Around the numeric-symbolic computation
of differential Galois groups

Joris van der Hoeven

Dépt. de Mathématiques (bât. 425)

Université Paris-Sud

91405 Orsay Cedex

France

Email: vdhoeven@texmacs.org

November 3, 2023



Abstract

Let be a linear differential operator, where is an effective algebraically closed subfield of . It can be shown that the differential Galois group of is generated (as a closed algebraic group) by a finite number of monodromy matrices, Stokes matrices and matrices in local exponential groups. Moreover, there exist fast algorithms for the approximation of the entries of these matrices.

In this paper, we present a numeric-symbolic algorithm for the computation of the closed algebraic subgroup generated by a finite number of invertible matrices. Using the above results, this yields an algorithm for the computation of differential Galois groups, when computing with a sufficient precision.

Even though there is no straightforward way to find a “sufficient precision” for guaranteeing the correctness of the end-result, it is often possible to check a posteriori whether the end-result is correct. In particular, we present a non-heuristic algorithm for the factorization of linear differential operators.


1Introduction

Let be a monic linear differential operator of order , where is an effective algebraically closed subfield of . A holonomic function is a solution to the equation . The differential Galois group of is a linear algebraic group which acts on the space of solutions (see section 2.2 and [Kaplansky, 1957; van der Put and Singer, 2003; Kolchin, 1973]). It carries a lot of information about the solutions in and on the relations between different solutions. For instance, the existence of non-trivial factorizations of and the existence of Liouvillian solutions can be read off from the Galois group. This makes it an interesting problem to explicitly compute the Galois group of .

A classical approach in this area is to let act on other vector spaces obtained from by the constructions from linear algebra, such as symmetric powers and exterior powers [Beke, 1894; Singer and Ulmer, 1993]. For a suitable such space , the Galois group consists precisely of those invertible matrices which leave a certain one-dimensional subspace of invariant [Humphreys, 1981, chapter 11]. Invariants in or under may be computed more efficiently by considering the local solutions of at singularities [van Hoeij and Weil, 1997; van Hoeij, 1997; van Hoeij, 1996]. More recently, and assuming (for instance) that the coefficients of are actually in , alternative algorithms appeared which are based on the reduction of the equation modulo a prime number [Cluzeau, 2004; van der Put, 1995; van der Put and Singer, 2003].

In this paper, we will study another type of “analytic modular” algorithms, by studying the operator in greater detail near its singularities using the theory of accelero-summation [Écalle, 1985; Écalle, 1987; Écalle, 1992; Écalle, 1993; Braaksma, 1991; Braaksma, 1992]. More precisely, we will use the following facts:

When using these facts for the computation of differential Galois groups, the bulk of the computations is reduced to linear algebra in dimension with multiple precision coefficients.

In comparison with previous methods, this approach is expected to be much faster than algorithms which rely on the use of exterior powers. A detailed comparison with arithmetic modular methods would be interesting. One advantage of arithmetic methods is that they are easier to implement in existing systems. On the other hand, our analytic approach relies on linear algebra in dimension (with floating coefficients), whereas modulo methods rely on linear algebra in dimension (with coefficients modulo ), so the first approach might be a bit faster. Another advantage of the analytic approach is that it is more easily adapted to coefficients fields with transcendental constants.

Let us outline the structure of this paper. In section 2, we start by recalling some standard terminology and we shortly review the theorems on which our algorithms rely. We start with a survey of differential Galois theory, monodromy and local exponential groups. We next recall some basic definitions and theorems from the theory of accelero-summation and the link with Stokes matrices and differential Galois groups. We finally recall some theorems about the effective approximation of the transcendental numbers involved in the whole process.

Before coming to the computation of differential Galois groups, we first consider the simpler problem of factoring in section 3. We recall that there exists a non-trivial factorization of if and only if the Galois group of admits a non-trivial invariant subspace. By using computations with limited precision, we show how to use this criterion in order to compute candidate factorizations or a proof that there exist no factorizations. It is easy to check a posteriori whether a candidate factorization is correct, so we obtain a factorization algorithm by increasing the precision until we obtain a correct candidate or a proof that there are no factorizations.

In section 4 we consider the problem of computing the differential Galois group of . Using the results from section 2, it suffices to show how to compute the algebraic closure of a matrix group generated by a finite number of given elements. A theoretical solution for this problem based on Gröbner basis techniques has been given in [Derksen et al., 2003]. The main idea behind the present algorithm is similar, but more emphasis is put on efficiency (in contrast to generality).

First of all, in our context of complex numbers with arbitrary precisions, we may use the LLL-algorithm for the computation of linear and multiplicative dependencies [Lenstra et al., 1982]. Secondly, the connected component of is represented as the exponential of a Lie algebra given by a basis. Computations with such Lie algebras essentially boil down to linear algebra. Finally, we use classical techniques for finite groups in order to represent and compute with the elements in [Sims, 1970; Sims, 1971; Murray and O'Brien, 1995]. Moreover, we will present an algorithm for non-commutative lattice reduction, similar to the LLL-algorithm, for the efficient computation with elements in near the identity.

The algorithms in section 4 are all done using a fixed precision. Although we do prove that we really compute the Galois group when using a sufficiently large precision, it is not clear a priori how to find such a “sufficient precision”. Nevertheless, we have already seen in section 3 that it is often possible to check the correctness of the result a posteriori, especially when we are not interested in the Galois group itself, but only in some information provided by . Also, it might be possible to reduce the amount of dependence on “transcendental arguments” in the algorithm modulo a further development of our ideas. Some hints are given in the last section.

Remark 1The author first suggested the main approach behind this paper during his visit at the MSRI in 1998. The outline of the algorithm in section 4.5 came up in a discussion with Harm Derksen (see also [Derksen et al., 2003]). The little interest manifested by specialists in effective differential Galois theory for this approach is probably due to the fact that current computer algebra systems have very poor support for analytic computations. We hope that the present article will convince people to put more effort in the implementation of such algorithms. We started such an effort [van der Hoeven et al., 2002–2005], but any help would be appreciated. Currently, none of the algorithms presented in this paper has been implemented.

