Integral transseries

 Joris van der Hoeven

 Dépt. de Mathématiques (bât. 425) Université Paris-Sud 91405 Orsay Cedex France

 March 22, 2005
 It is well known that the operation of integration may lead to divergent formal expansions like as soon as one leaves the area of formal power series for the area of formal transseries. On the other hand, from the analytic point of view, the operation of integration is usually “regularizing”, in the sense that it improves convergence rather than destroying it. For this reason, it is natural to consider so called “integral transseries” which are similar to usual transseries except that we are allowed to recursively keep integrals in the expansions. Integral transseries come with a natural notion of “combinatorial convergence”, which is preserved under the usual operations on transseries, as well as integration. In this paper, we lay the formal foundations for this calculus.

## 1.Introduction

A natural way to solve a differential equation like

 (1.1)

for large is to rewrite it in integral form

 (1.2)

and recursively replace the left-hand side by the right-hand side. This yields a convergent expansion for as an “integral transseries”

 (1.3)

More generally such infinite sums can occur recursively in the exponents. The aim of this paper is to develop a systematic calculus for integral transseries.

We will work in the context of complex grid-based transseries [vdH01], which is briefly recalled in section 2. An “integral transseries” is an infinite linear combination of “integral transmonomials”, which are finite expressions formed from certain “elementary monomials” using multiplication and integration. The elementary monomials are exponentials of integrals of “simpler” integral transseries. The representation of a complex transseries by an integral transseries is far from being unique, which provides a lot of flexibility for the computation with integral transseries.

A major aim of the theory is to lift the usual operations on complex transseries, such as differentiation, truncation, division, etc., to their representations by integral transseries. Moreover, we want these operations to preserve “combinatorial convergence”. For instance, even though the transseries represented by in (1.3) is not convergent, the expansion is said to be combinatorially convergent as an integral transseries. Indeed, the main reason of being for integral transseries is that they allow us to maintain a formal notion of convergence during our computations, there where the represented transseries themselves are divergent.

Before introducing integral transseries in their full generality, we first introduce the simpler notion of integral series in sections 3, 4 and 5. A first technical difficulty is to impose suitable conditions on the supports of integral series. Since we work in the grid-based context, we need a suitable analogue of the grid-based finiteness property. This involves two ingredients: finite generation (cf. regular language specifications in section 5.1) and an asymptotic descent condition (cf. the cycle condition in section 5.2), which states roughly speaking that later terms in the expansion are smaller and smaller from an asymptotic point of view. Moreover, because integral monomials represent series and not merely monomials, we need this descent condition to be sufficiently uniform. This motivates the introduction of the span of an integral series in section 4.3 and frameworks in section 4.4. These notions allow to obtain quick and rough bounds for the support of the transseries represented by an integral series or monomial.

In order to lift computations with classical complex transseries to integral transseries, the key step is a mechanism for rewriting integral transseries in a form with a clear asymptotically dominant part. For instance, in order to compute a fraction

it is necessary to first expand the denominator in such a way that it can be inverted. Using integration by parts, one has

 (1.4) (1.5)

whence

Now admits a natural “integral transseries” inverse . A more systematic procedure for obtaining expansions like (1.4) and (1.5) will be the object of sections 7 and 8. In section 6, we prepare this material by introducing the integral transseries analogue of transbases and differentiation. Putting all techniques together, we finally construct the field of integral transseries in section 9.

Integral transseries can be seen as a natural generalization of Écalle's arborified moulds [EV04]. One advantage of the approach in this paper is that a systematic calculus for integral transseries avoids the process of “arborification”. Even though the latter technique also has a large degree of generality, our technique works in a context as general as that of complex transseries, for which a satisfactory theory of accelero-summation is not even known yet (but under development). Furthermore, with more work, we think that our technique may be generalized so as to include other types of operators, like infinite summation, and parameterized transseries. Unfortunately, we also have to pay the price of a certain technicity in sections 7 and 8. It remains an interesting question how far the ideas in this paper may be further simplified. A few ideas in this direction will be mentioned in the conclusion. We plan to further develop the topic of integral transseries and its link with the theory of accelero-summation in a forthcoming paper.

## 2.Complex transseries

### 2.1.Construction of complex transseries

The field of complex grid-based transseries has been constructed and studied in [vdH01]. Below we will quickly present a classical variant of the construction. We first endow with the following total ordering:

This gives the structure of a totally ordered -vector space, although the ordering is not compatible with the multiplication.

Remark 2.1. In fact, it is possible to consider more general orderings on and vary the orderings during the construction [vdH01]. However, in this paper, we will assume the above ordering, for simplicity.

Now consider the totally ordered monomial groups and corresponding fields , which are inductively defined by

We call

the field of purely exponential complex transseries. Setting

we have , because of the grid-based property. For each , let denote the field obtained from when replacing by . We call

the field of complex transseries. Setting

we again have , because of the grid-based property.

Remark 2.2. In the above construction, we essentially close the field under exponentiation and next under logarithm. In [vdH01], we proceed exactly the other way around.

### 2.2.Differentiation and integration

We recursively define a strong derivation on , by setting

for monomials and extending by strong linearity. Next, we set

for . It is classical to verify that is a strong derivation which satisfies

 (2.1)

For all , the logarithmic derivative of is denoted by .

