − φ. 1 One has ω(δ) = 1 = v(t−1 δ(t)) = v(x( x1 − x−1 )). 4. F = C((x−1 )), v as above and t = 1/x. Let φ the C−automorphism defined by φx = qx, q ∈ C∗ and δ = 1 − φ. In this case ω(δ) = 0 = v(t−1 δ(t)) = 1 v(x( x1 − qx ).

3.2

Regular Pseudo-linear Systems

By a system of pseudo-linear equation over (F, φ, δ) we mean an equation of the form δY = AφY

where A is an n × n matrix in F and where Y is an unknown n−dimensional column vector. We shall refer to system (8) by δ,φ [A]. We note the following well-known examples of systems of pseudo-linear equations 1. Differential equations: F = C((x)), or C((x−1 )), φ = d 1F and δ = dx . 2. Difference equations: F = C((x−1 )), φ the C−automorphism defined by φx = x − 1 and δ = 1 − φ. 3. q−Difference equations: F = C((x)) or C((x−1 )), φ the C−automorphism defined by φx = qx, q ∈ C∗ and δ = 1 − φ. Consider a system of the form (8). The gauge transformation Y = T Z with T ∈ GL(n, F), leads to the equivalent system δZ = BφZ

(9)

B = T −1 AφT − T −1 δT =: Tδ,φ [A].

(10)

for some local parameter τ . Proof Since δ is continuous one has ω(δ) = v(f −1 δ(f )) for some nonzero element f ∈ F. Let m = v(f ). If m = 1 then

(8)

where B is given by

Definition 3.1. A pseudo-linear system δ,φ [A] is called regular (or (δ, φ)−regular) if there exists a gauge transformation T ∈ GL(n, F) such that v(Tδ,φ [A]) ≥ ω(δ).

In the sequel we define the span of a transformation T ∈ GL(n, F) as the nonnegative integer σ(T ) := −v(T ) − v(T −1 ) = αn − α1 .

The first problem we want to solve is the following: Given a pseudo-linear system δ,φ [A], to decide whether δ,φ [A] is regular or not and in case δ,φ [A] is regular to compute a gauge transformation T which takes the iven system into an equivalent system Tδ,φ [A] with v(Tδ,φ [A]) ≥ ω. This is the subject of the following section.

2. The second ingredient used by Moser (see Lemma 2, p 387 in [19]) can be stated using our notation as:

3.3

Lemma 3.3. Let A ∈ Mn (O) A = A0 + A1 t + . . . , ; with A0 6= 0. A necessary and sufficient condition that there exists a matrix T ∈ GL(n, F) such that the matrix

Moser Reducibility for Pseudo-Linear Systems

B := T −1 AT = B0 + B1 t + . . .

We shall develop an algorithm to determine whether a given pseudo-linear system is regular. More generally this algorithm reduces a given system δ,φ [A] to an equivalent irreducible system δ,φ [B] with v(B) = max {v(P [A]) : P ∈ GL(n, F)}. The method followed is similar to the one used by Moser in the differential case [19]

3.3.1

belongs to Mn (O) and satisfies rank(B0 ) < n0 := rank(A0 ) is that the polynomial θ(λ) := π tn0 det t−1 A0 + A1 − λI

A Reducibility Criterion

vanishes identically in λ. Moreover T can be chosen with σ(T ) = 1.

We associate with the system δ,φ [A] the following rational numbers:  0) if vA) < ω ω − v(A) + rank(A n mδ,φ (A) = 0 if v(A) ≥ ω

It is important to notice here that in fact T can be chosen of form T = Ctα with C a “constant” matrix ( C ∈ Mn (R) with det C ∈ U ) and α as above with αi ∈ {0, 1}. Suppose now that φ 6= 1F , write φt = qt + O(t2 ) with 0 6= q ∈ R and keep the notation above. Then it can be proved that:

µδ,φ (A) = min {mδ,φ (Tδ,φ [A]) | T ∈ GL(n, F)} Definition 3.2. The matrix A is called irreducible w.r.t. (δ, φ) if mδ,φ (A) = µδ,φ (A), otherwise it is called reducible. We remark that the system δ,φ [A] is regular if and only if µδ,φ (A) = 0. The following result is the analogue of Theorem 1 in [19] which gives a reducibility criterion in the differential case.