2Preliminaries

In this section, we recall several classical results about differential Galois theory and its link with accelero-summation theory. We also recall previous work on the efficient evaluation of holonomic constants. The main result of this section is theorem 20.

2.1Notations

Throughout this paper, we will use the following notations:

An algebraically closed field of constants.

The algebra of matrices with coefficients in .

The subgroup of of invertible matrices.

The vector space generated by a subset of a larger vector space.

The algebra generated by a subset of a larger algebra.

Vectors are typeset in bold face and we use the following vector notations:

Matrices will also be used as mappings . When making a base change in , we understand that we perform the corresponding transformations on all matrices under consideration. We denote for the diagonal matrix with entries . The may either be scalars or square matrices. Given a matrix and a vector , we write for the vector with for all .

2.2Differential Galois groups

Consider a monic linear differential operator , where is an algebraically closed subfield of and . We will denote by the finite set of singularities of (in the case of , one considers the transformation ). A Picard-Vessiot extension of is a differential field such that

PV1

is differentially generated by and a basis of solutions to the equation .

PV2

has as its field of constants.

A Picard-Vessiot extension always exists: given a point and , let be the unique solution to with for . We call the canonical basis for the solution space of at , and regard as a column vector. Taking , the condition PV2 is trivially satisfied since and the constant field of is .

Let be a Picard-Vessiot extension of and let be as in PV1. The differential Galois group of the extension is the group of differential automorphisms which leave pointwise invariant. It is classical [Kolchin, 1973] that is independent (up to isomorphism) of the particular choice of the Picard-Vessiot extension .

Given an automorphism , any solution to is sent to another solution. In particular, there exists a unique matrix with for all . This yields an embedding of into and we define . Conversely, belongs to if every differential relation satisfied by is also satisfied by (with ). Since this assumption constitutes an infinite number of algebraic conditions on the coefficients of , it follows that is a Zariski closed algebraic matrix group. Whenever is another basis, we obtain the same matrix group up to conjugation.

Assume now that is a larger algebraically closed subfield of . Then the field is again a Picard-Vessiot extension of . Furthermore, the Ritt-Raudenbush theorem [Ritt, 1950] implies that the perfect differential ideal of all with is finitely generated, say by . But then is still a finite system of generators of the perfect differential ideal of all with . Consequently, (i.e. as an algebraic group over ) is determined by the same algebraic equations as . We conclude that .

Let be a Picard-Vessiot extension of . Any differential field with naturally induces an algebraic subgroup of automorphisms of which leave fixed. Inversely, any algebraic subgroup of gives rise to the differential field with of all elements which are invariant under the action of . We say that (resp. ) is closed if (resp. ). In that case, the extension is said to be normal, i.e. every element in is moved by an automorphism of over . The main theorem from differential Galois theory states that the Galois correspondences are bijective [Kaplansky, 1957, Theorem 5.9].

Theorem 2With the above notations:

  1. The correspondences and are bijective.

  2. The group is a closed normal subgroup of if and only if the extension is normal. In that case, .

Corollary 3Let . If for all , then .

2.3Monodromy

Consider a continuous path on from to . Then analytic continuation of the canonical basis at along yields a basis of solutions to at . The matrix with

(1)

is called the connection matrix or transition matrix along . In particular, if , then we call a monodromy matrix based in . We clearly have

for the composition of paths, so the monodromy matrices based in form a group which is called the monodromy group. Given a path from to , we notice that . Since any differential relation satisfied by is again satisfied by its analytic continuation along , we have and .

Remark 4The definition of transition matrices can be slightly changed depending on the purpose [van der Hoeven, 2005b, Section 4.3.1]: when interpreting and as row vectors, then (1) has to be transposed. The roles of and may also be interchanged modulo inversion of .

Now assume that admits a singularity at (if then we may reduce to this case modulo a translation; singularities at infinity may be brought back to zero using the transformation ). It is well-known [Fabry, 1885; van Hoeij, 1996] that admits a computable formal basis of solutions of the form

(2)

with , , and . We will denote by the set of finite sums of expressions of the form (2). We may see as a differential subring of a formal differential field of “complex transseries” [van der Hoeven, 2001a] with constant field .

We recall that transseries in are infinite linear combinations of “transmonomials” with “grid-based support”. The set of transmonomials forms a totally ordered vector space for exponentiation by reals and the asymptotic ordering . In particular, each non-zero transseries admits a unique dominant monomial . It can be shown [van der Hoeven, 2001a] that there exists a unique basis of solutions to of the form (2), with and for all . We call the canonical basis of solutions in and there is an algorithm which computes as a function of .

Let be the subset of of all finite sums of expressions of the form (2) with . Then any can uniquely be written as a finite sum , where . Let be the group of all automorphisms for which there exists a mapping with for all . Then every preserves differentiation and maps the Picard-Vessiot extension of into itself. In particular, the restriction of to is a subset of .

Proposition 5Assume that is fixed under . Then .

ProofAssume that and let be a “generalized exponent” with . Let be a supplement of the -vector space . Let be the mapping in which sends to for each and . Then we clearly have .

Let be the set of generalized exponents corresponding to the generalized exponents of the elements of the canonical basis . Using linear algebra, we may compute a multiplicatively independent set such that for certain and all .

Proposition 6With the above notations, the algebraic group is generated by the matrices where is chosen arbitrarily.

ProofLet be the group generated by the matrices . Notice that each individual matrix generates : assuming , the variety is irreducible of dimension and is not contained in an algebraic group of dimension . Now any is a diagonal matrix for some multiplicative mapping . Hence

Conversely, each element

determines a multiplicative mapping which may be further extended to using Zorn's lemma and the fact that is algebraically closed. It follows that .

Assume that and let be the transformation which sends to , to and to . Then preserves differentiation, so any solution to of the form (2) is sent to another solution of the same form. In particular, there exists a matrix with , called the formal monodromy matrix around . We have .