Using general strongly linear algebra, it follows from (2.1) that admits a distinguished strong right inverse , i.e. for all . More specifically, one has

 (2.2)

for all transmonomials . It can also be checked that and are stable under integration for all (and similarly for and ). In particular, we have

in the inductive construction of .

### 2.3.Flatness relations

In this paper, the flatness relations , , and are defined in an -expo-linear way:

For instance, and . A subset of is said to be flat if for all . We denote by the set of all flat subsets. We have and is stable under arbitrary intersections.

Remark 2.3. The above definitions can be generalized to the case of more general strong algebras , where is a partially ordered monomial group which admits powers in a totally ordered ring with . In that case, we set

The definitions remain valid if is only a partially ordered monomial monoid which can be embedded into a partially ordered group.

Given , we define the flattened dominance and neglection relations , , and by

We also define the derived relations , , and by

Notice that and

The differentiation and the distinguished integration on satisfy

for all (the first relation is easily checked and the second one follows from relation follows from (2.2)). We say that and are flat.

Remark 2.4. In [vdH01], the flatness relations were defined in a -expo-linear way using

Equivalently, we may define them using

When replacing and by the relations and defined by

it is also possible to recover the -expo-linear case.

### 2.4.Transbases

Classically, a transbasis is a tuple of transseries with and such that

TB1

for some .

TB2

for all .

The integer is called the level of the transbasis and is said to be a plane transbasis if . We recall that the flatness relations and were defined in an -expo-linear way, whence . We denote by the set of power products with . The level of a transseries is the highest such that .

In this paper, it will be convenient to consider a variant of the concept of transbases. Given , consider the differentiation

A differential transbasis of level is a tuple of transseries with and such that

DTB

for all .

We will sometimes denote . We notice that is stable under for all and it has been proved in [vdH01] that is stable under .

In what follows, all transbasis will be considered to be differential. The following incomplete transbasis theorem is proved in a similar way as the usual theorem for non-differential transbases [vdH97]:

Theorem 2.5. (Incomplete transbasis theorem) Let be a transbasis of level and a transseries of level . Then there exists a transbasis of level such that .

## 3.Truncation-closed representation modules

### 3.1.Representation modules

Let be a constant field, a monomial set and let be a strong -module of formal power series. Consider a set and a strong -module which admits as a strong basis. We will call elements of monomials, even though we do not necessarily have an ordering on . Assume that we have a strongly linear mapping

such that is regular for all . Then we call a representation module and we say that represents for each . We denote by the equivalence relation on defined by . If and are monoids and preserves multiplication, then we call a representation algebra.

Example 3.1. Given a set and a mapping such that is regular for all , let be the set of all mappings such that is a summable family in . A family is said to be summable if is a summable family in , in which case we set

The strong summation is well-defined, since is summable for all and . The mapping extends to by strong linearity, giving the structure of a representation module. Any other representation module with the same and on can be embedded into .

Example 3.2. Let be the strong -algebra of grid-based series over a monomial group . Let be infinitesimal monomials in and consider the formal group . We have a natural multiplicative mapping , which extends to according to the previous example. This gives the structure of a representation algebra.

An is called a Cartesian representation of . If is generated by its infinitesimal elements, then each series in admits a Cartesian representation for a suitable .

Example 3.3. Let be a representation module with representation mapping . A support function is a mapping with for all . Given , we set , so that .

A family to be -summable if is finite for all . The subset of of all such that is -summable is a representation module for the -summability relation. We have for the trivial support function with for all .

### 3.2.Truncation operators

Let be a representation module with representation mapping and a support function as in example 3.3. Given and , we denote

A strongly linear operator is said to be a truncation operator w.r.t. , if

TO1

.

TO2

For all , we have .

TO3

For all , we have .

Two truncation operators and are said to be compatible if they commute. In that case, is again a truncation operator.

Let be a set of subsets of , which is closed under complements and finite intersections. Then we say that is -truncation-closed if for each , there exists a truncation operator for each , and for all . In what follows, given , we will sometimes use the notations , , etc.

Example 3.4. In the case of Cartesian representations from example 3.2, we may simply take

for all subsets of , where

## 4.Integral series

### 4.1.Integral series

Let be a strong differential -algebra for the derivation . Assume that admits a regular and distinguished right inverse . Let be a monomial group together with a multiplicative mapping such that is regular for all . In what follows, we will assume that for .

We denote by the free formal structure generated by , and . In other words, each element is a tree whose leafs are labeled by elements in , whose unary nodes are labeled by , and whose binary nodes are labeled by . For instance, if , then the following tree is an element of :

We will denote by the size of , i.e. the number of leafs plus the number of integral nodes of the tree . We denote by the total size of , i.e. when we also count multiplicative nodes. For instance, the size of the above example tree is and its total size . We denote by the finite subset of of leafs of . Elements of are called integral monomials. It may sometimes be useful to assume the existence of a special integral node monomial of size .

