|
Abstract
Let 
and 
, where 
is an effective
field and 
and 
are
given a suitable asymptotic ordering 
.
Consider the mapping 
,
where 
. For 
, it is natural to ask how to solve the
system 
. In this paper, we
will effectively characterize 
and show how to
compute a so called distinguished right inverse 
of 
. We will also
characterize the solution space of the homogeneous equation 
.
A well-known theorem [Fab85] states that any linear
differential equation over 
admits a basis of
formal solutions of the form
with 
, 
, 
and 
. This theorem naturally generalizes to the case
when 
is replaced by an effective algebraically
closed field of coefficients 
.
If we also replace the coefficients by polynomials in 
, then several algorithms exist for the
computation of a basis of solutions [Mal79, DDT82,
vH97].
There are several directions in which the above theorem may be
generalized. In [vdH97, vdH01, vdH06],
it is shown how to deal with so called transseries coefficients (a
transseries is an object which is constructed from 
or and an infinitely large variable 
using exponentiation, logarithm and infinite summation).
In collaboration with M. Aschenbrenner and L.
van den Dries, we are currently working on a generalization to arbitrary
asymptotic fields (an asymptotic field is a differential field with a
total asymptotic ordering which is “naturally compatible”
with the derivation).
In this paper, we will be concerned with the generalization to the case
of linear partial differential equations. The asymptotic resolution of
systems of such equations can be decomposed into two subproblems: the
computation of analogues of the exponential parts 
and the computation of the corresponding coefficients. We intend to deal
with the first subproblem in a forthcoming paper and focus on the second
subproblem in what follows.
In the case of holonomic systems of linear differential equations,
algorithms are known for the computation of formal and convergent
generalized series solutions [SST00, Chapter 2] in what the
authors call “Nilsson rings” [Nil65]. On the
other extreme, there exists a method [AC01] to find
“fractional power series solutions” to a single
p.d.e. with coefficients in 
.
In this paper, we will search for formal series solutions to consistent
systems of linear differential equations in variants of Nilsson rings of
the form 
. One of the major
difficulties is to cope with the integrability constraints which arise
when considering more than one equation.
In fact, in the continuation of our previous work on transseries, we
will rather work with infinitely large variables 
and series in 
. In this
equivalent setting, our linear differential operators belong to and we consider series in 
where 
. More precisely, we
assume a total asymptotic ordering 
on 
and consider so called grid-based series [vdH97,
vdH06] with monomials in and
coefficients in 
.
In sections 2 and 3 we first recall classical
algorithms for the computation of “standard bases”, which
are used to reduce a system of equations like 
with 
and 
to suitable
normal forms. The first algorithm is a variant of the skew version [Cas84, Cas87, Gal85, Tak91]
of Buchberger's algorithm [Buc65, Buc85],
although we rather compute coherent autoreduced sets in the sense of
differential algebra [Ros59]. We also recall Mora's
standard cone algorithm [Mor83, MPT92].
However, we will systematically present them in the setting of
p.d.e.s with second members, so the reader might at least
want to take a look at the notations. Also, corollaries 7
and 12 characterize when a system of equations with second
members satisfies the necessary integrability constraints which ensure
the existence of a solution.
In section 4, we will start with the study of linear
p.d.e.s with constant coefficients in . It is classical that the resolution of such
equations in is equivalent to finding the roots
of a set of polynomial equations in 
.
In particular, solution sets in correspond to
radical ideals in 
. More
generally, we will show that there exists a correspondence between the
solution sets in 
and arbitrary ideals in .
An important technique that we will use is the computation of so called
“distinguished solutions” to systems of equations with
second members. More precisely, given 
,
we may consider 
as an operator 
. Denoting 
,
we will effectively construct a right-inverse 
of
. This right-inverse is
unique with the property that the coefficient of any 
in any 
vanishes, where 
denotes the set of dominant monomials of solutions 
to 
. Having constructed 
, we will also show how the space
of solutions 
to can be
obtained from .