Proposition 7Assume that is fixed under and . Then .

ProofWe already know that . Interpreting as a polynomial in with , we must have since

Consequently, is of the form and

We conclude that for every with , whence .

2.4The process of accelero-summation

Let be the differential -algebra of infinitesimal Puiseux series in for and consider a formal power series solution to . The process of accelero-summation enables to associate an analytic meaning to in a sector near the origin of the Riemann surface of , even in the case when is divergent. Schematically speaking, we obtain through a succession of transformations:

(3)

Each is a “resurgent function” which realizes in the “convolution model” with respect to the -th “critical time” (with and ). In our case, is an analytic function which admits only a finite number of singularities above . In general, the singularities of a resurgent function are usually located on a finitely generated grid. Let us describe the transformations , and in more detail.

1The Borel transform
We start by applying the formal Borel transform to the series . This transformation sends each to

where , and extends by strong linearity:

The result is a formal series in which converges near the origin of . The formal Borel transform is a morphism of differential algebras which sends multiplication to the convolution product, i.e. .

2Accelerations
Given , the function is defined near the origin of , can be analytically continued on the axis , and admits a growth of the form at infinity. The next function is obtained from by an acceleration of the form

where the acceleration kernel is given by

(4)

For large on an axis with , it can be shown that for some constant . Assuming that satisfies

(5)

it follows that the acceleration of is well-defined for small on . The set of directions such admits a singularity on is called the set of Stokes directions. Accelerations are morphisms of differential -algebras which preserve the convolution product.

3The Laplace transform
The last function is defined near the origin of , can be analytically continued on the axis and admits at most exponential growth at infinity. The function is now obtained using the analytic Laplace transform

On an axis with

(6)

the function is defined for all sufficiently small . The set of Stokes directions is defined in a similar way as in the case of accelerations. The Laplace transform is a morphism of differential -algebras which is inverse to the Borel transform and sends the convolution product to multiplication.

Remark 8Intuitively speaking, one has .

Given critical times in and directions satisfying (5), we say that a formal power series is accelero-summable in the multi-direction if the above scheme yields an analytic function near the origin of any axis on satisfying (6). We denote the set of such power series by , where . Inversely, given , we denote by the set of all triples such that and so that is well-defined. In that case, we write and .

The set forms a differential subring of and the map for is injective. If and are obtained from and by inserting a new critical time and an arbitrary direction, then we have . In particular, contains , where denotes the ring of convergent infinitesimal Puiseux series. Let be sets of directions such that each is finite modulo . Let be the subset of of multi-directions which verify (5). We denote , and .

Taking , the notion of accelero-summation extends to formal expressions of the form (2) and more general elements of as follows. Given , , and , we simply define . It can be checked that this definition is coherent when replacing by for some . By linearity, we thus obtain a natural differential subalgebra of accelero-summable transseries with critical times and in the multi-direction . We also have natural analogues and of and .

The main result we need from the theory of accelero-summation is the following theorem [Écalle, 1987; Braaksma, 1991].

Theorem 9Let be a formal solution to . Then .

Corollary 10Let be the canonical basis of formal solutions to at the origin. We have .

ProofHolonomy is preserved under multiplication with elements of .

Remark 11We have aimed to keep our survey of the accelero-summation process as brief as possible. It is more elegant to develop this theory using resurgent functions and resurgence monomials [Écalle, 1985; Candelberger et al., 1993].

2.5The Stokes phenomenon

We say that is stable under Stokes morphisms is for all , there exists a with , and if the same property is recursively satisfied by . We denote by the differential subring of which is stable under Stokes morphisms. The mappings will be called Stokes morphisms and we denote by the group of all such maps.

Proposition 12Assume that is fixed under . Then is convergent.

ProofAssume that one of the admits a singularity at and choose maximal and minimal. Modulo the removal of unnecessary critical times, we may assume without loss of generality that . Let with be a multi-direction satisfying (5), such that for all . Then

for all sufficiently small . Now is obtained by integration around along the axis . By classical properties of the Laplace integral [Candelberger et al., 1993, Pré I.2], the function cannot vanish, since admits a singularity in (if the Laplace integrals corresponding to both directions coincide, then the Laplace transform can be analytically continued to a larger sector, which is only possible if is analytic in a sector which contains both directions ). We conclude that , so is not fixed under .

Remark 13Let be a set of multi-directions satisfying (5), with for exactly one , and so that for all , we have either or and for some small . For every , we have

By looking more carefully at the proof of proposition 12, we observe that it suffices to assume that is fixed under all Stokes morphisms of the form , instead of all elements in .

We say that are equivalent, if for all , where is the denominator of . We notice that is finite modulo this equivalent relation. We denote by a subset of with one element in each equivalence class.

Let us now come back to our differential equation . Given , the map induces an isomorphism between and . We denote by the unique matrix with . Given a second , the vector is again in , whence by repeating the argument. In particular, the Stokes morphism induces the Stokes matrix .

We are now in the position that we can construct a finite set of generators for the Galois group in a regular point .

Algorithm Compute_generators

Input: an operator and a regular point

Output: a set of generators for

for each do

return

Theorem 14With constructed as above, the differential Galois group is generated by as a closed algebraic subgroup of .

ProofAssume that is fixed by each element of . We have to prove that . Given a singularity , let be the “continuation” of along (which involves analytic continuation until followed by “decelero-unsummation”). By proposition 7, we have . From proposition 12 and remark 13, we next deduce that is convergent. Indeed, since , its realization in the convolution model with critical time is a function in . Consequently, whenever and are equivalent. At this point we have shown that is meromorphic at . But a function which is meromorphic at all points of the Riemann sphere is actually a rational function. It follows that .

Remark 15Theorem 14 is essentially due to Ramis [Ramis, 1985; Martinet and Ramis, 1991]. Our presentation is a more constructive version of the one from [Martinet and Ramis, 1991]. Notice that both proofs by Ramis crucially build upon Écalle's theory of resurgent functions and accelero-summability. In the Fuchsian case, i.e. in absence of divergence, the result is due to Schlesinger [Schlesinger, 1895; Schlesinger, 1897].

