Concrete Resolution of Differential Problems using Tannakian Categories

Jacques-Arthur Weil

Département de Mathématiques, Université de Limoges

Algorithms Seminar

April 19, 1999

[summary by Frédéric Chyzak]

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Abstract
Given a linear ODE with polynomial coefficients, one easily finds local information about its solutions. To obtain global information of algebraic nature (operator factorization, explicit finite form, algebraic relations between solutions), one classically reduces the problem to determining rational or exponential solutions of auxiliary linear ODE's. The latter are often uneasy to compute in practice, and we show by a few examples how to advantageously substitute differential systems that are simpler to construct, solve or study.



1   Solving Linear Differential Equations

The main question when studying a linear differential operator L is how to ``solve'' for its solutions. ``Solving'', however, covers several meanings. Throughout this text, L denotes a differential operator acting on a function y in the variable x by L(y)=any(n)+...+a0y for polynomials ai in x with coefficients in a field C. This field is Q, Q_ or C in practice.

The simplest way to solve is the determination of local information, like a basis of formal solutions in the neighbourhood of 0. The general form of a formal solution is the formal series
y=x
a
 
(p0(ln x)+p1(ln x)x1/r+...+pi(ln x)xi/r+···)
for polynomials pi with uniformly bounded degrees. Here, r is a positive integer, the ramification, and p0 is assumed to be non-zero so as to ensure that the highest possible power has been incorporated into the generalized exponent aÎ C[x1/r]. The power xa is nothing but expòa/x dx, the formal solution of y'=(a/x)y. This approach by generalized exponents is due to Van Hoeij [12] and unifies regular and irregular singular expansions. A similar treatment was developed in the case of systems by Barkatou [1] and Pflügel [5].

Of course, the most generally understood acceptance of ``solving'' relates to resolution in closed form. By simultaneously considering the bases of formal solutions in the neighbourhood of all possible singularities of the operator L, namely, the zeroes of its leading coefficient an(x), several algorithms are available to search for solutions in various classes of closed form, like polynomial solutions yÎ C[x], rational solutions yÎ C(x), exponential solutions y for which y'/yÎ C(x), or liouvillian solutions y for which y'/y is algebraic over C(x). See [4, 13] and the references there.

Note that each solution s in the above classes supplies a first-order right-hand factor of the operator L, namely -s'/s where denotes the derivation operator with respect to x. A more general problem is that of the factorization of operators from the ring C(x)[] of linear differential operators with rational function coefficients, and the search for higher-order right-hand factors. This relates to differential Galois theory. More specifically, polynomial, rational, and exponential solutions correspond to factorization in this ring, whereas liouvillian solutions correspond to the more complex problem of absolute factorization [13], i.e., factorization of an operator LÎ C(x)[] with factors in K[] for an algebraic closure K of C(x). In any case, factorization relates to solving since any solution of any right-hand factor is a solution of the original operator. Furthermore, specialized algorithms exist for linear differential equations of small orders.

Right-hand factors of an operator are a first type of auxiliary operators or lower order that simplify solving. More generally, another form of ``solving'' the operator L is by looking for its solutions that can be viewed as powers, products, or wronskians of an auxiliary operator, or system of operators, of lower order. This is the main discussion of the next sections. Applications include the classification of solutions, connexion problems, number theory (by looking for differential equations of minimal order), and the search for first integrals of non-linear differential equations.

2   Lower Order Equations and Symmetric Power Solutions

As an example, consider the third-order equation y'''-4ry'-2r'y=0 (rÎ C(x)). It admits a basis of solutions of the form (z1=y12,z2=y22,z3=y1y2), where both y1 and y2 are solutions of the same second-order equation y''=ry. To obtain such special solutions of a higher-order operator L, the crucial relation to be used is z1z2=z32. Indeed, considering the formal solution z~i=xaiSi corresponding to the expansion of each actual function zi, we obtain that the formal expansion of the product z1z2 is the product of formal expansions z~1z~2=xa1+a2S1S2. Identifying those generalized exponents for L that can be a sum of two terms therefore supplies a set of candidate exponents for the auxiliary operator and the zi. Note that the original third-order equation has been replaced by a ``simpler system'' consisting of a second-order equation and a quadratic relation.