Each integral monomial induces a natural element . Indeed, this was already assumed if . If resp. , then we recursively set

We also recursively define a non-trivial support function by

We let denote the (non-associative) representation algebra from example 3.3. Elements of are called integral series and we call a representation algebra of integral series with underlying monomial group . For each , we denote .

Remark 4.1. The multiplication on is not associative nor commutative. In the sequel of this paper, this will not be a problem, since integral transseries are mainly used for the purpose of representation. Nevertheless, it is possible to define an associative (and/or commutative) variant of .

Let be the equivalence relation on generated by all relations of the form . This relation is compatible with the representation mapping as well as the size function . Assuming that , it follows that has the structure of a multiplicative monoid and has the structure of an associative representation algebra. The mapping naturally factors through .

### 4.2.Computations in

The mapping naturally extends by strong linearity to . Indeed, given there is at most one with . Similarly, the multiplication extends to a strongly bilinear mapping . Given , we denote by and the strongly linear operators on with

The operators , and (for monomials ) admit strongly linear left inverses , and whose action on monomials is given by

Let be such that . Then we claim that

is a well-define multiplicative inverse of modulo . Indeed, . Moreover, given and with , we must have .

### 4.3.The span of an integral series

Consider an exponential transmonomial with . Then

For arbitrary transmonomials it follows that

for a sufficiently large . For such an , we define the integral span of by

We have

Example 4.2. We have , and . Notice that

Assume now that . For integral monomials , we recursively define the span of by

We have

For such that is finite (we say that has finitely generated support), we define .

Example 4.3. We have .

### 4.4.Frameworks

A framework is a set of subsets of which is closed under arbitrary intersections and such that for each . The elements of will be called frames. For each subset there exists a smallest frame which contains and we denote it by .

Example 4.4. Assume that is a monomial group and define the flatness relations as in remark 2.3. Then the set

is a framework. Indeed, if is a family of elements in with , then for any . If is only a monomial monoid which can be embedded in a monomial group , then is again a framework.

A framework function on is a function which associates a frame to each integral monomial , so that

In particular, for all . We call the frame for . More generally, given a set of integral monomials, we call

the frame for . Given a representation algebra of integral series with a framework function as above, we call a framed representation algebra.

Example 4.5. Assume that . Taking

for all , we define a framework function on .

### 4.5.Combinatorial convergence

Given an integral series , the number of integral monomials with is finite for each fixed size . Therefore, we may define the majorating series for by

We say that is combinatorially convergent if is convergent. We will denote the set of combinatorially convergent series in by .

Proposition 4.6. The set is stable under , and .

Proof. This follows immediately from the facts that for all , we have

Here for if for all .

A family of elements in is said to be summable if is a summable family of power series in . In other words, is finite for each and . The set is a strong -module for this infinite summation operator.

Consider a strongly linear mapping . For each , let . We call the majorant series for . Notice that is uniquely determined by the restriction of to . We may also regard as a mapping on by setting . We have

for all . If maps into itself, then maps into , and we say that is uniformly strong. This is the case if and only if is a summable family in for each . In particular, if and there exist constants with for all , then is uniformly strong.

A uniformly strong mapping is said to be a rewriting if for all . In that case is said to be a rewriting of and we write . If maps into itself, then we call a monomial rewriting. In particular, consider a mapping such that there exist constants with , and for all . Then extends uniquely to a monomial rewriting.

Example 4.7. The mapping which recursively replaces multiplications by multiplications determines a monomial rewriting.

### 4.6.Mouldification

Assume that is the constant field for the derivation on . Then for any there exists a constant with

Let be the subset of of integral monomials of the form

with . We recursively define a product on by

for all and and extension by linearity. Here we understand that the products are taken in . We recursively define the mouldification of an integral monomial by

This definition extends to a strongly linear mapping .

For several purposes, it is more convenient to work with moulds in than general integral series in . However, the mapping generally destroys combinatorial convergence, i.e. is not a rewriting. The convergence can often be restored using the process of arborification [EV04], which corresponds more or less to inverting the mapping in a nice way. In this paper, we will show that most important operations can be carried out directly in , while preserving combinatorial convergence.

## 5.Grid-based integral series

Throughout this section, we assume that is a framed representation algebra of integral series with underlying monomial group and framework function .

### 5.1.Regular languages

A regular language specification is a finite set of formal language symbols, together with a rule of one of the following forms for each :

R1

, with ;

R2

, with ;

R3

, with ;

R4

, with .

The language symbols may regarded as subsets of as follows. Consider the set of mapping , such that for all we have , , , resp. , if is specified by R1, R2, R3 resp. R4. Then is again in . We will regard as the subset of . Subsets of this kind are called regular languages.

Example 5.1. For each finite subset of , the set generated by , and is a regular language. Notice that consists of the with . Inversely, given a regular language specification , let be the finite set monomials , such that for some . Then each is contained in and the set is finite for each .

Given a regular language specification , we say that directly depends on , and we write , if for some , or . The transitive closure of the direct dependency relation is denoted by ; we say that depends on , if . The dependency relation is reflexive and transitive, but not necessarily anti-symmetric, since distinct language symbols may mutually depend on each other.

Example 5.2. Reconsider the introductory equation (1.1). In its integral form (1.2), this equation gives rise to the regular language specification

where we may take , and the natural embedding of into for . We will show later that all integral monomials which occur in the expansion (1.3) belong to .

