Operators on generalized power series

In [Hah07], Hahn introduced an abstract framework for algebraic computations on power series with generalized exponents like

One of his main results states that, given a field

and a totally ordered monomial group

, the set

of series

with well-ordered support in

carries a natural field structure. This result was generalized by Higman [Hig52] to the case of partially ordered monomial monoids

More recently, Dahn and Göring [DG86] and Écalle [É92] constructed so-called fields of “transseries”, which are fields of generalized power series

in the sense of Hahn, with additional structure, such as exponentiation, differentiation, integration, composition, etc. Examples of transseries are

In [vdH97], we have shown how to differentiate, integrate and compose such transseries, and how to solve algebraic differential equations (whenever possible).

In this paper, we will be concerned with the development of an abstract operator theory for generalized power series, in the setting of partially ordered monomial sets introduced by Higman. We start by recalling some basic results about Noetherian orderings (also called well-quasi-orderings) in section 2. In Higman's setting, generalized power series have Noetherian support. For this reason, we shall actually call them Noetherian series.

In section 3, we recall the definition of Noetherian series and develop the theory of strongly linear and strongly multilinear operators. More precisely, it is possible to define a notion of infinite summation on algebras

of Noetherian power series. One may think of this as something analoguous to normal summable families in analysis. Strongly linear mappings will then be linear mappings which also preserve infinite summation.

The remainder of this article focuses on the resolution of certain functional equations. Translated into the terminology of operators, this comes down to the isolation of nice classes of operators on which some kind of implicit function theorem holds (actually, we will rather prove “parameterized fixed point theorems”). As a basic example, one would like to solve implicit equations like

in fields of transseries, where

is a sufficiently small parameter (say

) and

the unknown.

In section 4, we start by developing a theory of continuous and contracting functions for Noetherian series and we will prove the existence of a solution

to equations like (1) using the technique of fixed points. Actually, we will prove an implicit function theorem which is very similar to fixed point theorems from [PC90] and [PCR93], although our proof is more constructive.

A more natural and even more explicit way of getting solutions to (1) would be to replace the left hand side by the right hand side in a recursive manner, while expanding all sums. This would lead to a formal solution of the form

The main difficulty then resides in proving that the obtained formal expansion is indeed summable in our generalized sense. In sections 5 and 6, we will prove that this is indeed the case for a suitable class of “Noetherian operators”.

2Noetherian orderings

Throughout this paper, orderings are understood to be partial, except when we explicitly state them to be total. Actually, almost all ordered sets considered in this paper are monomial sets, and we denote them by fraktur letters

. We denote by

(or by

) the orderings on such monomial sets. Usually,

is even a monomial monoid or group, on which the multiplication is assumed to be compatible with the ordering, i.e.

Let

be a monomial set. A chain in

is a subset of

which is totally ordered for the induced ordering. An antichain is a subset of

of pairwise incomparable elements. The ordering on

is said to be well-founded, if there are no infinite sequences

of elements in

. A Noetherian ordering is a well-founded ordering without infinite antichains.

Remark 2. In the literature, an ordered set

is usually said to be well-founded, if there are no infinite sequences

of elements in

. This definition is compatible with ours, if one interprets a monomial set

to be ordered by the opposite ordering

(as we did).

Let

be a monomial set. A final segment is a subset

, such that

, for all

. Given an arbitrary subset

, we denote by

the final segment generated by

. Dually, an initial segment is a subset

, such that

, for all

. The following characterizations of Noetherian orderings are classical [Mil85], [Pou85].

The most elementary examples of Noetherian orderings are well-orderings, and orderings on finite sets. Proposition 3 allows us to construct more complicated Noetherian orderings from simpler ones:

The following theorem is due to Higman [Hig52]. We will recall a proof due to Nash-Williams [NW63], because a similar proof technique will be used in section 6.1.

Proof. We say that

is a bad sequence in

, if there do not exist

with

. An ordering is Noetherian if and only if there are no bad sequences. Now assume for contradiction that

is a bad sequence in

. Without loss of generality, we may assume that each

is chosen in

such that it has minimal length as a word. We say that

is a minimal bad sequence.