Remark 16We have tried to keep our exposition as short as possible by considering only “directional Stokes-morphisms”. In fact, Écalle's theory of resurgent functions gives a more fine-grained control over what happens in the convolution model by considering the pointed alien derivatives for . Modulo the identification of functions in the formal model, the convolution models and the geometric model via accelero-summation, the pointed alien derivatives commute with the usual derivation . Consequently, if is a solution to , then we also have . In particular, given the canonical basis of solutions to , there exists a unique matrix with

This equation is called the bridge equation. Since admits only a finite number of singularities and the alien derivations “translate singularities”, we have for some , so the matrices are nilpotent. More generally, if are -linearly independent, then all elements in the algebra generated by are nilpotent.

It is easily shown that the Stokes morphisms correspond to the exponentials of directional Alien derivations . This yields a way to reinterpret the Stokes matrices in terms of the with . In particular, the preceding discussion implies that the Stokes matrices are unipotent. The extra flexibility provided by pointwise over directional alien derivatives admits many applications, such as the preservation of realness [Menous, 1996]. For further details, see [Écalle, 1985; Écalle, 1987; Écalle, 1992; Écalle, 1993].

2.6Effective complex numbers

A complex number is said to be effective if there exists an approximation algorithm for which takes on input and which returns an -approximation of for which . The time complexity of this approximation algorithm is the time it takes to compute a -approximation for . It is not hard to show that the set of effective complex numbers forms a field. However, given the question whether is undecidable. The following theorems were proved in [Chudnovsky and Chudnovsky, 1990; van der Hoeven, 1999; van der Hoeven, 2001b].

Theorem 17Let , , and . Given a broken line path on , we have

  1. The value of the analytic continuation of at the end-point of is effective.

  2. There exists an approximation algorithm of time complexity for , when not counting the approximation time of the input data , and .

  3. There exists an algorithm which computes an approximation algorithm for as in (b) as a function of , and .

Theorem 18Let be regular singular in . Let , and . Then is well-defined for all sufficiently small on the effective Riemann surface of above , and

  1. is effective.

  2. There exists an approximation algorithm of time complexity for , when not counting the approximation time of the input data , and .

  3. There exists an algorithm which computes an approximation algorithm for as in (b) as a function of , and .

In general, the approximation of involves the existence of certain bounds. In each of the above theorems, the assertion (c) essentially states that there exists an algorithm for computing these bounds as a function of the input data. This property does not merely follow from (a) and (b) alone.

The following theorem has been proved in [van der Hoeven, 2005b].

Theorem 19Let be singular in . Let be as in theorem 10 and . Given with and , we have

  1. is effective.

  2. There exists an approximation algorithm of time complexity for , when not counting the approximation time of the input data , and .

  3. There exists an algorithm which computes an approximation algorithm for as in (b) as a function of , and .

If we replace by an arbitrary effective algebraically closed subfield of , then the assertions (a) and (c) in the three above theorems remain valid (see [van der Hoeven, 2005a; van der Hoeven, 2003] in the cases of theorems 17 and 18), but the complexity in (b) usually drops back to . Notice also that we may replace by the transition matrix along in each of the theorems. The following theorem summarizes the results from section 2.

Theorem 20Let be an effective algebraically closed constant field of . Then there exists an algorithm which takes and on input, and which computes a finite set , such that

  1. The group is generated by as a closed algebraic subgroup of .

  2. If , then the entries of the matrices in have time complexity .

ProofIt is classical that the set of effective complex numbers forms a field. Similarly, the set of effective complex numbers with an approximation algorithm of time complexity forms a field, since the operations , , and can all be performed in time . In particular, the classes of matrices with entries in resp. are stable under the same operations. Now in the algorithm Compute_generators, we may take broken-line paths with vertices above for the . Hence (a) and (b) follow from theorem 19(a) resp. (b) and the above observations.

Given , we may endow with an approximate zero-test for which if and only if . We will denote this field by . Clearly, this zero-test is not compatible with the field structure of . Nevertheless, any finite computation, which can be carried out in with an oracle for zero-testing, can be carried out in exactly the same way in for a sufficiently small . Given , we will denote by the “cast” of to and similarly for matrices with coefficients in .

Remark 21In practice [van der Hoeven, 2006], effective complex numbers usually come with a natural bound for . Then, given with , it is even better to use the approximate zero-test if and only if . Notice that the bound usually depends on the internal representation of and not merely on as a number in .

3Factoring linear differential operators

Let be an effective algebraically closed subfield of . Consider a monic linear differential operator , where . In this section, we present an algorithm for finding a non-trivial factorization with whenever such a factorization exists.

3.1Factoring and invariant subspaces under

Let be a basis of solutions for the equation , where is an abstract differential field. We denote the Wronskian of by

It is classical (and easy to check) that

(7)

When expanding the determinant in terms of the matrices

it follows that

Denoting by the logarithmic derivative of , it can also be checked by induction that

admits as solutions, whence , using Euclidean division in the skew polynomial ring .

Proposition 22

  1. If admits a factorization , then leaves invariant.

  2. If is an invariant subvector space of , then admits a factorization with .

ProofAssume that admits a factorization . Then, given and , we have , whence . Conversely, assume that is an invariant subvector space of and let be a basis of . Then we observe that for all . Consequently,

for all , so that , by corollary 3. Hence

(8)

is a differential operator with coefficients in which vanishes on . But this is only possible if divides .

3.2A lemma from linear algebra

Lemma 23Let be a non-unitary algebra of nilpotent matrices in . Then there exists a basis of in which is lower triangular for all .

ProofLet be a matrix such that is a non-zero vector space of minimal dimension. Given and , we claim that . Assume the contrary, so that . By the minimality hypothesis, we must have . In particular, and . Again by the minimality hypothesis, it follows that . In other words, the restriction of to is an isomorphism on . Hence admits a non-zero eigenvector in , which contradicts the fact that is nilpotent.