Remark 5.3. In practice, in order to specify a regular language, it will often be convenient to regroup non interdependent rules together. For instance, the language from example may simply be specified by

Remark 5.4. Consider a regular language specification and a language symbol such that for no with for some . Then as a subset of . A language specification which contains no symbols of the above type is said to be well-rooted.

### 5.2.Grid-based languages

Let be a regular language specification. An arc is a sequence

 (5.1)

such that and for all , we have , and either

• or and .

• or and .

• and .

If , then we call (5.1) a cycle in . A cycle (5.1) is said to be minimal if . Notice that each two language symbols and in a cycle mutually depends on each other.

We say that satisfies the cycle condition, if for every cycle (5.1). In that case, we say that is a grid-based language specification and the elements of are called grid-based languages. A grid-based subset of is an arbitrary subset of a grid-based language. Series in with grid-based support are called grid-based integral series. We denote by the representation algebra of such grid-based series.

Remark 5.5. In relation to remark 5.3 it is often useful to consider a slightly more general notion of arcs by allowing sub-arcs to be replaced by one-step arcs in the case when the languages are unique with the properties that and for all .

Example 5.6. The language specification from example 5.2 is grid-based. Indeed, it suffices to verify the cycle condition for minimal cycles. Up to cyclic permutation, such minimal cycles are necessarily of the form

It is easily verified that and for all . Consequently, and , whence . If we set

then we notice that the cycle condition is no longer satisfied.

Example 5.7. Any grid-based subset of is also a grid-based subset of .

### 5.3.Derivation trees

Let be a regular language specification and define

Consider a -labeled tree with children . Assume that the roots of are labeled by . We say that is a derivation tree if this is recursively the case for and either

• , and .

• or for some , and .

• , and .

• , and .

We denote by the set of all such derivation trees. Given , we define , and as follows:

• is the product of all where ranges over all labels of .

• is obtained from by substituting each label by , or , depending on whether , resp. , and by eliminating all nodes with .

• , where is the label of the root of .

Example 5.8. For the language specification from example 5.2, the tree

is a derivation tree for the triple with , and .

Proposition 5.9. For each triple with and , there exists a with , and .

Proof. We recursively construct as follows:

• If , then is reduced to its root labeled by .

• If , then we choose with , and set

• If , and with , then

• If , and with and , then

It is easily verified by recursion that , and .

If is a derivation tree constructed as in the proof of the proposition, then we say that is a derivation tree for the triple . For instance, is a derivation tree for in example 5.8.

Lemma 5.10. Consider a subtree of a derivation tree with respect to a grid-based language specification . If , then .

Proof. The derivation tree is of the form

 (5.2)

where the expanded subtrees are . Denoting by the label of the root of and for all , we have a cycle

 (5.3)

since . We conclude that .

### 5.4.Summability of grid-based integral series

Let be a grid-based language specification. We say that a derivation tree is cycle-free, if it does not contain a subtree of the form (5.2) with . Given , let be the set of monomials such that for some cycle-free derivation tree . Clearly, is finite.

Now consider with and . For , we define the set by

• If or , then .

• If or , then .

• If , then .

Clearly, for every , we have a cycle (5.1), so

Let be the finite union of all , where are as above.

Lemma 5.11. For every , we have

Proof. Let . We will prove that by induction over the minimal size of a derivation tree for . This is clear if is cycle-free. Otherwise, admits a subtree of the form (5.2) with . Modulo the replacement of by a subtree, we may assume without loss of generality that the and are all cycle-free. By the definition of , we now have . Now consider the derivation tree which is obtained from when replacing by . By the induction hypothesis, we have . We conclude that .

Theorem 5.12. Consider the set of integral series over a grid-based differential algebra and assume that is grid-based on . Then all strong linear combinations over grid-based subsets of are summable.

Proof. Let be a grid-based subset of , so that for some grid-based language specification . Let and be as in the previous section and notice that is grid-based.

Now give the natural ordering and order by Higman's imbrication ordering [vdH04, Section 1.4], with the additional requirement that the imbrication preserves roots. Then Kruskal's theorem implies that the set of -labeled trees is well-quasi-ordered for the opposite ordering of . We claim that the mapping

preserves the ordering .

So assume that and let us prove by induction over the size of that . Write

Since the imbrication of into preserves roots, we have

• , so that , and for all .

• Each admits a subtree of the form with .

Now the induction hypothesis implies that that for all . By lemma 5.10, we also have for all . It follows that

This completes the proof of our claim.

Now let and consider a family with . Let be the set of triples with and . Then the family refines . By proposition 5.9, each triple admits a (distinct) derivation tree . By what precedes, it follows that is a well-based family. From lemma 5.11, we conclude that is a grid-based family.

Theorem 5.12 implies in particular that is grid-based for all . If , then we denote . A truncation operator on is said to be compatible with the grid-based structure, if for every grid-based language , there exists a grid-based language so that maps into .

## 6.Differentiation

Throughout this section, we assume that is a representation algebra of grid-based series with underlying monomial group and framework function .