In the last section 5, we will study the case of linear
p.d.e.s with coefficients in 
(for
effective purposes) and 
(for theoretical
purposes). We will first show how to reduce systems of such equations to
suitable asymptotic normal forms. Given a system in normal form, we will
next show how to compute a distinguished right inverse in a
coefficientwise manner. We will also characterize the set in this context and give an explicit “strong
basis” for .
Remark
Consider the “monomial monoid” 
,
whose elements are of the form 
,
with 
. A total ordering on 
is called a monomial
ordering, if it is compatible with the multiplication,
i.e. 
. It is
classical [Rob85] that any such an ordering is non uniquely
determined by a finite sequence of vectors 
and
 |
(1) |
Here 
denotes the scalar product. Clearly, the
relation (1) allows to extend to
and even 
.
Moreover, this extension is unique so as to preserve the compatibility
with the multiplication.
We say that is admissible if 
for all 
. In
that case, extends the (partial) divisibility
ordering 
on .
In particular, from Dickson's lemma, it follows that
is well-ordered. Given a subset 
,
we will denote by 
the final segment of generated by 
for the divisibility
relation. We recall that each final segment is finitely generated.
Let be a constant field of characteristic zero.
Given a monomial ordering on ,
a non-zero polynomial 
and a monomial 
, we denote by 
the coefficient of 
in 
. We also denote by 
the
highest monomial for occurring in and by 
the corresponding
coefficient. We call the dominant
monomial of , its dominant coefficient and 
its dominant term. The relation
naturally extends to 
by 
. We denote by 
the
equivalence relation associated to ,
so that 
. Similarly, we write
if 
,
which is equivalent to 
.
Let 
be a differential field with derivations
and field of constants . Given formal variables 
, we denote by 
the
differential algebra of differential polynomials in
over . Any 
admits a unique decomposition
where each 
is homogeneous of degree . We denote by 
the
space of homogeneous polynomials of degree and
.
Given 
and a tuple 
,
the substitution of 
for 
(
) in 
yields a new tuple of differential polynomials in 
, called the composition of
and 
, and denoted by 
. If 
and
, then 
. In particular, 
is an
algebra for 
and 
is a
subalgebra of whenever 
and 
are subalgebras of
with 
. If 
, we will denote by 
the
unique elements of 
, such
that 
.
Example
then
Example 
are differential operators, then
In other words, is isomorphic to 
.
In the remainder of this paper, we will only study differential
polynomials with second members 
with and as above (and often 
). Formally speaking, the monomial
monoid 
for is isomorphic
to . Consequently, the sets
and 
are isomorphic as
vector spaces (but not necessarily as algebras, except when 
). This isomorphism induces natural definitions
of 
, 
and 
for 
and of , ,
and on . These definitions naturally extend to 
, by taking 
for all 
. For instance, 
, if 
.
In our context of linear differential polynomials with second members, a
differential ideal of is a -subvector space which is stable under (i.e. left composition with 
). Moreover, if 
,
then we require that 
. Any
tuple 
naturally generates a differential ideal
. If 
, then 
.
When seeing as a system of equations 
, where
belongs to any differential -algebra
, then these equations are
equivalent to 
. In
particular, we say that a second system 
is
equivalent to , if
.
In the sequel, it will be convenient to extend notation for sets to
tuples. For instance, 
and 
if is smallest with 
and
if no such exists.
Assume that we fixed an admissible monomial ordering on and denote 
. Let 
with 
, so that 
for some
. The partial
reduction 
of
w.r.t. is defined by
 |
(2) |
Given a system 
, let 
. A normal form for 
modulo is an 
, with 
or 
, and such that
for certain 
(where 
)
and 
with 
for all . In that case, we write 
. We say that
is autoreduced if 
for all . By using partial reductions of w.r.t. members of as
long as possible, one obtains a normal form with 
:
Algorithm
Input: 
and 
Output: a normal form of 
modulo 
while 
do
return
Given , let 
be such that 
and 
.
Setting 
, 
and 
, the 
-polynomial of and
is defined by
 |
(3) |
By construction, be have 
. We
also notice that 
, whenever
. We say that a system 
is coherent if 
for
all 
. A coherent and
autoreduced system will also be called a
standard basis. Given an arbitrary system 
, the following classical algorithm computes a
standard basis which is equivalent to .