Let us now prove the lemma by induction over . If or , then we have nothing to do, so assume that and . We claim that admits a non-trivial invariant subvector space . Indeed, we may take if and if . Now consider a basis of and complete it to a basis of . Then each matrix in is lower triangular with respect to this basis. Let and be the algebras of lower dimensional matrices which occur as upper left resp. lower right blocks of matrices in . We conclude by applying the induction hypothesis on and .

Let be a finite set of non-zero nilpotent matrices. If all matrices in the -algebra generated by are nilpotent, then it is easy to compute a basis for which all matrices in are lower triangular. Indeed, setting for all , we first compute a basis of . We successively complete this basis into a basis of , and so on until .

If not all matrices in are nilpotent, then the proof of lemma 23 indicates a method for the computation of a matrix in which is not nilpotent. Indeed, we start by picking an and set , where is smallest with . We next set and iterate the following loop. Take a matrix and distinguish the following three cases:

Set and continue.

Set , and continue.

Return the non-nilpotent matrix .

At the end of our loop, we either found a non-nilpotent matrix, or we have for all . In the second case, we obtain a non-trivial invariant subspace of as in the proof of lemma 23 and we recursively apply the algorithm on this subspace and a complement. In fact, the returned matrix is not even monopotent (i.e. not of the form , where is a nilpotent matrix), since it both admits zero and a non-zero number as eigenvalues.

3.3Computation of non-trivial invariant subspaces

Proposition 22 in combination with theorem 20 implies that the factorization of linear differential operators in reduces to the computation of non-trivial invariant subvector spaces under the action of whenever they exist.

In this section, we will first solve a slightly simpler problem: assuming that is an effective algebraically closed field and given a finite set of matrices , we will show how to compute a non-trivial invariant subspace of under the action of , whenever such a exists.

1Good candidate vectors
Given a vector it is easy to compute the smallest subspace of which is invariant under the action of and which contains . Indeed, starting with a basis , we keep enlarging with elements in until saturation. Since will never contain more than elements, this algorithm terminates. A candidate vector for generating a non-trivial invariant subspace of is said to be good if .

2The -algebra generated by
We notice that is an invariant subspace for , if and only if is an invariant subspace for the -algebra generated by . Again it is easy to compute a basis for . We start with a basis of and keep adjoining elements in to until saturation. We will avoid the explicit basis of , which may contain as much as elements, and rather focus on the efficient computation of good candidate vectors.

3-splittings
A decomposition , where are non-empty vector spaces, will be called an -splitting of , if the projections of on are all in . Then, given , we have if and only if for all . If we choose a basis for which is a union of bases for the , we notice that the are block matrices. In the above algorithm for computing the -algebra generated by it now suffices to compute with block matrices of this form. In particular, the computed basis of will consist of such matrices. The trivial decomposition is clearly an -splitting. Given , we notice that any -splitting is also an -splitting.

4Refining -splittings
An -splitting is said to be finer than the -splitting if is a direct sum of a subset of the for each . Given an -splitting and an arbitrary element , we may obtain a finer -splitting w.r.t as follows. Let and consider . If are the eigenvalues of , then is an -splitting of , where . Collecting these -splittings, we obtain a finer -splitting of . This -splitting, which is said to be refined w.r.t. , has the property that is monopotent on for each , with unique eigenvalue .

We now have the following algorithm for computing non-trivial -invariant subspaces of when they exist.

Algorithm Invariant_subspace

Input: a set of non-zero matrices in

Output: an -invariant subspace of or fail

Step 1. [Initial -splitting]

Compute a “random non-zero element” of

Compute an -splitting w.r.t. and each

Step 2. [One dimensional components]

For every with and , do the following:

Pick a and compute

If then return

Otherwise, set

If for all then return fail

Step 3. [Higher dimensional components]

Let be such that

Let

Let

If then go to step 4 and otherwise to step 5

Step 4. [Non-triangular case]

Let be non-monopotent on (cf. previous section)

Refine the -splitting w.r.t. and return to step 2

Step 5. [Potentially triangular case]

Choose and compute

If then return

Otherwise, let be in with

Refine the -splitting w.r.t.

If this yields a finer -splitting then return to step 2

Otherwise, set and repeat step 5

The algorithm needs a few additional explanations. In step , we may take to be an arbitrary element in . However, it is better to take a “small random expression in the elements of for . With high probability, this yields an -splitting which will not need to be refined in the sequel. Indeed, the subset of matrices in which yield non-maximal -splittings is a closed algebraic subset of measure zero, since it is determined by coinciding eigenvalues. In particular, given an -splitting w.r.t. , it will usually suffice to check that each is monopotent on each , in order to obtain an -splitting w.r.t. the other elements in .

Throughout the algorithm, the -splitting gets finer and finer, so the -splitting ultimately remains constant. From this point on, the space can only strictly decrease in step , so also remains constant, ultimately. But then we either find a non-trivial invariant subspace in step 5, or all components of the -splitting become one-dimensional. In the latter case, we either obtain a non-trivial invariant subspace in step 1, or a proof that for every (and thus for every ).

Remark 24Assume that is no longer an effective algebraically closed field, but rather a field with an approximate zero-test. In that case, we recall that a number which is approximately zero is not necessarily zero. On the other hand, a number which is not approximately zero is surely non-zero. Consequently, in our algorithm for the computation of , the dimension of can be too small, but it is never too large. In particular, if the algorithm Invariant_subspace fails, then the approximate proof that for every yields a genuine proof that there are no non-trivial invariant subspaces.

3.4Factoring linear differential operators

Putting together the results from the previous sections, we now have the following algorithm for finding a right factor of .