### 6.1.Stability properties under boolean operations

Proposition 6.1. The intersection of a grid-based and a regular language is a grid-based language.

Proof. Consider a grid-based and a regular language specification resp. . In view of remark 5.3, we may assume that each language symbol resp. is specified by a rule of the form

 (6.1) (6.2)

Let be the regular language specification, whose symbols are formal intersections with and as above, and so that each is specified by

Since any cycle in induces a cycle in , we conclude that is a regular language specification.

Proposition 6.2. The set difference between a grid-based and a regular language is a grid-based language.

Proof. Consider a grid-based and a regular language specification resp. , where each language symbol resp. is specified by a rule of the form (6.1) resp. (6.2). Let be the regular language specification, whose symbols are formal differences with as in (6.1) and where is a finite union, where each is specified by

Each formal symbol is specified by

where stands for the set of pairs with and . Since any cycle in induces a cycle in , we conclude that is a regular language specification.

### 6.2.Uniform restrictions on the support

Consider , , , and a relation among , , , , , , , , , , and . Then we define

These relations generalize to grid-based integral series , by requiring that they hold uniformly for all monomials in a grid-based language . More precisely, denoting by the set of all grid-based languages, we define

Notice that these definitions indeed extend the case when is a monomial, since is a grid-based language for every . We say that is regular, if for certain , and . In that case, we denote .

Given a subset of and , we recall that . More generally, if is a subset of , then we denote . These notations are particularly useful if is one of the sets

Restrictions on the support combine in a natural way. For instance, we have

because of proposition 6.1.

Proposition 6.3. Let and . Given with , we define

Then extends by strong linearity into an operator .

Proof. Given , there exists a grid-based language specification and with and . Then the support of is included in , so is well-defined.

Remark 6.4. It can be shown that one actually has .

### 6.3.Logarithmic derivatives

Let be an algebra of grid-based integral series. A logarithmic derivation on is a mapping such that

LD1

, for all .

LD2

for all with .

LD3

is regular for all with .

LD4

for all with .

LD5

for all and .

LD6

for all with .

In that case, and in virtue of the next sections, we say that is a differential representation algebra of grid-based series. We say that is finitely based if

is finite for all finite .

Given , we will also denote . Assuming that , let , and be such that . Then admits a natural inverse modulo given by

If is a grid-based language with , and , then is a grid-based language with , and . It follows that is also a regular grid-based series. In the sequel, we will denote and . If , then we set and . If , then we take .

Assume now that for some finite set , that for some plane differential transbasis of transmonomials, and that for certain . In that case we say that the differential representation algebra is triangular and the above notations may be extended to more general monomials : given with and , we let and (if ). Furthermore, given , consider . Then for all with , we have , and . Similarly, with as above, we have , and . Setting and , we may therefore assume that and for all with . If , then we take .

### 6.4.Differentiation

Let us now define a strong derivation . We first define the derivative of each monomial in :

 (𝔵∈𝔊ℝ)

Clearly, for all .

Proposition 6.5. The mapping extends to a strongly linear mapping , which represents the derivation on .

Proof. Consider a grid-based language specification . For each , we define a new language symbol by

• If , then .

• If , then .

• If , then .

• If , then .

Clearly, if with , then . Now consider a cycle which involves one of the languages of the form with . Since none of the and none of the depend on , it follows that the cycle has the form for certain and modulo remark 5.5. But then is also a cycle, whence . We conclude that the with are grid-based languages.

Given , the above discussion shows that belongs to the grid-based language . In order to prove that the mapping extends by strong linearity, we still need to show that for every there exists only a finite number of with . Indeed, by induction over the size of , it is easily seen that is necessarily obtained from through the replacement of a subtree of by or the replacement of a subtree of the form with by .

### 6.5.Support properties of differentiation

Assume that is triangular, with .

Proposition 6.6. There exist grid-based subsets , and of such that for all we have

Proof. It suffices to prove the support bounds for monomials .

If , then there exists an with . Consequently, we may take .

If , then is of the form and . Consequently, , so we may take . We notice that is a finite set.

Let us finally show that we may take , where is a finite union up to duplicates. We use induction over the size of . If , then have . If , then by the induction hypothesis . Finally, if , then .

Proposition 6.7. Let be such that . Then is stable under differentiation.

Proof. Let be a grid-based language specification with and for all . Given , let us prove by induction over the size of that . If , then implies . If , then implies . Finally, if , then .

### 6.6.Combinatorial convergence

Proposition 6.8. Assume that maps into . Then the set is stable under differentiation.

Proof. Given a finite subset of of , let

For all with , we claim that

Indeed, using induction over , we have

By strong linearity, we conclude that

for all with , whence . We conclude by recalling that is finite for each .

## 7.First order expansions

Throughout this and the next section, we assume that is a triangular differential representation algebra of grid-based series with underlying monomial group , framework function and logarithmic derivation . The objective of this and the next section is to construct truncation operators on for all , and .

In this section, we start with the construction of on , where

We assume that is fixed and perform the construction simultaneously for all possible values of .