Now for all

, we must have

, so we can factorize

, where

is the first letter of

. By proposition 3(d), we can extract a sequence

from

. Now consider the sequence

. By the minimality of

, this sequence is good. Hence, there exist

and

with

, or

with

. But then,

resp.

. This contradicts the badness of

3Noetherian series

3.1Noetherian series and infinite summation

Let

be a commutative additive group of coefficients and

a set of monomials. The support of a mapping

is defined by

is Noetherian for the induced ordering, then we call

a generalized power series or a Noetherian series. We denote the set of all Noetherian series with coefficients in

and monomials in

. We also write

for the coefficient of

in such a series and

for

. Each

with

is called a term occurring in

This gives

the structure of a commutative group. More generally, consider a family

of series in

. We say that

is a Noetherian family, if

is Noetherian and for each

there exist only a finite number of

such that

. In that case, we define its sum by

This sum is again a Noetherian series. In particular, given a series

, the family

is Noetherian and we have

in the sense of (2).

It is useful to see

as a strong commutative group, i.e. a commutative group with an additional “infinite summation structure” on it. In our case, this structure is reflected through the infinite summation of Noetherian families; it satisfies the following fundamental properties:

Any zero family is Noetherian, and .

For any , the family is Noetherian, and .

If and are Noetherian and , then is Noetherian and .

If is a Noetherian family, then for any bijective mapping , the family is Noetherian, and .

If is a Noetherian family and a decomposition of into pairwise disjoint subsets, then is a Noetherian family for each , is a Noetherian family, and .

Proof. All properties are straightforward to prove. For illustration, we will prove (e). Let

be a Noetherian family and let

a partition of

. For each

and

, let

and

, so that

Now

is a Noetherian family for all

, since

and

is finite for all

. Furthermore,

and for all

, the set

is finite, because of (3). Hence, the family

is Noetherian and for all

, we have

Remark 7. Given two monomial sets

and

, it is often convenient to identify

with

via the natural isomorphism

In particular multivariate operators

may actually be regarded as a univariate operators

. Similarly, given a monomial set

, the Noetherian families

may be identified with series in

, where

is strictly ordered by

. We may thus view an operator

as an operator “in infinitely many variables”, which assigns to each Noetherian family

a series in

3.2Algebras of Noetherian series

Assume now that

is a (not necessarily commutative) ring, and

a (not necessarily commutative) monomial monoid. Then we may naturally see

and

as subsets of

via

resp.

. Given

and

, we define their product by

The right hand side is well defined by propositions 4(e) and 4(b). Higman [Hig52] first observed that

is a ring for this product. Actually, it is even a strong ring, because the product is compatible with the infinite summation structure on

in the following way:

Proposition 8. For all Noetherian families and , the family is also Noetherian, and

is Noetherian. Given

, the set of couples

with

forms a finite anti-chain; let

denote those couples. Then

Remark 9. Also, if

is a Noetherian family, then so is

, for each family

of scalars.

3.3Extension by strong linearity

Let

be a ring and let

be monomial sets. In all what follows, we understand that

operates on the left on

-modules and

-algebras. A linear mapping

is said to be strongly additive, if for all Noetherian families

, the family

is also Noetherian and

Notice that this condition implies that

is strongly linear, i.e.

, for every Noetherian family

and every family

of scalars. Notice also that the composition of two strongly linear mappings is again strongly linear.

A mapping

is said to be Noetherian, if

is a Noetherian family for every Noetherian subset

Proposition 10. Let and be -modules of Noetherian series. Then any Noetherian mapping extends to a unique strongly linear mapping .

Proof. Let

. By definition,

is a Noetherian family, and so is

. We will prove that

Given

and

we clearly have

. Now let

be a Noetherian family and denote

. We claim that

is a Noetherian family. First of all,

is Noetherian. Secondly, given

, the set

is finite, since

is a Noetherian family. Finally, for each

with

, the set

is also finite, since

is a Noetherian family. Hence, the set

is finite, which proves our claim. Now our claim, together with proposition 6(d) proves that

is a Noetherian family and

In order to see that

is unique with the desired properties, it suffices to observe that for each

, we must have

by linearity and

by strong linearity.