3   Liouvillian Solutions

To solve an operator L for its liouvillian solutions, one looks for the possible irreducible polynomials P of the form Xm-bm-1Xm-1-...-b0 such that P(u)=0 implies L(expò u dx)=0 [10]. Given the order n of the operator, differential Galois theory shows that only finitely many degrees are possible for the polynomial P. There exists an algorithm to compute the list of the possible numbers m: for n=2, the list is 1, 2, 4, 6, and 12; for n=3, it is 1, 3, 6, 9, 21, and 36 [6, 7, 9]; for n=4 and higher, a formula is known for the maximum number of the list.

By construction, the roots ui of P are logarithmic derivatives yi'/yi of a solution of L, and bm-1=åiui=åiyi'/yi is the logarithmic derivative of the product Õiyi. A necessary and sufficient condition for the existence of a polynomial P of degree m above, which describes the liouvillian solutions of L is that there exists a polynomial of degree m in solutions of L whose logarithmic derivative is rational, and which is the product of linear factors. More specifically, for a solution basis (z1,...,zm) of L the product Õiyi is searched for under the form Õi(ci,1z1+...+ci,mzm).

The search for liouvillian solutions therefore reduces to the search for exponential solutions. To this end, the present work allows to avoid computing the equation for the symmetric power, which is too large, but prefers a more compact representation.

4   Factorization and Alternate Power Solutions

As another typical example, let us consider the search for a right-hand factor H=2-b1-b0 of order 2 of the operator L=4-a22-a1-a0 of order 4. For any solution basis (y1,y2) of H, the operator H is given by the determinantal representation
H(y)= ½
½
y1 y2
y1' y2'
½
½
-1

 
½
½
½
½
y y1 y2
y' y1' y2'
y'' y1'' y2''
½
½
½
½		 =y''-
w0,2
w0,1
y'+
w1,2
w0,1
y   where   wi,j= ½
½
y1(i) y2(i)
y1(j) y2(j)
½
½		 .
To obtain a factor of order 2, we now search for an exponential solution and show that it can be interpreted as a determinant w0,1. Let A be the companion matrix
A= æ
ç
ç
ç
è
0 1 0 0
0 0 1 0
0 0 0 1
a0 a1 a2 0
ö
÷
÷
÷
ø		 ,     and let     Y= æ
ç
ç
ç
è
y
y'
y''
y'''
ö
÷
÷
÷
ø		 ,     so that     Y'=AY.
Let us introduce the vector Z=(w0,1,w0,2,w0,3,w1,2,w1,3,w2,3)T. In view of their definition, the wi,j satisfy differential relations like w0,1'=w0,2, w0,3'=w1,3+a0w0,0+a1w0,1+a2w0,2, and so on. From them, we find a matrix
L2(A)= æ
ç
ç
ç
ç
ç
ç
è
0 1 0 0 0 0
0 0 1 1 0 0
a1 a2 0 0 1 0
0 0 0 0 1 0
-a0 0 a2 0 0 1
0 -a0 0 -a1 0 0
ö
÷
÷
÷
÷
÷
÷
ø		     such that Z'=L2(A)Z.
Again, we then only look for exponential solutions Z of the matrix L2(A), which is easy to construct and contains more information than the usual single auxiliary equation used for factorization. Finally, one has to check that the solution Z is a determinant. For this, a necessary and sufficient condition is the Plücker relation, which here simply reduces to w0,1w2,3-w0,2w1,3+w0,3w1,2=0.

To rephrase the method in a more formal way, introduce V, the solution space of L. The search for Z is indeed a search for objects in the 2-exterior power L2(V), i.e., the vector space of linear combination of formal 2-exterior products vÙ w, (v,w)Î V2, which satisfy the rule wÙ v=-vÙ w. Pure exterior product uÙ v are interpreted as determinants. The search for Z is therefore equivalent to the search for a pure exterior product w0,1ÎL2(V) such that the 1-dimensional vector space Cw0,1 is stable under the action of the differential Galois group of L.