Algorithm
Input:
Output: a standard basis which is equivalent to
while 
if 
then
return 
while 
do
if 
then
return
Remark 
, then under the natural
isomorphism of 
with , the notions of partial reduction and 
-polynomials correspond to reduction and 
-polynomials in Buchberger's
algorithm (up to details: Buchberger rather takes 
. Also, he not only reduces the dominant term of
in 
,
but all terms). Consequently, Buchberger's algorithm for computing a
Gröbner basis [Buc65, Buc85] corresponds
to the above algorithm for computing a coherent autoreduced set.
Coherent autoreduced sets were first introduced by Rosenfeld [Ros59]
and they are similar (although more effective) to the characteristic
sets introduced by Ritt [Rit50]. We opted for Hironaka's
name standard bases here [Hir64] in view of the
generalization in the next section.
Consider a standard basis 
.
Then the reduction of each -polynomial
with 
to zero yields a
relation
with 
for all 
.
This relation may be rewritten as
 |
(4) |
with 
. We call (4)
the critical relation for the pair 
. Notice that we may regard the set of all critical
relations as a tuple 
.
Lemma be a standard basis. Then the 
generate the space of all with 
. In other words, given
with , there exists a 
with 
.
Proof. Assume for contradiction that there exists a
relation which is not generated by the . We may choose 
such that 
is minimal, as well as the number of
with 
.
Since 
, there must be at
least two indices and 
with 
. Using the fact that
divides 
,
let 
be such that 
,
and 
.
By construction, 
and 
, so 
and 
. For all 
,
we also have 
, so 
. It follows that the relation 
is smaller than the original relation 
in the sense of the minimality hypothesis. This
contradiction completes the proof.
Consider a system 
of linear differential
polynomials in . Given a
tuple 
, we say that 
is compatible with ,
if for every relation 
with 
, we have 
.
The set of such tuples forms a subvector space of 
, which we denote by 
.
Corollary 
with 
and 
is a standard basis if and only if
is a standard basis and 
is compatible with .
Proof. Assume that 
is a
standard basis. Consider 
with 
and 
. Then 
if and 
.
It follows that is a standard basis with
critical relation 
for all 
. Given a relation 
,
lemma 5 now implies 
for a certain
. We conclude that 
, whence 
.
Assume now that is a standard basis and that
is compatible with .
Then is autoreduced, since 
for all 
. Furthermore, for
all , the relation 
implies 
. But
the relation 
precisely proves that 
reduces to zero modulo .
Hence is coherent.
Corollary 
and 
, we have 
if and only if
.
Let and .
A canonical form for modulo is an with
for certain and with
for all ,
and such that 
for each term 
occurring in 
. It is easy to
modify 
so that it computes a canonical form of modulo
with 
:
Algorithm
Input: and
Output: a canonical form of
modulo
while 
do
Choose highest for
return
Lemma be a standard basis. Then we have
where 
and 
.
Proof. Assume for contradiction that 
is such that 
. Replacing by , we
may assume without loss of generality that is a
canonical form w.r.t. .
Now choose with 
such
that is minimal, in the same sense as in the
proof of lemma 5. Since 
,
we have 
, so there must be at
least two indices and
with . Setting with the notations from the proof of lemma 5,
then yields a more minimal representation for
. This contradiction proves
that 
for all 
.
In classical polynomial elimination theory, the use of non-admissible
monomial orderings allows for the computation in localized rings and
completions, such as rings of power series. However, additional care is
needed in order to ensure termination. For instance, the naive reduction
of modulo 
would yield an
infinite sequence 
. The
tangent cone algorithm [Mor83, MPT92] allows
for the computation of standard bases in the case of localizations of
polynomial rings.
In this section, we will present the tangent cone algorithm in the
differential setting. In all what follows, is a
differential field with constant field .
Geometrically speaking, elements of 
or
localizations of 
can still be thought of as
operators. For instance, 
naturally operates on
.
Let 
and let
be a monomial ordering on .