### 7.1.Expansion of monomials

Given , we define , and by induction over the size of . We always take

If , then we take

If , then it will be convenient to denote and . We distinguish the following cases:

1. []. We take

2. []. We take

3. []. If , then we set

If , then we take

 (7.1)

Remark 7.1. The relation (7.1) needs some further explanations. We first observe that the definitions of , and coincide with those from proposition 6.3 in the case when . This ensures that is well-defined. We will show below (proposition 7.7) that . From proposition 6.7, it follows that . This ensures that is well-defined. Finally, it can be shown that , which justifies the simplification

### 7.2.Expansion of the language specifications

Consider a grid-based language . Given a grid-based language specification with , then we notice that only depends on languages with . Modulo removing all other languages, we may thus assume without loss of generality that for all .

In order to show that the definitions of , and for monomials extend by strong linearity to , we first have to specify regular languages , and with respect to which , and can be expanded for . We proceed along similar lines as for the construction of , and .

Since , we take . If , then we take

If , then we abbreviate , and distinguish the following cases:

1. []. We take

2. []. We take

3. []. We take

4. []. If , then we set

If , then we take

By induction over the size of , it is straightforward to verify that and for all .

Remark 7.2. If for all , then case 4 with never occurs, and it is easily checked that and for all . We will show in proposition 7.3 below that . It follows that and are well-defined grid-based languages.

### 7.3.Frames for the expanded languages

Proposition 7.3. For with , we have

Moreover, if , then .

Proof. The inclusions are clear if . So assume that and let . Let us prove by induction over the minimal size of a derivation tree for with that and . We distinguish the following cases:

We must have . Now and .

Let be such that . Then we must have , whence . By the induction hypothesis, we get and .

We obtain with and . By the induction hypothesis, we have , and . It follows that and .

If , then with . By the induction hypothesis, we have and . Since , it follows that and . If , then with and . By the induction hypothesis, we get and . Since , we also have and . It follows that and .

The other inclusions are proved in a similar way.

Proposition 7.4. Given and with , we have

Proof. The inclusions are clear if . Assume therefore that . If , then also , whence and . Otherwise, we must have , since and . We conclude by proposition 7.3.

### 7.4.Extension by strong linearity

Proposition 7.5. The languages , and are grid-based.

Proof. For languages of the form there is nothing to prove. The case when for all has also been dealt with in remark 7.2.

Consider a cycle which involves a language of the form . Then none of the can be in , since none of the languages in depend on . Consequently, the cycle is of the form

with for all . For all , let if and otherwise. Then, using proposition 7.4,

is again a cycle with .

Let us next consider a cycle which involves a language of the form . Then none of the can be a language of the form , , or , with . Consequently, the cycle is of the form

with for all . For all with or and or , let . For the other , let . Then, using proposition 7.4,

is again a cycle with .

Proposition 7.6. , and extend to by strong linearity.

Proof. The proposition is clear for . For with , we also have and . Given with , let

We also define

Let us prove by induction over the size of that and are finite. Now

By the induction hypothesis, the sets at the right hand sides are finite.

### 7.5.Properties of the truncation operators

Proposition 7.7. The operators , , on are projections with

Also, the restriction of to is a projection on .

Proof. The operator is clearly a projection with . By construction, we also have and for resp. . Given with , it follows that and similarly for . We conclude by proposition 7.3.

Proposition 7.8. Let and . Then the operators and on satisfy .

Proof. The result is clear when or . Assume therefore that . If , then . If , then and .

Let us denote by the flattened support of in . By proposition 6.6, there exists sets , and of with

Proposition 7.9. There exist a grid-based set such that the operators and on satisfy

 (7.2) (7.3)

Proof. It suffices to prove both bounds for monomials . The bound (7.2) directly follows from the fact that . Taking

we prove (7.3) by induction over the size of . In the case when , we have and we are done, so assume that . If , then and we are again done. If , then

Assume now that . If , then we denote , so that and . Let . We distinguish three cases:

1. [ and ] Since , we have . This shows that , since and .

2. [ and ] Since , we have

It follows that with and . Since , we also have with and . Hence .

3. [] Let , , , , , and be as in the following diagram:

with and , so that . If , then , whence and . If , then , whence and . Again, we obtain .

### 7.6.Combinatorial convergence

Proposition 7.10. Assuming that maps into , the with map into itself.

Proof. Let be such that and

for all with . We claim that for all , we have

Indeed, using induction over the size if , we are always in one of the following cases:

By strong linearity it follows that for all . Using a similar induction it can be checked that

for all . Indeed, in the product case, we have

We deduce that for all . Setting , with as in the proof of proposition 6.8, we finally claim that

for all . Indeed, using induction, we have

in the multiplicative case and

in the integral case. We conclude that for all .

## 8.Full expansions

### 8.1.Higher order expansions

Let be a grid-based subset of and . We will denote by the set of all series for which there exists a grid-based language specification with

• and for some .

• for all .

Given and , let us now define the grid-based operator

We proceed by induction over

The case has been dealt with in the previous section. Assume therefore that . Let be the maximal element of . Then . Moreover, , so we already have constructed on . Given we now define

Similarly, given a grid-based language , we define

so that for all . Since , we notice that and are well-defined. We also notice that the definitions extend the previous definitions on if resp. , since and are respectively the zero and the identity operator on . By strong linearity, we thus obtain the desired operator on , which extends its previous definition on . Since any belongs to for some and , it follows that is defined on .