Here the search for a second-order right-hand factor of a fourth-order equation has been reduced to solving a ``simpler'' system of six first-order equations.

5   Module and Dual Module Associated with an Operator

As an important tool for the study of a linear differential operator L, one classically associates a canonical module in the following way. For L in the algebra k[] of linear differential operators with coefficients in a field k, one considers the quotient M=k[]/k[]L of k[] by its left ideal k[]L. The left module M can be viewed as the module k[]y generated by a generic solution y of the operator L. Linear constructs on and between solution spaces of operators, like (direct or usual) sums, (symmetric or exterior or usual commutative) products, (indefinite) integration, and so on, correspond to constructs on and between the corresponding k[]-modules.

A variant module is obtained by endowing the dual k-vector space M* with a k[]-module structure. Let r be the order of L, then M is of dimension r and its dual M*=Homk(M,k) is isomorphic to kr. Now let A be the companion matrix of L and (b1,...,br) be the canonical basis of M*. The latter is turned into a k[]-module by defining an operator Ñ on M* by the action
(Ñ b1,...,Ñ br)T=-AT(b1,...,br)T
and letting  act by Ñ. Thus, Ñ(am)=aÑ m+a'm when aÎ k and mÎ M. From this Leibniz rule applied to the product y1b1+...+yrbr=(y1,...,yr)(b1,...,br)T, we derive the equality
Y'=AY    for     Y=(y1,...,yr)T
whenever Ñ(y1b1+...+yrbr)=0. Note that this k[]-module structure on M* usually does not make it the dual k[]-module Homk[](M,k), for the operator L usually has no solution in k.

The modules M* allow for a better description of the calculations suggested in the previous sections through a link between the solution space V=Sol(L) and the k[]-module M*. This link is obtained by introducing the map f from V to M* defined by f(y)=yb1+...+y(r-1)br. Calculations with elements of the C-vector space Sol(L) have their counterparts in the k[]-module M*. For example, one recovers the determinants of the previous sections from the following identity for exterior products in the module L2M*
f(y1)Ùf(y2)=
 
å
1£ i<j£ r
wi-1,j-1 biÙ bj     with     wi,j= ½
½
y1(i) y2(i)
y1(j) y2(j)
½
½		 .
Again, constructs at the level of solution spaces translate into constructs at the level of the corresponding k[]-modules.

6   Tannakian Definition of the Differential Galois Group

This section is based on my (Chyzak's) study and tentatively reflects what was not presented by the speaker for lack of time. It aims at defining differential Galois groups by the Tannakian viewpoint, as an alternative to Kolchin's more traditional and elementary definition by differential extension fields. Interestingly, some properties are easier to derive by the Tannakian viewpoint, for instance that it is a linear algebraic group (i.e., a subgroup of GLn(C) and an algebraic variety). Another consequence is the possibility to rephrase algorithms in such a way that differential Galois theory, in the sense of Kolchin, is only used as a classification tool to prove the correction of the algorithms, while calculations take place at the level of modules in a more efficient way. This presentation is based on a discussion with the speaker, on conference proceedings by Ramis and Martinet [8, Part 2, Chapter 1], and on unpublished notes by Churchill [2, 3]. More direct references may be works by Bertrand, Deligne, and Katz. The Tannakian construction has a natural counterpart in difference Galois theory [11, Section 1.4].

For comparison sake, Kolchin's definition of the differential Galois group of a linear differential operator LÎ k[] is as follows. Let C be the subfield of constants of k, n be the order of L, and consider the Picard-Vessiot extensions k' of k associated with L, i.e., the differential field extensions of k that contain an n-dimensional C-vector space of solutions of L and do not enlarge the constant field C. Then the differential Galois group of L is defined as the group G of differential field automorphisms (i.e., field automorphisms that respect the differential structure) of any Picard-Vessiot extension k' that additionally respect the action of k on k'. This mimics the classical Galois theory for a polynomial PÎ k[X], where one introduces the group of field automorphisms of a suitable extension k' of k which contains all solutions of P and restrict to the identity on k. While the (algebraic) Galois group of a polynomial is a subgroup of a permutation group  Sn, the differential Galois group of an operator is a subgroup of the linear group GLn(C) for the common field of constants C of k and k'.