T −1 φT = q α + O(t) and hence T −1 AφT = (T −1 AT )T −1 φT = B0 q α + O(t).

Theorem 3.1. Suppose that r := −v(A) > −ω and let n0 = rank(A0 ). Then the system δ,φ [A] is reducible if and only if the polynomial

θ(λ) := π tn0 det tr−1 A − λI vanishes identically in λ. Remark 3.1. It is easy to see that the polynomial θ(λ) depends only on A0 , A1

θ(λ) = π tn0 det t−1 A0 + A1 − λI This theorem can be proved in a quite similar way to that used for Theorem 1 in [19]. For lack of place we shall not give a detailed proof here, we just want to point out some useful facts which may help the reader to understand why the approach of Moser can be adapted to our situation 1. The first ingredient used in the proof by Moser (see Lemma 1 of [19]), namely the property that any gauge transformation T can be written in Smith normal form, remains valid in our situation: Any matrix T ∈ GL(n, F) can be written in Smith normal form : T = P tα Q where P, Q ∈ Mn (O) with det P, det Q ∈ U, tα = diag(tα1 , · · · , tαn ) αi in Z with α1 ≤ α2 ≤ · · · ≤ αn . This result follows from the fact that O is a pricipal ideal domain (the ideals of O are of the form tm O).





Now since rank(B0 q α ) = rank(B0 ) it follows that the above lemma 3.3 remains valid if we define B as B = T −1 AφT . 3. Finally, the key for establishing Theorem 3.1 is (as in the proof by Moser) to remark that, using a transformation T as above, the reduction of the rank of A0 is not affected by the term T −1 δT in Tδ,φ [A].

3.4

Super-Reduction for Pseudo-Linear Systems

In [15], Hilali and Wazner introduced the concept of the super-irreducible form, which can be seen as a generalisation of the Moser-irreducible form. in this section we extend this concept to pseudo-linear systems. Consider again a pseudo-linear system δ,φ [A] of the form (8) where the matrix A is given by its t−adic expansion (7) and suppose that r = −v(A) > −ω(δ). Mimicking the differential case we define, for 1 ≤ k ≤ r + ω, the rational number mkδ,φ (A) by mkδ,φ (A) = r + ω +

n0 n

+

n1 n2

+ ··· +

nk−1 nk

where ni = ni (A) is the number of rows of A with valuation v(A) + i. Now define µkδ,φ (A) = min{mkδ,φ (Tδ,φ [A]) | T ∈ GL(n, F)}.

Definition 3.3. The matrix A or system (8) is said to be k-irreducible (w.r.t. (δ, φ)) if mkδ,φ (A) = µkδ,φ (A). Otherwise A is called k-reducible. The matrix A or system (8) is said to be super–irreducible, if it is k-irreducible for every k, or equivalently if msδ,φ (A) = µsδ,φ (A).

that C[A] = A˜ has a corresponding L-matrix with the following structure:   11 L L12 0 21 22   L (12) L 0 L31 L32 L33 − λ

A criterion for k−reducibility is obtained in exactly the same way as in the differential case. One defines sk and Θk (λ) as where

where L11 ,L22 and L33 are square matrices of dimension d, s − q and q respectively with the additional condition that:  11   11  L L L12 rank + s − q = rank (13) 21 21 22 L L L

sk = sk (A) = kn0 + (k − 1)n1 + · · · + nk−1 . Θk (λ) := tsk det(tr−k A − λIn ) and verifies that Θk (λ) belongs to O[λ]. Then one can define the polynomial θk (λ) ∈ R[λ] as   θk (λ) = π tsk det(tr−k A − λIn ) (11) We then have the following Theorem 3.2. The the matrix A is k–irreducible, if and only if the polynomials θj (λ), (j = 1, . . . , k), do not vanish identically in λ.

4.

TWO REDUCTION ALGORITHMS

In this section we give two reduction algorithms for systems of linear functional equations. When specialising these algorithms to the differential case, they correspond to the Moser- and super-reduction algorithms as reviewed in Sec. 4.1 and Sec.4.3. Our approach is to replace the differential equivalence (5) by the pseudo-linear transformation (10) and to show that this results in valid reduction algorithms. The main task for the first algorithm is to prove that Lemma 2.3 in [11] remains valid, for the second algorithm one has to show that the block-reduction algorithm can also be used when using pseudo-linear transformations.