Actually, the above proposition generalizes to the “strongly multilinear” case. If

and

are monomial sets, then we call a multilinear mapping

strongly multilinear (or strongly multi-additive), if for all Noetherian families

, the family

is also Noetherian and

In particular, if

is a monomial monoid, then the multiplication on

is strongly bilinear, by proposition 8. Also, compositions

Recall that a mapping

is Noetherian, if

is a Noetherian family for every Noetherian subset

. The following proposition is proved in a similar way as proposition 10:

Remark 12. In a similar way as we identified

with

in remark 7, we may see

as the strong tensor product of

and

. We have a natural strongly bilinear mapping

. Furthermore, for any strongly bilinear mapping

, there exists a unique strongly linear mapping

, such that

3.4Applications of strong linearity

Corollary 13. Let and be monomial monoids and let be a Noetherian mapping which preserves multiplication. Then preserves multiplication.

Proof. The mappings

and

are both strongly bilinear mappings from

into

, which coincide on

. The result now follows from the uniqueness of strongly bilinear extensions in proposition 11.

Corollary 14. Let be a monomial monoid and a Noetherian mapping, such that for all . Then is a (strong) derivation on .

Proof. The mappings

and

are both strongly bilinear mappings from

into

, which coincide on

. The result again follows from the uniqueness of strongly bilinear extensions in proposition 11.

Proof. This still follows from the uniqueness of extensions by strong linearity, since

and

coincide on

Assume that

is a monomial monoid. We call a series

infinitesimal, if

for all

. Then extension by strong linearity may in particular be used to define the composition

of a multivariate power series

with infinitesimal series

. Indeed, if

is the multiplicative mapping which sends each

, then we define

. Then corollaries 13 and 15 yield the following result:

4The topological implicit function theorem

4.1Truncation of Noetherian series

Let

be a monomial set and

. Given a subset

, we define the restriction

Given two series

, we say that

is a truncation of

(and we write

), if there exists an initial segment

, such that

. Thus

is an ordering on

Let

be a non-empty family of series. A common truncation of the

is a series

, such that

for all

. A greatest common truncation of the

is a common truncation, which is greatest for

. Such a greatest truncation actually always exists and we denote it by

Proof. Fix some

and consider the set

of initial segments

, such that

for all

. We observe that arbitrary unions of initial segments of a given ordering are again initial segments. Hence

is an initial segment of each

. Furthermore, for each

and

, there exists an

with

. Hence

for all

. This proves that

is a common truncation of the

. It is also greatest for

, since any common truncation is of the form

for some initial segment

with

Let

again be a family of series. A common extension of the

is a series

, such that

for all

. A least common extension of the

is a common extension, which is least for

. If such a least common extension exists, then we denote it by

Now consider a directed index set

. In other words, we have an ordering on

, such that for any

, there exist a

with

and

. Let

be a

-increasing family of series in

, i.e.

whenever

. If

is Noetherian or totally ordered, then there exists a least common extension of the

Proposition 18. Assume that is Noetherian or totally ordered. Then any directed -increasing family of series in admits a unique least common extension and .

Proof. Let

. We claim that

is Noetherian. This is clear if

is Noetherian. Assume that

is totally ordered and that

is an infinite sequence of monomials in

. Since

is directed and

whenever

, there exist

with

for each

. But we also have

for each

, so that

. Since

is Noetherian, the sequence

therefore stabilizes.

Given

, we claim that the coefficient

is independent of the choice of

, under the condition that

. Indeed, let

be such that

and

, then there exists a

with

and

. Hence,

and

, so that

. Now the series

is the least common extension of the

4.2Stationary limits

Let

be a directed index set and

a family of series. We call

a pseudo-limit of the

, if for each final segment

and for all

, we have

Equivalently, we may require that for each inital segment

and for each

, we have

Assume from now on that

is either Noetherian or totally ordered. Below, we will show that the stationary limit of the

, which is defined by

Proposition 19. Let be a family of series and let be an initial segment of .