Algorithm Right_factor

Input:

Output: a non-trivial right-factor of or fail

Step 1. [Compute generators]

Choose and let

Compute a finite set of generators for (cf. theorem 20)

Step 2. [Initial precision]

while do

Step 3. [Produce invariant subspace]

Let

If fail then return fail

Let be a column basis of

Let

Step 4. [Produce and check guess]

Let

Divide by , producing with

If then go to step 5

Reconstruct from and with precision

If we obtain no good approximations or then go to step 5

Return

Step 5. [Increase precision]

Go to step 3

The main idea behind the algorithm is to use proposition 22 in combination with Invariant_subspace so as to provide good candidate right factors of in . Using reconstruction of coefficients in from Laurent series in with increasing precisions, we next produce good candidate right factors in . We keep increasing the precision until we find a right factor or a proof that is irreducible. Let us detail the different steps a bit more:

Step 2

We will work with power series approximations of terms and approximate zero-tests in . The degree of a rational function is defined by . The initial precisions and have been chosen as small as possible. Indeed, we want to take advantage of a possible quick answer when computing with a small precision (see also the explanations below of step 5).

Step 3

If Invariant_subspace fails, then there exists no factorization of , by remark 24. Effective power series and Laurent series are defined in a similar way as effective real numbers (in particular, we don't assume the existence of an effective zero-test). Efficient algorithm for such computations are described in [van der Hoeven, 2002].

Step 4

The reconstruction of and from and contains two ingredients: we use Padé approximation to find rational function approximations of degree and the LLL-algorithm to approximate numbers by numbers in .

Step 5

Doubling the precision at successive steps heuristically causes the computation time to increase geometrically at each step. In particular, unsuccessful computations at lower precisions don't take much time with respect to the last successful computation with respect to the required precision. Instead of multiplying the precisions by two, we also notice that it would be even better to increase by a factor which doubles the estimated computation time at each step. Of course, this would require a more precise complexity analysis of the algorithm.

The problem of reconstructing elements in from elements in is an interesting topic on its own. In theory, one may consider the polynomial algebra over generated by all coefficients occurring in and the number we wish to reconstruct. We may then apply the LLL-algorithm [Lenstra et al., 1982] on the lattice spanned by monomials of smallest total degree (for instance) and search for minimal -digit relations. If is the algebraic closure of , then we may simply use the lattice spanned by the first powers of .

At a sufficiently large precision , the LLL-algorithm will ultimately succeed for all coefficients of a candidate factorization which need to be reconstructed. If there are no factorizations, then the algorithm will ultimately fail at step 3. This proofs the termination of Right_factor.

Remark 25In practice, and especially if , it would be nice to use more of the structure of the original problem. For instance, a factorization of actually yields relations on the coefficients which we may try to use. For high precision computations, it is also recommended to speed the LLL-algorithm up using a similar dichotomic algorithm as for fast g.c.d. computations [Moenck, 1973; Pan and Wang, 2002].

Remark 26Notice that we did not use bounds for the degrees of coefficients of possible factors in our algorithm. If a bound is available, using techniques from [Bertrand and Beukers, 1985; van Hoeij, 1997; van der Put and Singer, 2003], then one may take instead of in step 5. Of course, bounds for the required precision are even harder to obtain. See [Bertrand and Beukers, 1985] for some results in that direction.

4Computing differential Galois groups

4.1Introduction

Throughout this section, will stand for the field of effective complex number with the approximate zero-test at precision . This field has the following properties:

EH1

We have an effective zero-test in .

EH2

There exists an algorithm which takes on input and which computes a finite set of generators for the -vector space of integers with .

EH3

There exists an algorithm which takes on input and which computes a finite set of generators for the -vector space of integers with .

EH4

is closed under exponentiation and logarithm.

Indeed, we obtain EH2 and EH3 using the LLL-algorithm. Some of the results in this section go through when only a subset of the conditions are satisfied. In that case, we notice that EH2EH1, EH3EH1 and EH4(EH2EH3).

Given a finite set of matrices , we give a numerical algorithm for the computation of the smallest closed algebraic subgroup of which contains . We will represent by a finite set and the finite basis of a Lie algebra over , such that

and each corresponds to a unique connected component of . We will also prove that there exists a precision such that the algorithm yields the theoretically correct result for all .

4.2The algebraic group generated by a diagonal matrix

Let be the group of invertible diagonal matrices. Each matrix has the form , where is the vector in of the elements on the diagonal of . The coordinate ring of is the set of Laurent polynomials in .

Now consider the case when consists of a single diagonal matrix . Let be the ideal which defines . Given a relation between the , any power satisfies the same relation , whence . Let be the ideal generated by all , such that .

Lemma 27We have .

ProofWe already observed that . Assuming for contradiction that , choose

such that is minimal. If , then , since otherwise and has less than terms. In particular, the vectors with are linearly independent. But this contradicts the fact that for all .

By EH2, we may compute a minimal finite set of generators for the -vector space of with . We may also compute a basis for , where . Then is the connected component of , since for all , and cannot be further enlarged while conserving this property.

Let . We construct a basis of , by taking to be shortest in (if ) or (if ), such that . This basis determines a toric change of coordinates with such that with respect to the new coordinates. Similarly, we may construct a basis of , by taking each to be shortest in such that is maximal. This basis determines a second toric change of coordinates with such that () with respect to the new coordinates.

After the above changes of coordinates, the ideal is determined by the equations . Setting

it follows that . Rewriting with respect to the original coordinates now completes the computation of .

4.3The algebraic group generated by a single matrix

Let us now consider the case when consists of a single arbitrary matrix . Then we first compute the multiplicative Jordan decomposition of . Modulo a change of basis of , this means that

where and are the semi-simple and unipotent parts of :

where

Proposition 28We have .

RemarkNotice that is well-defined for power series and nilpotent matrices ; in this case, and .

ProofThe assertion is clear if , so assume . Let . We clearly have , since is a closed algebraic group which contains . Moreover, the set is infinite, so . Since is irreducible and , we conclude that .

Proposition 29We have .

ProofSince is a commutative group, [Humphreys, 1981, Theorem 15.5] implies that , where and are closed subgroups of . Now and are closed subgroups of resp. , so is a closed subgroup of . Since , it follows that .

Corollary 30If , then .

4.4Membership testing for the connected component

In order to compute the closure of the product of a finite number of algebraic groups of the form , an important subproblem is to test whether a given matrix belongs to .