4.1

Differential Reduction Algorithms

We consider a differential system [A] of the form (1) with A given by its x − adic expansion (2). Here we are working with the field F = K((x)) equipped with φ = 1F , δ = ϑ = xd/dx and we take t = x as local parameter. We have ω(ϑ) = 0. In [11], the first and third author of this paper have introduced a new method that implement the Moser-reduction for linear systems of differential equations. We shall now briefly outline this method, assuming that A0 is nilpotent and in Jordan canonical form A0 = diag(J, 0s ) where J has d Jordan blocks of dimension ni ≥ 2 with n1 ≥ ... ≥ nd < nd+s = 1. Define li (ci respectively) for i = 1, .., d + s, as the position of the ith zero row (column respectively) of A0 . The L-matrix L(A, λ) ∈ k[λ](d+s)×(d+s) is then defined by L(A, λ) = ((αi,j )) − diag(0d , λIs ) where ((αi,j )) ∈ k(d+s)×(d+s) is defined by setting αi,j as the entry of A1 of row li and column cj . It was then shown in [11] that the system (1) is Moser-reducible if and only if detL(A, λ) ≡ 0. At each reduction step, this algorithm uses the following normalisation: construct a constant transformation C such

and L33 is upper-triangular. The diagonal transformation which carries out the reduction step is then of the form [11] S = diag(In1 −1 , x, ..., Ind −1 , x, xIs−q , Iq )

4.2

(14)

The First Reduction Algorithm

Here again we are working with the field F = K((x)) equipped with the x−adic valuation. The residue field of F can be identified with K. We take t = x as local parameter. We fix φ and δ and assume that φ 6= 1F , δ 6= 0. Recall that δ is of the form δ = γ(1F − φ) for some nonzero element γ ∈ F. Without any loss of generality we can assume that ω(δ) = 0 (to achieve this it suffices to replace δ by δ˜ := x−ω(δ) δ). We write φx = qx + O(x2 ) for some nonzero element q ∈ K and we obtain for k 6= 0 φ(xk ) = q k xk + O(xk+1 ).

(15)

We consider a pseudo-linear system δ,φ [A] of the form (8) with A given by its x − adic expansion (2) and suppose that r = −v(A) > 0 = −ω(δ). The matrix A defines also a differential system ϑ,1F [A] of the fom (1). Our assumption ω(δ) = 0 = ω(ϑ) implies that mδ,φ (A) = mϑ,1F (A). We shall refer to this quantity simply by m(A). Notice also that the two polynomials θ(λ) associated with the systems δ,φ [A] and ϑ,1F [A] coincides. Therefore we have the following interesting property: the matrix A is reducible w.r.t. (δ, φ) if and only if it is w.r.t. (ϑ, 1F ). Suppose now that A is reducible and let T be a transformation such that m(Tϑ,1F [A]) < m(A). One question arises naturally : is it true that we have also m(Tδ,φ [A]) < m(A)? The answer to this question is, in general, no! However, the answer is yes for transformations T of the form T = CS with C and S = diag(xd1 , . . . , xdn ) such that σ(S) = max di − min di ≤ 1. This is precisely, as we have seen in the previous section, the kind of transformation used in each reduction step in the algorithm for the Moser-reduction in the differential case. We shall now prove this result. The constant transformation C which is used for the normalisation of the system can be used similarly as in the differential case since Cφ,δ [A] = C −1 AφC − C −1 δC = C −1 AC = Cϑ,1F [A] for any constant transformation C, due to the fact that φ is a K-automorphism. We will show that for the diagonal transformation S we have m(Sφ,δ [A]) < m(A).