Proof. We first observe that for all

we have

. In particular, this ensures that

exists in (b).

Now assume that

and let

. Then

, whence

, for all

. This shows that

is a common truncation of the

. Inversily, assume that

is such that

for all

. Then also

for all

, so that

. Hence

. This shows that

is the greatest common truncation of the

Assume now that

is directed and

-increasing and let

. Then

, whence

, for all

. Consequently,

is a common extension of the

. Furthermore, its support

is the same as the support of the least common extension of the

. Hence

Proof. Since

, we have

. On the other hand, given

, we have

for some

. Choosing

with

and

, we then have

and

Proposition 21. For any directed family , its stationary limit is a pseudo-limit.

Proof. Let

be an initial segment of

and let

be such that

for all

. Then proposition 19 implies that

Given

and

, we will write

, if for all

, there exists an

with

. The following properties of

will be used frequently in the next section:

Then our hypothesis means that

for all

. Now

, by proposition 21. But this means that

4.3The implicit function theorem

A final segment

of a monomial set

is said to be attractive, if for each

there exists an

with

. If

is totally ordered, then all non-empty final segments are attractive. The intersection of two attractive final segments is again an attractive final segment and arbitrary non-empty unions of attractive final segments are again attractive final segments. In other words, the attractive final subsets

together with the empty set are the open sets of a topology on

Now let

be a commutative additive group. The attractive open subsets of

are the subsets of the form

, where

and where

is an attractive final segment of

. These sets form a basis for the open subsets of the natural or attractive topology on

. We notice that the attractive topology makes

an additive topological group. Given another monomial set

, we also notice that the attractive topology on

(remember remark 7) coincides with the usual product topology on

(if

and

are given the attractive topologies).

Consider a mapping

, where

. We call

contracting, if for all

, we have

. A contracting mapping is in particular continuous at each point

, since for any attractive open neighbourhood

, the set

is an open neighbourhood of

with

Theorem 23. Assume that is Noetherian or totally ordered and let be a continuous mapping, such that the mapping is contracting for each . Then there exists a unique mapping with for each , and is continuous.

The sequence decreases for . Let us prove by (weak) transfinite induction over

that

for all ordinals

. This is clear for

. Assume that

is a successor ordinal. Since

is contracting, the induction hypothesis then implies that

for all

is a limit ordinal and

, then let us prove by a second (weak) transfinite induction over

that

for all

. This is indeed true for

, by the first induction hypothesis. Assuming that

, we also have

again by the first induction hypothesis and proposition 22(c). If

is a limit ordinal, then the second induction hypothesis implies that

for all

. Hence,

In a similar way, one proves that

. Since

is contracting,

also implies that

. Consequently,

, by proposition 22(c).

Existence and uniqueness. Having shown that the sequence

is decreasing for

, we now claim that we must have

for some sufficiently large

. Otherwise, each of the sets

-maximal monomials of

would be non empty, so that

for some

. Indeed, this will happen as soon as the monomials in

get exhausted, i.e. for some

such that the cardinality of

is the one larger than the cardinality of

. Now let

. Since

, there exists an

with

. But this contradicts the

-maximality of

. This shows our claim and we conclude that the

with

satisfies

Assume now that two Noetherian series

and

both satisfy

and

. Then

, since

is contracting. But we can only have

. This establishes the existence and the uniqueness of the mapping

Continuity. In order to prove that

is continuous in any given

, let

be an attractive open neighbourhood of

. Then there exists an attractive open subset of

of the form

, such that

. We claim that

. Indeed, let

. Taking

in our sequence above, it suffices to prove that

for all

. We prove this by transfinite induction.

For

and

, we are already done. If

, then

implies that

, whence

. If

is a limit ordinal, then we have seen above that

for all

. Taking any such

, we also have

by the induction hypothesis, whence again

and

. This completes the induction and the proof of the theorem.

