Concrete Resolution of Differential Problems using Tannakian
Categories
JacquesArthur 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)=a_{n}y^{(n)}+...+a_{0}y for polynomials a_{i} 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 

(p_{0}(ln x)+p_{1}(ln x)x^{1/r}+...+p_{i}(ln x)x^{i/r}+···) 
for polynomials p_{i} with uniformly bounded degrees. Here, r is
a positive integer, the ramification, and p_{0} is assumed to be
nonzero so as to ensure that the highest possible power has been
incorporated into the generalized exponent aÎ C[x^{1/r}].
The power x^{a} 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 a_{n}(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
firstorder righthand 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 higherorder righthand 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 righthand factor is a solution
of the original operator. Furthermore, specialized algorithms exist
for linear differential equations of small orders.
Righthand 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 nonlinear differential equations.
2 Lower Order Equations and Symmetric Power Solutions
As an example, consider the thirdorder equation y'''4ry'2r'y=0
(rÎ C(x)). It admits a basis of solutions of the
form (z_{1}=y_{1}^{2},z_{2}=y_{2}^{2},z_{3}=y_{1}y_{2}), where both y_{1} and y_{2}
are solutions of the same secondorder equation y''=ry. To obtain
such special solutions of a higherorder operator L, the crucial
relation to be used is z_{1}z_{2}=z_{3}^{2}. Indeed, considering the formal
solution z^{~}_{i}=x^{a}_{i}S_{i} corresponding to the
expansion of each actual function z_{i}, we obtain that the formal
expansion of the product z_{1}z_{2} is the product of formal expansions
z^{~}_{1}z^{~}_{2}=x^{a}_{1}^{+a}_{2}S_{1}S_{2}.
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 z_{i}. Note that the original thirdorder
equation has been replaced by a ``simpler system'' consisting of a
secondorder 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 X^{m}b_{m1}X^{m1}...b_{0} 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 u_{i} of P are logarithmic
derivatives y_{i}'/y_{i} of a solution of L,
and b_{m1}=å_{i}u_{i}=å_{i}y_{i}'/y_{i} is the logarithmic derivative
of the product Õ_{i}y_{i}. 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 (z_{1},...,z_{m}) of L the
product Õ_{i}y_{i} is searched for under the
form Õ_{i}(c_{i,1}z_{1}+...+c_{i,m}z_{m}).
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
righthand factor H=¶^{2}b_{1}¶b_{0} of order 2 of the
operator L=¶^{4}a_{2}¶^{2}a_{1}¶a_{0} of order 4. For
any solution basis (y_{1},y_{2}) of H, the operator H is given by
the determinantal representation
H(y)= 
½ ½ 
y_{1} 
y_{2} 
y_{1}' 
y_{2}' 

½ ½ 

½ ½ ½ ½ 
y 
y_{1} 
y_{2} 
y' 
y_{1}' 
y_{2}' 
y'' 
y_{1}'' 
y_{2}'' 

½ ½ ½ ½ 
=y'' 

y'+ 

y
where
w_{i,j}= 
½ ½ 
y_{1}^{(i)} 
y_{2}^{(i)} 
y_{1}^{(j)} 
y_{2}^{(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 w_{0,1}. Let A be the companion matrix
A= 
æ ç ç ç è 
0 
1 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
a_{0} 
a_{1} 
a_{2} 
0 

ö ÷ ÷ ÷ ø 
,
and let
Y= 
æ ç ç ç è 

ö ÷ ÷ ÷ ø 
,
so that
Y'=AY. 
Let us introduce the
vector Z=(w_{0,1},w_{0,2},w_{0,3},w_{1,2},w_{1,3},w_{2,3})^{T}.
In view of their definition, the w_{i,j} satisfy differential
relations like w_{0,1}'=w_{0,2},
w_{0,3}'=w_{1,3}+a_{0}w_{0,0}+a_{1}w_{0,1}+a_{2}w_{0,2},
and so on. From them, we find a matrix
L_{2}(A)= 
æ ç ç ç ç ç ç è 
0 
1 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
a_{1} 
a_{2} 
0 
0 
1 
0 
0 
0 
0 
0 
1 
0 
a_{0} 
0 
a_{2} 
0 
0 
1 
0 
a_{0} 
0 
a_{1} 
0 
0 

ö ÷ ÷ ÷ ÷ ÷ ÷ ø 
such that Z'=L_{2}(A)Z. 
Again, we then only look for exponential solutions Z of the
matrix L_{2}(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 w_{0,1}w_{2,3}w_{0,2}w_{1,3}+w_{0,3}w_{1,2}=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 2exterior power L_{2}(V), i.e., the vector space
of linear combination of formal 2exterior products vÙ w,
(v,w)Î V^{2}, 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 w_{0,1}ÎL^{2}(V) such that the
1dimensional vector space Cw_{0,1} is stable under the action
of the differential Galois group of L.
Here the search for a secondorder righthand factor of a fourthorder
equation has been reduced to solving a ``simpler'' system of six
firstorder 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 kvector
space M^{*} with a k[¶]module structure. Let r be the
order of L, then M is of dimension r and its
dual M^{*}=Hom_{k}(M,k) is isomorphic to k^{r}. Now
let A be the companion matrix of L and (b_{1},...,b_{r}) 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 b_{1},...,Ñ b_{r})^{T}=A^{T}(b_{1},...,b_{r})^{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 y_{1}b_{1}+...+y_{r}b_{r}=(y_{1},...,y_{r})(b_{1},...,b_{r})^{T}, we
derive the equality
Y'=AY for Y=(y_{1},...,y_{r})^{T}
whenever Ñ(y_{1}b_{1}+...+y_{r}b_{r})=0. Note that this
k[¶]module structure on M^{*} usually does not make it
the dual k[¶]module Hom_{k[¶]}(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)=yb_{1}+...+y^{(r1)}b_{r}. Calculations with elements of
the Cvector 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 L^{2}M^{*}
f(y_{1})Ùf(y_{2})= 