Lemma 4.1. The rank-reduction in the differential algorithm is achieved by merely using the similarity transformation S −1 AS. Proof Using the fact that S is a diagonal transformation S = diag(xd1 , . . . , xdn ) =: xD , where D = diag(d1 , . . . , dn ) with di ∈ N. S −1 = x−D and ϑS = DxD hence B = S[A] = S

−1

Proof We have Sφ,δ [A]

˜ is a constant diagonal matrix. Using Lemma 4.3, where C we have S −1 AφS = B = x−r (B0 + O(x)),

We compute

with rank B0 < rank A0 . But we have assumed ω = 0 > −r hence, the proposition follows. 2

AS − D.

We can see that the matrix D does not affect the leading matrix B0 of the transformed system since r > 0. Hence the reduction in rank will have to be achieved by S −1 AS alone. 2

Lemma 4.2. Let S and D be as in Lemma 4.1 and its proof, with di ∈ {0, 1} (i = 1 . . . n). Then φS = Sq D + O(x2 ).

The first reduction algorithm is then Pseudo Linear Reduction 1(A, φ, δ) Input: A ∈ Mn (F ), φ a K-automorphism, δ a Pseudo-derivative w.r.t. φ Output: Transformation T such that Tφ,δ [A] is Moser (δ, φ)– irreducible 1. T := In ;

Proof Using (15), we obtain  1 if di = 0 φ(xdi ) = q di xdi + O(xdi +1 ) otherwise

2. while (r(A) > −ω) and (θ(λ) ≡ 0) do (a) Compute a constant transformation C to normalise A;

(16)

(b) A := Cφ,δ [A]; T := T C;

and, using di ∈ {0, 1} we compute φS

= S −1 Aφ(S) − S −1 δ(S) ˜ ω + O(x1+ω ) = S −1 Aφ(S) − Cx

=

diag(q d1 xd1 , . . . , q dn xdn ) + O(x2 )

(c) Compute a diagonal transformation S as in the differential case;

=

Sq D + O(x2 ).

(d) A := Sφ,δ [A]; T := T S; 2

3. return T ;

4.3 Lemma 4.3. The operation S −1 AφS with S as in the previous lemma achieves a rank-reduction. Proof Using Lemma 4.2, we find   S −1 AφS = S −1 A (Sq D + O(x2 ) =

S

−1

ASq

D

+S

−1

(17) 2

A · O(x ).

Using Lemma 4.1, we know that S −1 AS has a leading matrix with reduced rank, hence this is also true for S −1 ASq D =: B0 x−r + O(x−r+1 ). But it is clear that the second term in the last sum in (17) is S −1 A · O(x2 ) = O(x−r+1 ) since v(S −1 ) = −1, implying that this term does not affect B0 . 2

Proposition 4.1. Suppose that A is Moser-reducible and the diagonal transformation S = diag(xd1 , . . . , xdn ) with di ∈ {0, 1} satisfies m(S[A]) < m(A). Then we also have m(Sφ,δ [A]) < m(A).

The Second Reduction Algorithm

In [11] it was shown that the computation of a superirreducible system can be reduced to the computation of several Moser-irreducible systems of smaller size, using a block-reduction algorithm. In this section, we assume that the first reduction algorithm has been applied to the system (8). We will show that the Direct Block Reduction Algorithm as introduced in [12] can be used for systems of linear functional equations in order to obtain a second reduction algorithm which can be characterised as computing a new system of the form δY = BφY where is a block-triangular super-reduced matrix where each diagonal block is Moser-reduced. As explained in [12], Section 3, the block-reduction is achieved by using Elementary Operations using transformations of the form   1 ..     . α     . .   . Ei,j (α) =     1     ..   . 1 where the entry at position (i, j) is α ∈ K[[x]]. We recall that transforming a given differential system (1) with Ei,j (α) results in a new system whose coefficient matrix A˜

is obtained from A by adding to the jth column the ith column multiplied by α, then subtracting the jth row multiplied by α from the ith row, and adding ϑ(α) to the entry in the (i, j) position. The effect of using Ei,j (α) as a pseudo-linear transformation is very similar to the differential case, with the difference that the jth column of the transformed systems results from adding to the jth column the ith column multiplied by φ(α) and also adding δ(α) to the entry in the (i, j) position. The concept of normalised Moser-irreducible forms as introduced in [12] can be easily extended to the case of systems of linear functional equations as the normalisation is carried out using a constant transformation. Proposition 4.2. The direct Block-Reduction algorithm of [12] can be adapted for our use by replacing the elementary transformation Eij (α) with Eij (q −h α) where h = v(α). Proof By reviewing the process of eliminating terms in A it becomes apparent that the elimination is achieved using linear combinations of leading coefficients of elements in A, multiplied by the leading coefficients of α. Let α = cth + ..., it follows φ(α) = q h cth + · · · , hence the leading coefficient of φ(q −h α) equals c. Hence using Eij (q −h α) for a pseudo-linear transformation carries out an identical elimination process on the corresponding leading terms. 2