Proposition 8.1. Assuming that maps into , the with map into itself.

Proof. Assume that . Let us show by induction over that . We have already treated the case in proposition 7.10. In particular, , with as above. Now

 (♦∈{≍,≺})

We conclude by the induction hypothesis.

### 8.2.Properties of higher order expansion

Proposition 8.2. The operators , and on are projections with

Furthermore, the restriction of to is a projection on .

Proof. Given , let us prove by induction over that implies . If , then we are done by proposition 7.7. Otherwise, we have either or . In the first case, we obtain and . In the second case, we get and , by induction. Similarly, implies and implies . It follows that , and are the identity operators on , resp. .

Given a grid-based language , let us now prove by induction over that , , and whenever . If , then we are done by proposition 7.7. Otherwise, we have (where is the maximal element of ), so the induction hypothesis implies

and if . Since , we conclude that .

Proposition 8.3. Let and . Then the operators and on satisfy .

Proof. Without loss of generality, we may assume that . We prove that on by induction over . If , then we are done by proposition 7.8. So assume that . Let if and if . Setting , we have . The induction hypothesis combined with strong linearity implies

 (8.1)

Now we distinguish three cases:

If , then

If , then we get

If , then

If , then , so that

If , then , whence

Let if and if , so that . Then

and similarly . Now and commute, since we have either or or . From (8.1), we conclude that .

### 8.3.Recursive expansions

We finally have to extend , and from to . By induction, we may assume that we have done this for with and that propositions 8.5 and 8.6 below already hold for and instead of and . In particular, is a projection of on .

Now consider a monomial which is not in . Then we set

Similarly, given a language but , we set

so that for each and . Notice that the defining relations remain valid if or , by the induction hypotheses. By strong linearity, the operators , and on therefore extend to .

Proposition 8.4. Assuming that maps into , the with map into itself.

Proof. If is the largest element of , then we are done by proposition 8.1. Otherwise, we use induction and assume that we have proved the assertion for all larger . Given , it follows that . Now

 (♦∈{≻,≺})

We conclude by proposition 8.1.

### 8.4.Properties of recursive expansion

Proposition 8.5. The operators , and on are projections with

Furthermore, the restrictions of and to are projections on resp. .

Proof. If , then we are done by proposition 8.2. If , then the induction hypotheses at the beginning of this section imply that the above properties are already satisfied for the operators , and .

Assume now that is such that and . If , then proposition 8.2 implies . Otherwise, the induction hypothesis implies and . This shows that . In a similar way, if is such that and , then . Finally, if is such that and , then , so , by proposition 8.2.

Given a grid-based language , the induction hypothesis and proposition 8.2 also imply

Finally, if , then we obtain

This completes the proof.

Proposition 8.6. Let , and with . The the operators , on satisfy .

Proof. If , then we are done by proposition 8.3. If , then the induction hypotheses at the beginning of this section imply that we have commutation when is replaced by . Notice that we do not necessarily have contrary to before, but merely .

Let us first assume that . Let and denote if and otherwise. Then , so the induction hypothesis implies

Moreover, by proposition 8.3, and commute on the image of . We conclude that and commute everywhere.

Assume now that . Let and denote if and otherwise. If , then , and the induction hypothesis implies

Assume now that . We have already shown above that (or ) and (or or or ) commute. We also have . Consequently,

This completes the proof.

### 8.5.Extension of the transbasis

Let us now consider an extension of the transbasis with a new element between and . Likewise, assume that we have extensions of , of with and of , together with a logarithmic derivation on which extends the one on .

Denote the truncation operators on as constructed above by for all , and . If and , then we want to show that the restriction of to coincides with . Denoting by the canonical injection of into , we thus have to show that .

In the cases when or and , the definitions of and clearly coincide. Assume therefore that . If , then

Now for we have and , whence and . It follows that and , whence . If , then

Indeed, in a similar way as above, we have

Furthermore, for we have , since .

## 9.Integral transseries

### 9.1.The set of exponential integral transseries

In this section, we will construct the representation algebra of integral exponential transseries, with . Using induction over , we first construct a differential representation algebra of grid-based integral series, with underlying monomial group , and such that

IH1

For each , there exists a “privileged” with .

IH2

For all , there exists a triangular differential representation subalgebra of grid-based series with .

In IH2, we call a triangular set for . We notice that the union of two triangular sets is again a triangular set.

Given with , we claim that the operators with are naturally defined on . Indeed, given , there exists a triangular set for , and . In sections 7 and 8, we have shown that is a well-defined element of . Moreover, because of section 8.5, the value of does not depend on the choice of .

Taking and , the induction hypotheses IH1 and IH2 are clearly satisfied for . Assume that has been constructed and let

The set

is clearly a monomial group with -powers and we have a natural mapping defined by

We define a logarithmic derivative on as follows. Given with , we may write with , and . Now we set

Since , we notice that .

We extend the mapping into a mapping as follows. If is such that there exists an with , then we arbitrarily take such an and set . Otherwise, we simply take .