We first observe that implies . After the computation of and with it therefore suffices to check that and . In fact, it suffices to check whether , where is the unique matrix in with . Modulo a suitable base change, we have thus reduced the general problem to the case when is a diagonal matrix whose eigenvalues are all roots of unity.

Assume that and are such that . Since and commute, it follows that and can be diagonalized w.r.t. a common basis. The elements of this basis are elements of the different eigenspaces of . In other words, if with pairwise distinct , then is diagonal for some block matrix with for each . It follows that for certain . Without loss of generality, we may therefore replace by the intersection of with .

From now on, we assume that the above two reductions have been made. Let be a diagonal matrix in . By lemma 27, we have if and only if any -linear relation induces a relation , where . Now consider a random matrix in , i.e. a linear combination of the basis elements with small random integer coefficients. We compute its blockwise Jordan normal form so that and let be the restriction of to the diagonal. We have . Computing a basis for the -linear relations of the form using EH3, the above criterion now enables us to check whether .

If the check whether succeeds, then we are clearly done. Otherwise, since was chosen in a random way, the relation is very likely to be satisfied for all possible choices of (up to permutations of coordinates inside each block). Indeed, the for which this is not the case lie on a countable union of algebraic variety of a lower dimension, so has measure . Heuristically speaking, we may therefore conclude that if the check fails (at least temporarily, modulo some final checks when the overall computation of will be completed).

Theoretically speaking, we may perform the above computations with instead of , where is a basis of and the are formal parameters. We then check whether the relation is still satisfied for the analogue of . If so, then we are sure that . Otherwise, we keep trying with other random elements of .

It is likely that a more efficient theoretical algorithm can be designed for testing -linear relations between the eigenvalues of elements in . One of the referees suggested to use similar methods as in [Masser, 1988; Bertrand, 1995; Compoint and Singer, 1998]. However, we did not study this topic in more detail, since our final algorithm for the computation of Galois groups will be based on heuristics anyway. We also notice that a “really good” random number generator should actually never generate points which satisfy non-trivial algebraic relations.

4.5Computing the closure of

A Lie algebra is said to be algebraic, if it is the Lie algebra of some algebraic group, i.e. if is an algebraic subset of . It is classical [Borel, 1991, Corollary 7.7] that the smallest Lie algebra generated by a finite number of algebraic Lie algebras is again algebraic. The Lie algebras we will consider in our algorithms will always assumed to be algebraic. Given a finite number of algebraic Lie algebras and a basis for , it is easy to enrich so that is a Lie algebra: as long as for two elements , we add to . By what precedes, the computed Lie algebra is again algebraic.

Putting together the ingredients from the previous sections, we now have the following algorithm for computing the smallest closed algebraic group which contains .

Algorithm Closure()

Input: A subset of

Output: a numeric approximation of

Step 1. [Initialize algorithm]

Compute for each

Let (notice that )

Let

Step 2. [Closure]

While there exists an with set

While there exists an with set

While there exists with do

Compute

If then set , quit loop and repeat step 2

Otherwise, set

Return

The termination of this algorithm relies on a lemma, whose proof was kindly communicated to the author by J.-Y. Hée.

Lemma 31Let be a closed algebraic subgroup of and let be a finite number of matrices in the normalizer of . Denote by the group generated by and . If all elements in have finite order, then is finite.

ProofIn the case when , the result is classical [Dixon, 1971, Theorem 9.2]. In the general case, the normalizer of is a closed algebraic subgroup of and is a normal subgroup of . By [Borel, 1991, Theorem 6.8 and Proposition 1.10], it follows that is an affine algebraic group which is isomorphic to a closed algebraic matrix group. This reduces the general case to the special case when .

Theorem 32There exists an such that, for every with , the set returned by Closure, considered as a subset of , coincides with the smallest closed algebraic subgroup of which contains .

ProofClearly, the dimension of increases throughout the execution of the algorithm, so it remains ultimately constant. At this point, the set will keep growing and the lemma implies that ultimately stabilizes. When this happens, is closed under multiplication modulo , as well as under multiplicative inverses, since each element in has finite order modulo . We conclude that is indeed the smallest closed algebraic subgroup of which contains , provided that the approximate zero-test always returns the right result.

In order to prove the correctness at a sufficient precision, we assume that we use the theoretic membership test from section 4.4 and that the random number generator successively generates the same random numbers each time we relaunch the algorithm at a higher precision. Now consider the trace of the execution of our algorithm when using an infinite precision. Let be a sufficient precision such that all zero-tests in this execution tree are still correct when we replace the infinite precision by a precision . Then the trace of the execution any finite precision coincides with the trace of the execution at infinite precision. This completes the proof.

Remark 33The main improvement of the algorithm Closure w.r.t. the algorithm from [Derksen et al., 2003] lies in the more efficient treatment of the connected component (using linear algebra). On the other hand, the mere enumeration of representatives in each connected component can be very unefficient (although a Gröbner basis might be of the same size). Fortunately, we will see in the next sections how to remove this drawback.

Assume now that is the set of generators for as computed in theorem 20. Assume that we have computed a reasonable candidate for , expressed in the original basis corresponding to . We still have to reconstruct and with such that .

In the case of , by selecting a suitable basis of , we may consider as a big matrix whose first columns are linearly independent. We compute the row-echelon form of this basis:

The entries of must be in : provided that is indeed generated by a basis of matrices with entries in , the row-echelon form of this second basis coincides with . It therefore suffices to reconstruct the entries of using the LLL-algorithm.

In the case of a matrix , the set is an algebraic variety of dimension over . Now choose close to in such a way that independent coordinates of are all in . Then the other coordinates of , considered as elements of , are easily found using Newton's method. Since is an algebraic variety, these other coordinates are actually in , and we reconstruct them using the LLL-algorithm.