A polynomial solution Y ∈ K[x]n of degree ν can be viewed as local formal solution (at x = ∞) of the form X −i+ν x Yi , (19) Y (x) = i≥0 n

where Yi ∈ K , Y0 6= 0 and Yi = 0 for i > ν . The idea is to work with F = K((x−1 )), the completion of K(x) w.r.t. to the t−adic valuation (here t = x−1 ). Define φ and δ by φ(t) = qt and δ = 1F − φ. Our system (18) can then be written as the pseudo-linear system: δY = A(t)φY

(20)

where A(t) = M (q −1 t−1 ) − In ∈ Mn (F). Multiplying this system on the left by the diagonal matrix D := diag(tα1 , . . . , tαn ), where αi = − min (v(Ai,. ), 0), Ai,. being the ith row of the matrix A, yields the equation DδY = CφY,

(21)

where C = DA. By definition, one has D, C ∈ Mn (K[[t]]). Put X X C= Ci ti , D = Di ti . i≥0

i≥0

We look for formal solutions of the form: Remark 4.2. This does not imply that the final result of the block-reduction is identical in both of the differential and general pseudo-linear case. Crucially, each isolated step of the algorithm does indeed an identical elimination in both cases, but the transformations introduces also additional terms of higher order that are different.

Y =

+∞ X

ti+ν Yi ν ∈ K, Yi ∈ K n , Y0 6= 0.

i=0

One has φY =

+∞ X

q i+ν ti+ν Yi

i=0

5.

APPLICATIONS FOR Q-DIFFERENCES

Let q ∈ K with q 6= 1 and consider a linear q−difference system with coefficients in K(x):

and δY = Y − φY =

+∞ X

(1 − q i+ν )ti+ν Yi .

i=0

Y (qx) = M (x)Y (x), M (x) ∈ Mn (K(x)).

5.1

(18)

Polynomial Solutions

We are interested in this section by the problem of of computing all the polynomial solutions of a system of he form (18). Algorithms for solving this problem in the differential and the difference cases have been proposed in [7, 1]. Here we shall show that the same approach remains valid for the q−difference case. For sake of brevity we shall consider here only the problem of computing a bound on the degree of polynomial solutions. Such a bound can be obtained from the so-called indicial equation (at x = ∞). Unfortunately the indicial equation is not immediately apparent for a given general system. The idea consists in reducing the given system to a simple from which the indicial equation can be immediately obtained. We will show that such a simple form can be derived from a super-irreducible form in exactly the same way as in the difference and the differential cases.

5.2

The indicial equation

Replacing D, C, φY and δY by their t−adic expansions in (21) and identifying coefficients of the same powers of t (after simplifying the factor tν ) yields , in particular, the equation (1 − q ν )D0 Y0 = q ν C0 Y0 . Thus in order that the system (21) admits a formal solution of the form (19), ν and Y0 must satisfy the equation
C0 − (q −ν − 1)D0 Y0 = 0 which implies that (q −ν −1) must be a root of the polynomial E(λ) := det (C0 − λD0 ). As a consequence: • If Y ∈ K[x]n is a nonzero polynomial solution of (18) of degree ν then E(q −ν − 1) = 0. • The degree of polynomial solution can be bounded by the biggest nonnegative integer ν such that q −ν − 1 is a root of E(λ).