Let us prove IH2, which will complete the induction. Given , the set is finite. By the induction hypothesis, there exists a triangular differential representation subalgebra of grid-based series with . Taking , we then obtain a triangular differential representation subalgebra of grid-based series with .

We clearly have . The set

is called the set of exponential integral transseries. For , the smallest with is called the exponential height of . Setting , the finiteness property implies that and .

### 9.2.The set of general integral transseries

By induction over the exponential height , let us now construct a strongly linear mapping which maps integral monomials to integral monomials. So assume that or that and that we already have mapping . Given a monomial , we recursively define by

(f=0∨f∈𝔼
 ∗ n-1,∫
)

This definition has a natural analogue for language specifications:

Hence, extends by strong linearity to . Now for each , the representation algebra

of with derivation is formally isomorphic to the representation algebra of with derivation . By what precedes, we may therefore embed each into (for instance, is identified with ), so that

The set

is called the set of integral transseries. Setting

we have .

Proposition 9.1. Let , and . Then

1. There exists a with .

2. There exists an with .

Proof. In the case when , , we may take . Similarly, if and , then we take . The general case is reduced to one of the above cases modulo a sufficient number of upward shiftings.

Corollary 9.2. Each with admits a multiplicative inverse modulo .

Proof. Writing with and , we may take

where is such that .

### 9.3.Combinatorial convergence

An integral transseries in is said to be combinatorially convergent if is combinatorially convergent as an integral series and if is recursively combinatorially convergent for every . The inclusions

together with proposition 4.6 imply:

Proposition 9.3.

The set is stable under , and .

Proposition 9.4. The set is stable under postcomposition with .

Proof. Given , let us prove by induction over the exponential height of that .

For each , we have either and , or and , where has exponential height . In the second case, the induction hypothesis implies that . Since

it follows that for all .

Furthermore, the postcomposition with send monomials to monomials with . It follows that

This shows that .

Proposition 9.4 implies that and more generally

We call

the set of combinatorially convergent integral transseries. We denote , , etc. the images of , , etc. under . The following is an immediate consequence of proposition 9.3:

Proposition 9.5.

The set is stable under , and .

Proposition 9.6. Given , there exists a triangular set with and such that maps into .

Proof. We prove the proposition by induction over the maximal exponential height of . If , then the result is clear, so assume . The set is finite. By the induction hypothesis, there exists a triangular set with and such that maps into . Taking , we have . Let us show that maps into . So let with . Writing , where , we have

By the construction of , we have . Since , proposition 8.4 also implies that .

Proposition 9.7. Let and with . Then maps into itself.

Proof. Given , let be a triangular set with and such that maps into . By proposition 8.4, we now have .

In a similar way as proposition 9.1 and its corollary, one may now prove:

Proposition 9.8. Let , and . Then

1. There exists a with .

2. There exists an with .

Corollary 9.9. Each with admits a multiplicative inverse modulo .

## 10.Conclusion

In this paper, we have laid the foundations for the formal calculus with integral transseries. We intend to further develop this theory in a forthcoming paper. Let us briefly mention a few points which still have to be investigated in more detail.

First of all, from the foundational point of view, we have chosen to work with “uniform” arcs and cycles. It is also possible to consider “individual” arcs and cycles of the form

such that and for all , we have , and either

• or , and .

• or , resp. and .

• , and .

We expect that languages which verify the weaker “individual” cycle condition can always be rewritten into languages which verify the usual uniform cycle condition, using the technique of “loop unrolling”.

More generally, given an arbitrary regular language , one may define its “descending part” : it consists of those integral monomials such that any “cycle in the derivation tree of satisfies the individual cycle condition. Again, may be computed using the process of loop unrolling. Another application of loop unrolling in combination with truncation is to rewrite an arbitrary grid-based language specification into an equivalent language specification modulo , such that for every , we have for all . In particular, in the cycle condition, this implies that for all .

Another interesting topic from the foundational point of view is to systematically work with operators of either one of the forms

instead of usual integration. These operators have the advantage of being closer to arborified moulds and may make it easier to develop the theory of accelero-summation. Secondly, operators of the form with are naturally “infinitesimal” on suitable frames, and it should be possible to rewrite arbitrary grid-based languages as a tree whose leafs are languages which are constructed using products, infinitesimal operators and repetition.

Finally, it remains to be shown that the set of integral transseries is stable under many other operations, such as composition and functional inversion (when defined), formal alien differentiation, the resolution of quasi-linear differential equations, and so on. Of course, the consideration of additional operators besides integration, such as infinite summation, is another interesting topic.

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[EV04]

J. Écalle and B. Vallet. The arborification-coarborification transform: analytic, combinatorial, and algebraic aspects. Technical Report 2004-30, Univ. Paris-Sud, 2004.

[vdH97]

J. van der Hoeven. Automatic asymptotics. PhD thesis, École polytechnique, France, 1997.

[vdH01]

J. van der Hoeven. Complex transseries solutions to algebraic differential equations. Technical Report 2001-34, Univ. d'Orsay, 2001.

[vdH04]

J. van der Hoeven. Transseries and real differential algebra. Technical Report 2004-47, Université Paris-Sud, Orsay, France, 2004. Submitted to Springer-Verlag's series of Lecture Notes in Mathematics.