Remark 24. The theorem still holds for monomial sets

without “infinite combs” [PCR93]. Our proof also generalizes to this setting, because it can be shown in this case that the stationary limit of a sequence

exists, whenever

is strictly decreasing for

Remark 25. Although the above topological implicit function theorem may be very useful to solve certain parameterized functional equations over Noetherian series, one of its major drawbacks is that we needed the very strong Noetherianity assumption on

in the partial context. Even the slightly weaker condition about the absence of infinite combs is usually not satisfied. The functional equation

with

is an example which shows that there is not much hope for a stronger implicit function theorem in the same spirit. Indeed, the natural “solution” to this equation, which is obtained by recursively replacing the left hand side by the right hand side in the equation, does not have a Noetherian support.

Remark 26. Another drawback of theorem 23, is that it does not provide us with any additional information about the solutions. The solutions may even be quite pathological: consider the monomial group

with

. Given

, we denote

. We define a linear (but not strongly linear) operator

Then it is easily verified that

is contracting (whence continuous) on

. The equation

will therefore admit a unique solution, which happens to be

. However, we do not have

5Noetherian operators and combinatorial representations

5.1Noetherian operators

Let

and

be sets of monomials. A Noetherian operator is a mapping

, such that there exists a family

of strongly multilinear mappings

with

for all Noetherian families

. In particular, this assumes that the family of summands

is Noetherian. We will call

a multilinear decomposition of

. The number

is the arity of

By regrouping the

of the same arity, it actually suffices to consider the case when

and there is exactly one

for each arity

. In this case, we may write

, with

for all

and

. In section 5.4, we will see that this representation is unique, under the assumption that

and that the

are symmetric (we may always take the

to be symmetric if

). However, for the purpose of combinatorial representations in the next section, it is natural to consider more general multilinear decompositions. Notice also that the space of Noetherian operators from

has a natural strong group structure.

Remark 27. The formula (5) should hold in particular for families that consist of only one element. In other words, we should have

for all

. However, the more complicated assumption (5) is essential, as you will notice in example 31 below.

Remark 28. In view of remark 7 the present definition of Noetherian operators also provides a definition of multivariate Noetherian operators.

Example 31. Let

and

be two Noetherian operators. Then we claim that

is also a Noetherian operator. Indeed, let

resp.

be multilinear decompositions of

and

. Then for each Noetherian family

we have

This establishes our claim, since the operators

are strongly multilinear. Notice that example 30 may be looked at as a combination of the present example and the last two cases in example 29.

One obtains interesting subclasses of Noetherian operators by restricting the strongly multilinear mappings involved in the multilinear decompositions to be of a certain type. More precisely, let

be a monomial monoid and let

be a set of strongly multilinear mappings

. We say that

is a multilinear type if

is of type

, if for

, there exists a mapping

, such that

coincides with the restriction of the domain and image of

resp.

. We say that a Noetherian operator

is of type

, if it admits a multilinear decomposition consisting of strongly multilinear mappings of type

only. In examples 30 and 31, we may then replace “Noetherian operator” by “Noetherian operator of type

”.

Example 32. For any set

of strongly linear mappings

, there exists a smallest multilinear type

which contains

. Taking

to be the field of transseries whose logarithmic and exponential depths are bounded by

, interesting special cases are obtained when taking

. Noetherian operators of type

resp.

may then simply be called differential resp. integral Noetherian operators. Given a finite subset

of positive infinitely large transseries in

, another interesting case is obtained by taking

, where

stands for right composition with

5.2Combinatorial representations of Noetherian operators

Let

be a Noetherian operator with a multilinear decomposition

. Then

is uniquely determined by the action of the

on monomials in

. For the deeper theory of Noetherian operators, it is convenient to represent this action in a combinatorial way.

Abstractly speaking, a set of -labeled structures is a set

, together with a map that assigns to each

a labeling

, where

stands for the size or arity of

; for simplicity, we denote such a set of

labeled structures also by

. For each subset

, we denote the subset of

-labeled structures in

We strictly order couples in

. A mapping

is called a choice operator. We say that

is Noetherian, if for any Noetherian subset

, the subset

Example 33. Let

be a strictly increasing

-ary operation and let

, with

for all

and

. Then

is a Noetherian choice operator.