4 . We assign φ and δ in Maple: 3 > phi:= proc(a,x) return subs(x=x-1,a) end: > delta:=proc(a,x) return a-phi(a,x) end: Calling the super-reduction on this example gives a list of results, the first entry being the transformed system that is super-irreducible: > tmp := super reduce(A, x, 1/x, lambda, phi, delta, 1, S, invS): tmp[1];   6 x2 −9 x+4 0 (x − 1)−1 (x−1)3     −1 x2   5 x 0   (x−1)3   3 2 5+x x +10 x−5 x−2 −3 2 x −6 x(x−1) x(x−1)3

But, it may happen that the determinant E(λ) vanishes identically in λ in which case it is quite useless to us. This motivates the following definition Definition 5.1. The system (21) is said to be simple if det (C0 − λD0 ) 6= 0 (as a polynomial in λ). In this case the polynomial E(λ) := det (C0 − λD0 ) is called the indicial polynomial of (21) . As an example of simple systems, take a system of the form (20) with v(A) ≥ 0. In this case D = In and C = A. Hence E(λ) = det (A0 − λIn ) 6≡ 0. Consequently, the system is simple and its indicial polynomial has degree n. Proposition 5.1. Every q−difference system (20) can be reduced to an equivalent system (21) which is simple.

This resulting system has v(A) = 1 = ω hence, it is a regular system and the associated Moser-invariant is µδ,φ = 0. As part of the third entry of the results, one obtains in this example the indicial equation of the system: >tmp[3];

Proof Since every q−difference system (20) is equivalent to a super-irreducible one, it suffices to prove that every superirreducible system is simple. Consider a system of the form (20) and put r = −v(A). If r ≤ 0 then the system is simple. Suppose that r > 0 (notice that ω(δ) = 0) and let D and C = DA be defined as above then

[[0, λ3 − 12 λ2 + 47 λ − 60]]

E(λ) = det(C0 − λD0 ) = θr (λ).

2. As a second example, consider the q-difference system

Indeed, one easily verifies that det(D) = tsr (see Section 3.4 for the definition of sr and θr ) hence tsr det(A − λIn )

δ(Y ) = A(x)φ(Y ) where, φ and δ are as in Example 2.2 fourth case, with ω(δ) = 0 and the matrix A(x) is given by ! q 3 x4 −x A(x) = . x3 −1

= det D det(A − λIn ) = det(DA − λD).

Hence θr (λ)

= tsr det (A(t) − λIn )|t=0 = det (C(t) − λD(t))|t=0 = det(C0 − λD0 ).

We keep δ = 1 − φ and define φ by he following: > phi:= proc(a,x) return subs(x=q*x,a) end: This system is Moser-irreducible but not super-irreducible. We call the super-reduction on this example: > tmp := super reduce(A, x, 1/x, lambda, phi, delta, 1, S, invS): tmp[1];  3 4  q x −1 0   3 x3 − −1+q 3 q

Now if (20) is super–irreducible then, by Theorem 3.2, the polynomial θr (λ) is not identically zero and (20) is simple. 2

6.

EXAMPLES

Our reduction algorithms are implemented in the Maple package ISOLDE [10] and we shall now present two examples using this package. After installing ISOLDE, the function super reduce is available. It takes as input a matrix A and as additional parameters the automorphism φ, pseudo-derivative δ and the quantity ω. The command ?super reduce displays additional user information.

Additional output returns a list of polynomials: tmp[3]; [[0, 1 − q 3 − q 3 λ], [1, 1], [2, 1], [3, 1], [4, λ − q 3 ]]. This system admits a polynomial solution ! 1 Y = x3

1. As a first example, consider the following difference system:

of degree ν = 3 and one indeed verifies that E(q −3 − 1) = 0 where E(λ) is given by the first polynomial in the above output.

δ(Y ) = A(x)φ(Y ) where F = C((x−1 )), δ and φ are as in Example 2.2, third case, with ω(δ) = 1 and the matrix A(x) is given by   5 x−1 x2 0   . A(x) =  0 x−3   0 x−3 0 3 x−1 + 5 x−2 The valuation of A is v(A) = −2 < ω = 1 and hence, we calculate the number mδ,φ = ω−v(A)+rank A0 /n =

7.

CONCLUSION

In this paper, we have shown that existing algorithms that we have previously implemented for efficient calculation of Moser- and Super-irreducible forms of linear differential systems can be extended relatively easily to handling general systems of linear functional equations. This has led to an implementation in ISOLDE available for use with future applications such as computing closed form solutions.

8.

REFERENCES

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