w_{i1,j1} b_{i}Ù b_{j}
with
w_{i,j}= 
½ ½ 
y_{1}^{(i)} 
y_{2}^{(i)} 
y_{1}^{(j)} 
y_{2}^{(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 GL_{n}(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 PicardVessiot extensions k' of k
associated with L, i.e., the differential field extensions of k
that contain an ndimensional Cvector 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 PicardVessiot 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 S_{n}, the
differential Galois group of an operator is a subgroup of the linear
group GL_{n}(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 abovementioned
transformations have to be compatible with taking solutions.
At the heart of the Tannakian construction are kvector 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
finitedimensional kvector 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 k^{r} of the equation D y=0
where D=d/dzA for (Ñ b_{1},...,Ñ
b_{r})^{T}=A^{T}(b_{1},...,b_{r})^{T} once a basis (b_{1},...,b_{r})
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
nonsingular point aÎC of the operator L, and introduce the
field M_{a} of germs of meromorphic functions at a, which
is isomorphic to the field of convergent Laurent
series C{za}[(za)^{1}]. By Cauchy's theorem, the Cvector
space kerD, where D is now viewed as acting
on ( M_{a})^{r}, is of dimension r and supplies with
horizontal vectors of Ñ in M_{a}Ä_{C}_{(z)}M.
Note that kerD and kerÑ usually have no more structure
than that of Cvector spaces.
As an example, let us consider the C(z)[¶]module generated
by the Bessel function of the first kind J_{0}(z). It is a
twodimensional C(z)vector space with basis (J_{0}(z),J_{1}(z)),
and J_{0}'=J_{1}. With the above notation,
æ è 

ö ø 
= 
æ è 

ö ø 
= 
æ è 

ö ø 
æ è 

ö ø 
,
so that
A= 
æ è 

ö ø 
. 
As a result of a simple computation, h=f_{1}J_{0}+f_{2}J_{1} is a
horizontal vector of Ñ if and only if
æ è 

ö ø 
ÎkerD
=C z 
æ è 

ö ø 
ÅC z 
æ è 

ö ø 
, 
where Y_{n}(z) are the Bessel functions of the second kind, and
where J_{n} and Y_{n} now denote germs of the corresponding
functions (their local expansions at a¹0). This simplifies
to hÎC z(Y_{0}J_{1}J_{0}Y_{1}), whence h is a constant by the
Wronskian relation Y_{0}J_{1}J_{0}Y_{1}=2/p z.
To the module M above and any nonsingular point aÎC, we have
just associated the Cvector space of horizontal vectors
of Ñ in the form of local expansions at a.
Denote h_{a}(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 

=

æ è 

ö ø 
,
Ñ 

=Ñ_{V}Ä1_{W}+1_{V}ÄÑ_{W}.

This class becomes a category for the usual k[¶]module
morphisms. The map h_{a} extends to {M} by
h_{a}(V)=ker(Ñ_{ M}_{a}_{Ä}_{C}_{(z)}_{V}). In
particular, h_{a}(VÅ W)=h_{a}(V)Åh_{a}(W)
and h_{a}(VÄ W)=h_{a}(V)Äh_{a}(W). The crucial fact
is that h_{a} 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 Clinear homomorphism h_{a}(h)
between h_{a}(V) and h_{a}(W). This makes h_{a} a functor,
in the sense that the diagram
To relate horizontal vectors at two nonsingular points a and b,
consider the maps s that associate with any
k[¶]module V a Clinear
map s(V):h_{a}(V)®h_{b}(V), subject to the
constraint that
Such a map s (from {M} to the linear morphisms in the
category of Cvector space) is called a morphism from (the
functor) h_{a} to (the functor) h_{b}. 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 h_{a} into itself form a group (the group of
automorphisms of the functor h_{a}). This group is the
differential Galois group of L, following the DeligneKatz
definition.
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 [1]

Barkatou (M. A.). 
An algorithm to compute the exponential part of a formal fundamental
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 [2]

Churchill (R. C.). 
Connections on modules. 
February 1997. Unpublished Notes for the Kolchin
Seminar in Differential Algebra.
 [3]

Churchill (R. C.). 
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differential Galois group. 
February 1998. Unpublished Notes for the Kolchin
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 [4]

Pflügel (Eckart). 
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 [5]

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 [6]

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 [7]

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 [8]

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Ulmer (Felix). 
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Ulmer (Felix). 
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van der Put (Marius) and Singer (Michael F.). 
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 [12]

Van Hoeij (Mark). 
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Weil (JacquesArthur). 
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This document was translated from L^{A}T_{E}X by
H^{E}V^{E}A.