Returning to our Noetherian operator

, we may see each tuple

as an

-labeled combinatorial structure with

and

for all

. Let

denote the set of such structures. We get a natural Noetherian choice operator

by taking

. Graphically speaking (see figure 1), we may represent the action of

by a box with (a tuple of) “inputs” in

and (a set of) “outputs” in

Inversely, given a Noetherian choice operator

and an operator

with

for all

, we define a Noetherian operator by

5.3Composition of choice operators

In example 31, we have shown that the composition of two Noetherian operators

and

is again Noetherian. Let us now show how to interpret the composition

in a combinatorial way. Denote the natural choice operators associated to

and

resp.

. We first define the composition

of the choice operators

and

. Then

and

will be given by (6) and similar formulas, for certain mappings

resp.

. Here we may assume that

and

are given and we have to construct

Let

be given together with a tuple

, such that

for each

. Then these data determine a unique

-labeled structure

, with

and

, for all

and

. We define

to be the set of all such combinatorial structures (see figure 2). Then we claim that the choice operator

is Noetherian.

So let

be a Noetherian subset of

. We will prove that for any sequence

of elements in the set

there exist

with

. Since

is Noetherian,

is a Noetherian subset of

, and we observe that

for each

. Since

is Noetherian, we may therefore assume that

, modulo the extraction of a subsequence. If

for some

, then we have

and we are done. Hence, we may assume that

. We conclude by the observation that given

there exist only a finite number of

, such that

. Indeed, for each

, there are only a finite number of

with

, since

is Noetherian.

5.4Canonical multilinear decompositions

We already noticed that each Noetherian operator

has a multilinear decomposition of the form

, such that

has arity

for each

. Setting

for all

and

, we then have

Now assume that

(so that

is in particular torsion-free). Then, modulo replacing each

by the operator

with

we may assume without loss of generality that the

are symmetric. Under this additional symmetry assumption, the decomposition (8) is actually unique, and we call

the homogeneous part of

of degree

Proposition 34. Let be a Noetherian operator with a multilinear decomposition , such that is symmetric and of arity for each . If is torsion-free and , then for each .

Proof. We observe that it suffices to prove that

for each

, since the

are symmetric and

is torsion-free. Assume the contrary and let

be such that

for some

. Choose

is Noetherian. The Noetherianity of

implies that there exist only a finite number of indices

, such that

. Let

be those indices.

Let

for all

. For any

, we have

, by multilinearity. On the other hand,

for each

, so that

The matrix on the left hand side admits an inverse with rational coefficients (indeed, by the sign rule of Descartes, a real polynomial

cannot have

distinct positive zeros unless

). Consequently, an integer multiple of the vector on the right hand side vanishes. We infer that

, since

is torsion-free. This contradiction completes the proof.

6The algebraic implicit function theorem

Let

and

be monomial sets and let

be a Noetherian operator. We call

strictly extensive in if there exists a multilinear decomposition

, such that for all

and

, we have

. In particular, such a

is contracting in

. The main objective of this section will be to prove the following theorem:

Theorem 35. Let be a Noetherian operator, which is strictly extensive in . Then for each the operator on has a unique fixed point , and the operator is Noetherian.

6.1Iteration of choice operators with parameters

Let

be as in theorem 35 and let

be the natural Noetherian choice operator associated to

. The fact that

is strictly extensive in

implies that

may be assumed to be strictly extensive on

, i.e.

Also, let

be the natural Noetherian choice operator associated to the identity mapping

. Actually, we take

, with

and

for all

Now consider the sets

-labeled combinatorial structures, where the

are defined by

For each

, the minimal

with

is called the depth of

. We have a natural choice operator

, which is defined componentwise by

Here

stands for the choice operator which coincides with

and with

. Similarly, the componentwise definition of

means that we take

. In figure 3 one finds an illustration of the action of

on a structure in

. We will also call

the iteration of with parameters in .

Theorem 36. Let be a set of -labeled structures and a Noetherian choice operator which is extensive on . Then is Noetherian.