Given , we define 
, 
and 
as in (3). As a special case, 
is given by (2) if . Now let 
be the opposite
ordering of . Given 
, we denote the dominant monomial
of for by 
for and we define 
,
, 
, 
,
etc. in a similar way. We will also write 
for the element of with 
. If is admissible, 
and 
for all , then we notice that 
(i.e. 
for the natural extension of
the ordering to 
).
Moreover, if 
, then 
.
In the sequel, we will assume that the vectors 
which determine using (1) are all
in 
. In that case is called a tangent cone ordering. Notice that it
is possible to consider more general tangent cone orderings [MPT92],
but we have chosen to keep the exposition as simple as possible. Given
, let
Given 
with 
,
we denote
Notice that 
and 
(for a
dummy 
). Now let 
and 
be such that 
and 
. Then we have 
and we define the -th
ecart of by 
.
We call 
the ecart of
and recall that 
is well-ordered by the
lexicographical ordering. The definition extends to the case when 
by taking 
for all .
Given , some easy properties
of the ecart are
 |
(5) |
Moreover, if 
, then
where the inequality is strict whenever 
.
It follows that
 |
(6) |
In particular, if , then
 |
(7) |
The following lemma will guarantee the termination of the tangent cone algorithm.
Lemma 
and 
be such that for all
, we have
for some 
.
Whenever 
for some 
, then 
.
Then the sequence 
is finite.
Proof. Assume for contradiction that there exist
infinitely many with 
. By Dickson's lemma, we may find two such indices
with and 
. But then
which contradicts our assumption (b). It follows that 
is strictly decreasing for sufficiently large . We conclude by the fact that
is well-ordered.
Given 
, a
normal form for modulo
is an , with
or , and such that
for certain 
and 
with
and for all . Notice that this notion extends the previous
notion of normal forms, since 
if is admissible. In our new context, we may use the
following algorithm to compute a normal form:
Algorithm
Input: and
Output: a normal form of
modulo
while do
Take with 
such that
is minimal
return
Indeed, the sequence 
of successive values of
during the algorithm fulfills the conditions of
lemma 9, so this sequence is finite. Moreover, using
induction, it is easily checked that there exist 
and 
and with 
and 
for all .
So the last term of the sequence is indeed a normal form for modulo .
Defining the notions of autoreduced systems, coherent systems and
standard bases as in section 2.3, the same algorithm 
may be used to compute an equivalent standard basis
for a given system. Given a standard basis 
and
, we have a relation
with and for all . As before, we may rewrite this
relation as a critical relation of the form 
.
In order to generalize lemma 5, let 
be the set of series 
with well-ordered support
. If
is admissible, then 
coincides with . If is admissible then
elements of are power series in
applied to 
. The set is naturally stable under composition. We denote
.
Lemma be a standard basis. Then the
generate the space of all 
with .
Proof. We have to construct 
with , where 
corresponds to the critical relation .
For each 
, let 
. Writing 
,
let us construct 
by transfinite induction over
. Given an ordinal 
, the induction hypothesis is as
follows:
has been constructed for all 
in a final segment 
of 
for .
for all 
.
Denoting 
and 
,
we have 
for all and
.
If 
or is a limit
ordinal, then we may take 
.
If 
and 
,
then we are done. So assume that and 
. Let 
and
let be minimal such that 
. Let 
,
with
let 
and take 
for all
with 
.
By construction,
for all 
. Since 
, it follows that as
well. This proves the last induction hypothesis. By transfinite
induction, we conclude that there exists an with
, whence 
.
Consider a system of linear differential
polynomials in . Assume also
that naturally operates on a subring of (for instance, we may take
). Given a tuple 
, we say that is
compatible with , if for
every relation with 
, we have .
The set of such tuples forms a (strong) subvector space 
of 
. The following
consequences of the above lemma is proved in a similar way as
corollaries 6 and 7.
Corollary
Corollary
Let 
and .