For its part, instead of a single extension k' of k, the Tannakian presentation simultaneously considers a whole collection of k[]-modules, and introduces the differential Galois group as a group of internal transformations on this collection. Crucially, each transformation has to transform all the modules in a way compatible with the linear maps between the modules. Moreover, each module M is associated with a solution set that can be viewed as the kernel of the derivation on M, and the above-mentioned transformations have to be compatible with taking solutions.

At the heart of the Tannakian construction are k-vector spaces V that are closed under the action of an operator Ñ which extends the action of the derivation on k by the Leibniz rule:
Ñ(af)=aÑ(f)+a'f,    when aÎ k and fÎ V.
This makes V a k[]-module with acting by Ñ.

From now on, we restrict to k[]-modules that are finite-dimensional k-vector spaces. Fundamental examples are the modules M=k[]/k[]L discussed in the previous section. We also restrict to k=C(z). An element hÎkerÑ is called a horizontal vector. As has been explained when discussing dual modules M*, horizontal vectors in M* correspond to solutions yÎ kr of the equation D y=0 where D=d/dz-A for (Ñ b1,...,Ñ br)T=-AT(b1,...,br)T once a basis (b1,...,br) of M* has been chosen. Rather than enlarging the space M where we have a solution for L, as is the case in the traditional differential Galois theory, we now enlarge the coefficient field of M* so as to ensure the existence of a solution to D and thus of horizontal vectors for Ñ. To this end, consider a non-singular point aÎC of the operator L, and introduce the field  Ma of germs of meromorphic functions at a, which is isomorphic to the field of convergent Laurent series C{z-a}[(z-a)-1]. By Cauchy's theorem, the C-vector space kerD, where D is now viewed as acting on ( Ma)r, is of dimension r and supplies with horizontal vectors of Ñ in  MaÄC(z)M. Note that kerD and kerÑ usually have no more structure than that of C-vector spaces.

As an example, let us consider the C(z)[]-module generated by the Bessel function of the first kind J0(z). It is a two-dimensional C(z)-vector space with basis (J0(z),J1(z)), and J0'=-J1. With the above notation,
æ
è
Ñ J0
Ñ J1
ö
ø		 = æ
è
J0'
-J0''
ö
ø		 = æ
è
0 -1
1 -1/z
ö
ø		 æ
è
J0
J1
ö
ø		 ,     so that     A= æ
è
0 -1
1 1/z
ö
ø		 .
As a result of a simple computation, h=f1J0+f2J1 is a horizontal vector of Ñ if and only if
æ
è
f1
f2
ö
ø		 ÎkerD =C z æ
è
-J1(z)
J0(z)
ö
ø		 ÅC z æ
è
-Y1(z)
Y0(z)
ö
ø		 ,
where Yn(z) are the Bessel functions of the second kind, and where Jn and Yn now denote germs of the corresponding functions (their local expansions at a¹0). This simplifies to hÎC z(Y0J1-J0Y1), whence h is a constant by the Wronskian relation Y0J1-J0Y1=2/p z.

To the module M above and any non-singular point aÎC, we have just associated the C-vector space of horizontal vectors of Ñ in the form of local expansions at a. Denote ha(M) this vector space. We now proceed to associate horizontal vectors to more involved module constructions. Denote {M} the smallest class of k[]-modules containing M and closed under finite direct sums and products, finite symmetric and exterior products, dualization, and taking the module of homomorphisms between two modules, and submodules. One can extend Ñ from M to any VÎ{M} in a canonical way; in particular:
Ñ|
 