4.6Fast computations with the connected components

The algorithm Closure from the previous section is quite inefficient when the set becomes large. It is therefore useful to seek for a better computational representation of . For finite groups , one classical idea is to search for a sequence of subgroups

(9)

such that the indices are small. Then we may represent elements in by sequences with for each . This representation is particularly useful if operates on a set and if there exists points in such that

is the stabilizer of the set for each . Then the set corresponds to the orbit of while leaving fixed [Sims, 1970; Sims, 1971].

In the case of matrix groups, one often takes for [Murray and O'Brien, 1995]. However, this approach only yields interesting results when there exist non-trivial invariant subspaces under the action of the group, which will usually not be the case for us (otherwise we may factor and consider smaller problems). A theoretical way out of this is to also consider the action of on exterior powers . However, this approach is very expensive from a computational point of view. In our more specific context of matrices with complex entries, we will therefore combine two other approaches: non-commutative lattice reduction and the operation of on via conjugations .

The algebra admits a natural (multiplicative) norm, given by

where stands for the Euclidean norm on . If is finite, this enables us to construct as in (9) as follows. Assuming that have been constructed, we consider a matrix for which is minimal, and let be the set generated by and in . This construction allows us to rapidly identify a big commutative part of . More precisely, we have

Proposition 34Let be such that and . Then we have

ProofWriting and , we expand and in . This yields a non-commutative power series in and whose terms in and vanish. It follows that

The proposition implies that whenever . Now take and with , where the are as above. Then it follows that and commute whenever . What is more, proposition 34 shows that taking commutators is a convenient way to construct matrices close to identity from a set of non-commutative generators.

However, from the effective point of view, we will not compute the exact computation of minimal representatives in cosets of algebraic groups in detail. We will rather simulate such a computation in a way which is sufficient for our purpose. If is such that is small, then we will also try to use the fact that the centralizer of is often a big subgroup of , so the orbit of is small.

4.7Non-commutative lattice reduction

Let be a closed algebraic subgroup of with associated Lie-algebra . In this section, we will show how to compute efficiently with elements of the finite group . Until the very end of this section, we assume that is included in the connected component of the normalizer of in . We denote by the Lie algebra of this connected component. By [Humphreys, 1981, Theorem 13.3], we have .

1Orthogonal projection
Let and recall that belongs to the normalizer of . If , then also belongs to the normalizer of . Since lies in the connected component of this normalizer, we have . Now consider the orthogonal supplement of for the Hermitian product on . We define , where is the orthogonal projection of on . From , it follows that , and we denote . Since is connected, the function may actually be analytically continued to a multivalued function . After choosing branch cuts (the way this is done is not crucial for what follows, provided that we make the standard choice for with ), this allows us to extend the definitions of and to the case when .

2Representation of the elements in
Let and be such that

Let be the prime-decomposition of the order of modulo , with . Let and for all . Let be the subgroup of of elements which commute with modulo , so that . For , we represent elements in the quotient by elements in the orbit of the action modulo . Since , the set is a Lie algebra whose normalizer contains . Consequently, , and we represent elements in by products , with and . The elements in are represented in a similar way as the elements in , using recursion. The successive matrices for , , etc. will be called a basis for . A basis is said to be sorted if .

3Adding new elements to a basis
Let be a sorted basis for and assume that we want to compute the extension of by a new matrix . Whenever we hit an element with during our computations, then we start the process of basis reduction, which is described below. Whenever we find an element in , then we abort all computations and return this element (indeed, in that case, we may continue with the closure of the connected component in Closure).

Let , , etc. be as above. We start by computing the orbit of modulo for . Whenever we hit an element (modulo ) with or , then we start the process of basis reduction. Otherwise, we obtain a finite orbit, together with a finite number of matrices by which we have to extend . We keep doing this using the same method for until .

At the end, we still have to show how to extend with a new matrix . Now recursive application of the algorithm to and yields a sorted basis . When keeping track of the corresponding powers of during the computations, we also obtain a finite system of generators for . Using g.c.d. computations we either obtain a minimal generator or a new element in the connected component. In the first case, we return if and apply basis reduction otherwise.

4Basis reduction
Let be in with . We call a raw basis. In the above algorithms, raw bases occur when we are given an ordered basis for , and we find a new element with .

Using the above base extension procedure, we may transform a raw basis into a basis for : starting with , we successively add . However, it is more efficient to reduce first. More precisely, let us now describe a procedure which tries to replace by a better raw basis , with , and whose elements are closer to identity. Of course, we may always return the original basis if a better one could not be found.

We first test whether all basis elements are roots of unity modulo . If not, then we found a new element in the connected component. We next test whether there exist with , in which case we keep adding the smallest such commutator to the basis. Whenever this stops, we write with and consider all lattice reductions proposed by the LLL-algorithm in the commutative vector space . Whenever , for one such reduction, then we perform the corresponding reduction on our basis and keep repeating the basis reduction process.

5The general case
We still have to show how to deal with the case when is not included in the connected component of the normalizer of in . In that case, we start with the computation of a basis for , using linear algebra. Since is a normal subgroup of , we have . Now we have explained above how to compute with elements in . If , then may use recursion for computations in the finite group . If , then elements in have necessarily small order, so we simply list the elements of .

5Conclusion and final notes

We hope that we provided convincing evidence that analytic methods may be used for the efficient computation of differential Galois groups and related problems like the factorization of linear differential operators.

The two main remaining challenges are the concrete implementation of the algorithms presented here (as part of a more general library for the computation with analytic functions such as [van der Hoeven et al., 2002–2005]) and the development of a priori or a posteriori methods for ensuring the correctness of the computed result. Some ideas into this direction are as follows:

Besides the above ideas for improving the algorithms, this paper also raises a few other interesting questions:

Of course, a better mastering of the algorithms in this paper may also lead to more efficient algorithms for other computations which rely on differential Galois theory, like the computation of Liouvillian or other forms of solutions. More generally, our algorithms may be used for other computations with algebraic matrix groups over and other fields of characteristic . We also expect all results to generalize to holonomic systems of linear partial differential equations.

6Acknowledgment
The author would like to thank J.-Y. Hée, W. de Graaf, F. Zara, J. Écalle and the referees for several helpful comments and references.

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