A canonical form for modulo is an 
with
for certain 
and with
and for all , and such that for
each term 
occurring in . Although we have no algorithm to compute canonical
forms, like in section 2.5, the existence of canonical
forms can be proved using a similar transfinite induction as in the
proof of lemma 10. Using another transfinite induction,
lemma 8 also generalizes to the current setting:
Lemma
In this section, we consider systems of linear
partial differential equations in one unknown
with coefficients in a field of constants of
characteristic zero. We will consider the resolution of such systems in
the algebras
where 
(Kronecker symbol). We will first consider
homogeneous linear differential equations, but we will also study linear
differential equations with second members. In the latter case, we will
allow the second members to belong to or . Throughout this section stands for an admissible tangent cone ordering on .
in
In this section, we will only consider linear p.d.e.s
without second members. Let 
be a homogeneous
linear differential polynomial. We may represent
as
where 
is a polynomial in . Inversely, each polynomial 
gives rise to a homogeneous linear differential polynomial 
. Denoting 
,
we have
for all 
and in particular
Let denote the set of all 
with 
. We have
for all 
, where 
denotes the -vector
space generated by . Given
, we will denote 
.
More generally, given a set 
of homogeneous
linear differential polynomials, a subset 
of
, a subset 
of 
and a subset 
of 
, we denote
and
Because of the natural isomorphisms
all algebraic geometry properties of the correspondences 
and 
induce analogue properties for
the correspondences 
and 
. In particular, Hilbert's Nullstellensatz implies
Theorem be a coherent and autoreduced system with 
. If is
algebraically closed, then admits a solution
.
in
Recall that stands for an admissible
tangent cone ordering on .
Consider a standard basis 
for . We may regard as an
operator from into 
,
whose image is in 
. We denote
by the set of monomials 
, such that 
for all . The aim of this section is to
construct a right inverse 
of , which is “distinguished” in the
sense that 
for all and
.
The relation on induces
a relation 
on by
Whenever 
are such that 
and 
, it follows that
Indeed, if , then 
for at least one with 
.
Proposition for 
and
, let
be such that 
is maximal for . Then 
does not depend
on the choice of and 
.
Proof. We will first show that 
whenever 
is another index with 
. Let 
and 
be such that 
and consider the
associated critical relation
with 
for all 
.
Since is compatible with , it follows that
For each , we have
It follows that
 |
(8) |
Hence
It follows that
Now 
implies 
,
so we conclude that .
It remains to be proved that ,
i.e. 
for all . If 
,
then 
and for all 
with
, we have 
. By strong linearity, it follows that 
. Furthermore
implies 
, whence . If 
,
then the relation (8) remains valid. Moreover, if 
is such that 
,
then
and
Since 
, it follows that 
, whence . By construction, we therefore have .
Given , let 
be the term as in proposition 15. Now consider the sequence
defined by 
and 
.
This sequence is finite, since 
and is a well-ordering on .
Consequently, 
is a solution to 
with for all .
We set 
and call the
distinguished right inverse of .
Let . Since 
for all , it follows that
. Consequently 
is a solution to 
with 
. Inversely, implies
, since otherwise 
for some and 
. We claim that the 
form a basis for the solution space of in . Indeed,
given an arbitrary solution ,
consider the sequence defined by 
and 
as long as 
.
This sequence is necessarily finite, since 
and
is well-ordered. Hence, 
. We call 
the
distinguished basis of .
We notice that 
, where 
, so that 
decomposes into an isomorphism on 
with left
inverse and the zero map on . We also notice that the distinguished right
inverse is uniquely determined by the fact that
for all and 
. Indeed, assume that
and 
and 
for all . Then for
and 
for all . It follows that 
.
Let us now consider an arbitrary system .
Using the tangent cone algorithm, may be
rewritten into an equivalent system 
which is a
standard basis. Then the sets 
and 
are independent from the particular choice of , since is precisely
the set of elements which cannot occur as dominant monomials of elements
in 
, by lemma 13.
Consequently, the construction of the distinguished right-inverse and
the distinguished basis 
do not depend on the
choice of , and we may define
, 
, etc.
in
Let us now consider a general system 
as an
operator 
. Then acts “by spectral components” 
. More precisely, given 
, let 
be the unique operator
such that
for all . Considering as an operator in 
,
we obtain from by
substituting 
for each 
. Given
with 
, it follows that
Hence, denoting by 
the solution space of 
for 
, the
solution space of for 
is
given by
Denoting by 
the distinguished inverse of as an operator on ,
the mapping
is a right-inverse of .