VÅ W
= æ
è
Ñ|V 0
0 Ñ|W
ö
ø		 ,      Ñ|
 
VÄ W
=Ñ|VÄ1|W+1|VÄÑ|W.
This class becomes a category for the usual k[]-module morphisms. The map ha extends to {M} by ha(V)=ker(Ñ| MaÄC(z)V). In particular, ha(VÅ W)=ha(V)Åha(W) and ha(VÄ W)=ha(V)Äha(W). The crucial fact is that ha is compatible with the maps that are natural between modules on the one hand and horizontal vector spaces on the other hand. Specifically, any k[]-module homomorphism h between two modules V and W induces a C-linear homomorphism ha(h) between ha(V) and ha(W). This makes ha a functor, in the sense that the diagram

To relate horizontal vectors at two non-singular points a and b, consider the maps s that associate with any k[]-module V a C-linear map s(V):ha(V)®hb(V), subject to the constraint that Such a map s (from {M} to the linear morphisms in the category of C-vector space) is called a morphism from (the functor) ha to (the functor) hb. The collection of such morphisms when a and b vary is a semigroup for composition. The corresponding notion of isomorphisms (of functors) is obtained when each of the linear maps s(V) is invertible. Two cases are of interest: when a¹ b, one of those isomorphisms is provided by analytic continuation along a path from a to b; when a=b, the isomorphisms from ha into itself form a group (the group of automorphisms of the functor ha). This group is the differential Galois group of L, following the Deligne-Katz definition.

References

[1]
Barkatou (M. A.). -- An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system. Applicable Algebra in Engineering, Communication and Computing, vol. 8, n°1, 1997, pp. 1--23.

[2]
Churchill (R. C.). -- Connections on modules. -- February 1997. Unpublished Notes for the Kolchin Seminar in Differential Algebra.

[3]
Churchill (R. C.). -- A comparison of the Kolchin and Deligne-Katz definitions of a differential Galois group. -- February 1998. Unpublished Notes for the Kolchin Seminar in Differential Algebra.

[4]
Pflügel (Eckart). -- ISOLDE, a package for computing invariants of systems of ordinary linear differential equations. In Salvy (Bruno) (editor), Algorithms Seminar, 1997-1998, INRIA Research Report, pp. 79--82. -- 1998.

[5]
Pflügel (Eckhard). -- An algorithm for computing exponential solutions of first order linear differential equations. In Küchlin (W.) (editor), ISSAC'97 (July 21--23, 1997. Maui, Hawaii, USA). pp. 164--171. -- ACM Press, 1997.

[6]
Singer (Michael F.). -- Liouvillian solutions of nth order homogeneous linear differential equations. American Journal of Mathematics, vol. 103, n°4, 1981, pp. 661--682.

[7]
Singer (Michael F.) and Ulmer (Felix). -- Liouvillian and algebraic solutions of second and third order linear differential equations. Journal of Symbolic Computation, vol. 16, n°1, 1993, pp. 37--73.

[8]
Tournier (E.) (editor). -- Computer Algebra and Differential Equations. -- Academic Press, Computational Mathematics and Applications, 1989.

[9]
Ulmer (Felix). -- On Liouvillian solutions of linear differential equations. Applicable Algebra in Engineering, Communication and Computing, vol. 2, n°3, 1992, pp. 171--193.

[10]
Ulmer (Felix). -- Linear differential equations and liouvillian solutions. In Salvy (Bruno) (editor), Algorithms Seminar, 1993--1994, INRIA Research Report, pp. 41--44. -- 1994.

[11]
van der Put (Marius) and Singer (Michael F.). -- Galois Theory of Difference Equations. -- Springer, 1997, Lecture Notes in Mathematics.

[12]
Van Hoeij (Mark). -- Formal solutions and factorization of differential operators with power series coefficients. Journal of Symbolic Computation, vol. 24, n°1, 1997, pp. 1--30.

[13]
Weil (Jacques-Arthur). -- Absolute factorization of differential operators. In Salvy (Bruno) (editor), Algorithms Seminar, 1996--1997, INRIA Research Report, pp. 33--36. -- 1997.
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