Proof. Let

be a Noetherian subset of

. Assume that there exists a bad sequence

with

and

for each

. We may assume that we have chosen this bad sequence minimally in the sense that the depth of each

is minimal in the set of all bad sequences with fixed

. Writing

for each

, we claim that the induced ordering on

is Noetherian.

be a bad sequence. Notice that

for all

, since

is strictly extensive on

. Hence, taking

such that

is minimal, the sequence

At this point we have proved that

is Noetherian. In particular,

is Noetherian. Hence, there exist

with

, since

. If

for some

, then

and we are done. Otherwise,

. Now for every

, the

with

are finite in number, since they form an antichain. Consequently,

can only take a finite number of values and there exist

with

. This contradicts the badness of (9).

6.2Proof of the implicit function theorem

With the notations from the previous section, let

be a mapping, such that

for all

, and such that (6) holds for all

. We now define

componentwise as follows:

We can now prove the following more explicit version of the implicit function theorem.

Theorem 37. Let be a Noetherian operator, which is strictly extensive in . Then the Noetherian operator defined by (10) is unique with the property that for all .

for all

. The uniqueness of

follows in the same way as in the proof of theorem 23, since

is contracting in

6.3Applications

Example 39. Let us first show that the classical implicit function theorem for bivariate power series follows from theorem 23. So let

be a bivariate power series with

and

. Then we have to prove that there exists a unique power series

with

Modulo division of

and passing

to the other side of the equation, the problem can be reduced to solving the equation

for

with

. Under these assumptions, the series

corresponds to an operator

. Theorem 23 then provides us with a unique mapping

with

. Taking

, we thus find the unique solution to (11).

Moreover, theorem 37 actually tells us that the “natural solution” to (11), which is obtained by recursively plugging in the left hand side of the equation in the right hand side, is indeed a solution. We also notice that by applying theorem 37 to the operator

instead of the previous

, we actually get a solution

in terms of the coefficients of

Example 40. The above example naturally generalizes to the multivariate case. What is more, we may consider non commutative power series in several variables. Given symbols

, we order the free monomial monoid

by the ordering

from example 1.4. Then the ring of non commutative power series in

over

is given by

. Now consider the equation

for

with

. Then it may be proved in a similar way as in the previous example that this equation admits a unique infinitesimal solution. Again, this solution is equal to the natural expression which is obtained when repreatedly plugging in the left hand side of (12) into the right hand side. Again, the solution may be expressed naturally in terms of the coefficients of the equation.

Example 41. Let

be the field of transseries in

, whose logarithmic and exponential depths are bounded by some integer

[vdH97]. The transseries

is an example of an element in

. Now consider the integral equation

for

and where

. Taking

we may consider the operator

. Theorem 23 then implies that there exists a unique function

, such that

satisfies (13) for all

. Theorem 37 and its corollary imply that

is actually an integral Noetherian operator. Modulo regrouping terms, this means that the series

Example 42. Let

now be the field of transseries in

, whose exponential and logarithmic depths are bounded by

. Consider the functional equation

for

and

. Taking

, theorem 37 yields a Noetherian operator

, such that

is a solution to (14). Moreover,

is what one could call a “differential compositional Noetherian operator”.

Example 43. For independent infinitely large variables

consider the monomial group

admits a unique solution

, which can be expressed as a “partial differential series”. Theorem 23 can not be directly applied in this case.

Bibliography

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1Introduction

2Noetherian orderings

3Noetherian series

3.1Noetherian series and infinite summation

3.2Algebras of Noetherian series

3.3Extension by strong linearity

3.4Applications of strong linearity

4The topological implicit function theorem

4.1Truncation of Noetherian series

4.2Stationary limits

4.3The implicit function theorem

5Noetherian operators and combinatorial representations

5.1Noetherian operators

5.2Combinatorial representations of Noetherian operators

5.3Composition of choice operators

5.4Canonical multilinear decompositions

6The algebraic implicit function theorem

6.1Iteration of choice operators with parameters

6.2Proof of the implicit function theorem

6.3Applications

Bibliography