Moreover, 
is unique with the property that 
, where
Remark on 
to
in any way which preserves spectral components
(i.e. if 
, then
for all 
),
the space 
coincides with the set of all 
such that 
for all 
; see the next section.
Theorem 
be the set of differential ideals of and let 
the set of subsets of which occur as zero-sets of systems 
. Then the correspondences
are mutually inverse bijections.
Proof. Let 
and 
be two differential ideals with the same set of solutions 
. Then the differential ideal 
generated by and still
has the same set of solutions. Assuming for contradiction that 
, the set
strictly contains or , say .
Now consider the differential ideal 
.
By theorem 14, there exists an with
. Since 
and 
(here 
stands for
, where
is any system which generates 
),
it follows that 
. But then
and 
.
Remark (which is
necessarily radical and even prime) and “linear differentially
algebraic” zero-sets in .
Via the isomorphism 
,
arbitrary ideals of 
are therefore also in a
geometric correspondence with zero-sets of linear differential
operators. This provides a geometrical reason why the existence of
Ritt-Rosenfeld-Buchberger-type algorithms for the computation with
ideals, and not merely radical ideals, is important.
The study of the linear p.d.e.s with
coefficients in is equivalent to the study of
equations with coefficients in 
modulo the
substitutions 
, 
and multiplication with a suitable 
. Since the ordinary partial derivatives preserve
the “valuation” in 
,
it will be more convenient to work with coefficients in .
Assume that we have fixed an admissible ordering
on , determined by 
. Assume also that we have fixed a
total ordering on which gives
the structure of a totally ordered 
-vector
space. Then we also have a natural ordering on
:
A subset 
of is said to
be grid-based if there exist 
with 
and 
.
Given a ring of coefficients the set of series
with grid-based support 
forms an -algebra [vdH97,
vdH06]. We denote this algebra by 
and its elements are called grid-based series. This still goes
through for coefficients in 
,
since such operators act by spectral components. In this section, we
will consider systems of linear p.d.e.s in 
and study their solutions in 
.
The admissible orderings on and on may
be combined into a total admissible ordering 
on
using
Consequently, an element can also be regarded as
a series 
with anti-well-ordered support in (the support is not necessarily grid-based, although
we might have required this). Similarly, elements in 
can be seen as series with monomials in 
.
The ordering 
is extended to 
by understanding that 
for all 
. We will use 
in order
to emphasis when a notation should be understood with respect to the
relation .
Consider a system 
such that 
for all . Given with 
and 
, let 
,
, 
and
We say that is a standard basis for
if for each there exists
a critical relation
 |
(9) |
where 
is such that 
for
all .
Given with 
(or 
with 
), let us
denote 
(resp. 
).
Lemma 
be a standard basis and let 
be such that 
. Then 
is a standard basis and 
.
Proof. Since 
for all , the system 
is autoreduced. For all , the
relation (9) implies
so 
is a standard basis for the relations 
. Now consider a relation 
. Then we have 
for some 
. Now 
implies 
. We
conclude that 
, so .
Lemma 
be a standard basis and .
Then is again a standard basis.
Proof. Any 
satisfy the
relations
Hence, any critical relation 
for induces a critical relation 
for
. So we may take 
.
Given an arbitrary system 
,
an equivalent standard basis can be “computed” by a variant
of Hironaka's infinite division “algorithm”. If the
dependency of in is only
polynomial, then a fully effective method can be devised, by adapting
the algorithms from section 2.3.
In this subsection and in this subsection only, let 
, 
,
and 
.
The set 
is formally isomorphic (as a vector
space) to 
by sending each 
to and to 
. Moreover, the ordering
on corresponds to a tangent cone ordering on
. Consequently, the
definition of ecart in section 3.2 transposes to elements
in 
.
Unfortunately, we do not necessarily have 
for
and 
(for instance 
). Nevertheless, this relation does
hold if 
. For this reason, we
adapt the definition of partial reduction by setting
for all 
with 
.
Because of the twist, we again have
We also notice that 
coincides with the usual
partial reduction “up to lower order terms”, since 
. We obtain the following version
of :
Algorithm
Input: and 
Output: an “asymptotic normal form” of
modulo
while 
do
Take with 
such that
is minimal
return
The termination of the modified version of NF is proved in the same way
as before. Again, the successive values of in the algorithm verify relations
for certain 
, 
and 
with 
and 
for all 
.
Example 
and 
. Then
Hence divides ,
from the asymptotic point of view.
In a similar way, we may define the twisted -polynomial of by
Given a system , the
corresponding algorithm now computes an
equivalent system 
, such that
for all we have a relation
where 
, 
and 
are such that 
and
for all .
But 
admits 
as inverse in
, which leads to the relation
 |
(10) |
Moreover, each 
induces an element
with 
. When rewriting (10) in terms of 
and 
, we obtain a critical relation for and in the sense of section 5.1. Modulo this normalization of the result, SB therefore
computes a skew standard basis.
Let again , 
and . Let 
. Using the isomorphism 
, we observe that 
,
, etc. are
well-defined. Given 
it is also convenient to
extend the notation by setting 
if and only if 
.
Lemma be a standard basis for 
and
let be such that 
.
Then there exists a 
with 
and 
. In particular, if , then there exists a 
with 
.
Proof. Let 
with be such that 
.
Then 
is stable under composition. For each 
with 
, let
us show how to construct 
,
such that 
satisfies 
. We use weak induction over 
.
So let and assume that 
has been constructed for all 
.
Let 
. Since 
for all 
with ,
and 
, we have 
. Setting 
,
we have 
, so 
for some 
. Taking 
, it follows that 
.
By induction, we conclude that 
is well-defined
and we have 
for all , so .
Corollary and 
, we have if and only if
.
Proof. Similar to the proofs of corollaries 6
and 7.
Corollary is a standard basis for
and let 
be non-zero. Then 
.
Proof. Let be such that 
. Modulo division of by , we may
assume without loss of generality that .
Let be as in the above lemma, so that 
. In fact, 
, since 
implies 
. We conclude that 
.
in
Consider a standard basis for . Given 
,
we may regard 
as an operator on . We denote
and write 
for the distinguished right inverse of
.
Proposition be a standard basis for .
Then 
admits a unique right inverse such that 
for all .
Proof. Let with and 
be such that 
and 
. For any with 
, it
follows that 
. Let us show by
well-ordered induction over 
how to construct
such that for 
.
Given , we assume that 
has been constructed for all 
with . Denoting 
, we also assume that 
for all with .
By construction, we first observe that 
,
whence 
. Now we take 
, which is well-defined by lemmas
20 and 19. Setting 
, it follows that 
.
For all with ,
we also have 
. We infer that
. By induction, we obtain a
series 
with and 
for all .
We conclude that . The
uniqueness is proved as usual.
Proposition be a standard basis for .
For each 
, let 
. Then
for all solutions 
to .
Proof. Setting
we have
Now 
for all ,
by the distinguished property of and the fact
that 
. Consequently, 
and 
. But
this is only possible if 
.
Let us now consider an arbitrary system 
and let
be an equivalent standard basis. By corollary 24, we notice that the differential ideal 
does not depend on the particular choice of , and similarly for the twisted differential ideals
. Consequently, the spaces
, 
and
the operator are independent of the particular
choice of . We may therefore
define the distinguished right inverse 
of 
by 
.
Putting everything together from the effective point of view, we have:
Theorem 
and 
, computes the asymptotic expansion of 
.
Proof. Using the algorithm from section 5.2,
we start by computing an equivalent standard basis 
for and make the corresponding change 
for . We next
test whether is compatible with
using corollary 23. If so, and assuming that 
, we determine the dominant term 
of and compute the dominant term
of 
using the method from
section 4.2. Setting 
and continuing
the same procedure with 
instead of , we obtain the asymptotic expansion of 
.
Remark 
, where
.
Remark and
not merely the first 
terms (as done by the above
algorithm).
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and 
is a standard basis if and only if 
,
if and only if 
.