# Ilyashenko fields based on convergent transmonomials

(Joint work with Zeinab Galal and Tobias Kaiser)

I describe here the generalization of this construction to arbitrary convergent transmonomials, based on the Continuation Theorem.

### Definition

A germ $f \in \I$ is simple if $\level(f) = \eh(f)$.

For example, every convergent transmonomial is simple, as is every purely infinite germ.

Recall that $$\la := \bigcup_{n \in \NN} \M \circ \log_n$$ is the set of all convergent transmonomials, while $$\U:= \log \circ \la$$ is the set of all purely infinite convergent transseries. We call a germ $f \in \H$ a convergent transmonomial (resp., purely infinite) if its image $T(f)$ is.

The following upper bound is crucial for generalizing our construction:

### Corollary 1

Let $f,g \in \I$ be simple. Then $$\level(f)-\level(g) \le \eh(f \circ g^{-1}) \le \max\{0, \level(f)-\level(g)\}.$$ in particular, if $\level(f) \le \level(g)$, then $\level(f \circ g^{-1}) \le 0$.

### Application 1

Let $\S_0$ be the set of all simple germs in $\I$ of level $0$. Then $\S_0$ is a group under composition. $\qed$

### Application 2

To generalize our construction, recall its schematic:

$$\begin{matrix} \RR & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \F_0 \\ e^{-x} \circ (x) \end{bmatrix} \\ & \swarrow \circ\log \swarrow & \\ \begin{bmatrix} \F_1′ \\ e^{-x} \circ (\log) \end{bmatrix} & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \F_1 \\ e^{-x} \circ (x,\log) \end{bmatrix} \\ & \swarrow \circ\log \swarrow & \\ & \vdots & \\ & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \F_{k-1} \\ e^{-x} \circ (x,\log, \dots, \log_{k-1}) \end{bmatrix} \\ & \swarrow \circ\log \swarrow & \\ \begin{bmatrix} \F_k’ \\ e^{-x} \circ (\log, \dots, \log_{k}) \end{bmatrix} & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \F_k \\ e^{-x} \circ (x,\log, \dots, \log_{k}) \end{bmatrix} \\ \end{matrix}$$

In this schematic, each bracket $\begin{bmatrix} \F_i \\ e^{-x} \circ (x,\log, \dots, \log_{i}) \end{bmatrix}$ indicates the field $\F_i$ constructed at stage $i$ whose asymptotic expansions have support generated by the germs $e^{-x} \circ (x,\log, \dots, \log_{i})$. The $\swarrow \circ\log \swarrow$ line indicates shifting this field by composing on the right by $\log$, to obtain the field $\begin{bmatrix} \F_{i+1}’ \\ e^{-x} \circ (\log, \dots, \log_{i+1}) \end{bmatrix}$ on the left of the next lower row. Finally, the $\xrightarrow{\text{(UP)}}$ arrow indicates constructing the next field with asymptotic expansions in the monomials generated by $e^{-x}$ with coefficients in $\F_{i+1}’$, using the Uniqueness Principle, and then replacing each of the coefficients in $\F_{i+1}’$ by their previously constructed asymptotic expansions.

To adapt this schematic to a general tuple $f = (f_0, \dots, f_k) \in \I^{k+1}$ with $f_0 > \cdots > f_k$, it would have to look like this:

$$\begin{matrix} \RR & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \F_0 \\ e^{-x} \circ (x) \end{bmatrix} \\ & \swarrow \circ \left(f_k \circ f_{k-1}^{-1}\right) \swarrow & \\ \begin{bmatrix} \K_1′ \\ e^{-x} \circ \left(f_k \circ f_{k-1}^{-1}\right) \end{bmatrix} & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \K_1 \\ e^{-x} \circ \left(x,f_k \circ f_{k-1}^{-1}\right) \end{bmatrix} \\ & \swarrow \circ\left(f_{k-1} \circ f_{k-2}^{-1}\right) \swarrow & \\ & \vdots & \\ & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \K_{k-1} \\ e^{-x} \circ \left(x,f_2 \circ f_{1}^{-1}, \dots, f_k \circ f_{1}^{-1}\right) \end{bmatrix} \\ & \swarrow \circ \left(f_1 \circ f_{0}^{-1} \right) \swarrow & \\ \begin{bmatrix} \K_k’ \\ e^{-x} \circ \left(f_1 \circ f_{0}^{-1}, \dots, f_k \circ f_{0}^{-1}\right) \end{bmatrix} & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \K_k \\ e^{-x} \circ \left(x,f_1 \circ f_{0}^{-1}, \dots, f_k \circ f_{0}^{-1}\right) \end{bmatrix} \\ & \swarrow \circ f_0 \swarrow & \\ \begin{bmatrix} \K_{k+1}’ \\ e^{-x} \circ \left(f_0, \dots, f_k\right) \end{bmatrix} &\end{matrix}$$

Provided the construction can be carried out like this for $f$, we then set $\K_f:= \K_{k+1}’$.

To carry out the construction for $f$, it has to satisfy the following two conditions:

1. for $1 \le i < j \le k$ and every standard power domain $V$, there exist a standard power domain $U$ and a holomorphic continuation $\f_{ij}$ of $f_j \circ f_i^{-1}$ on $U$ such that $\f_{ij}(U) \subseteq V$;
2. for $0 \le i \le k$, the set of transmonomials generated by $e^{-x} \circ \left(f_i \circ f_{i}^{-1}, \dots, f_k \circ f_{i}^{-1}\right)$ is a scale on standard quadratic domains.

### Definition

The tuple $f$ is called admissible if the above two conditions hold.

By Corollary 1 and the Continuation Theorem, the tuple $f$ is admissible whenever each $f_i$ is simple and $\level(f_0) > \cdots > \level(f_k)$ (as is the case for $f_i = \log_i$, for instance). Therefore, we are at least able to extend our construction to the case where each $f_i$ is purely infinite (so that each $e^x \circ f_i$ is a convergent transmonomial) and $\level(f_0) > \cdots > \level(f_k)$.

Our statement of the Continuation Theorem does not appear to allow generalizing the construction to the cases where some of the $f_i$ have the same level. However, our statement is not actually the full statement that can be proved by induction on terms. If one works with the full statement instead, one can prove the

If each $f_i$ is simple and $f_0 \succ \cdots \succ f_k$, then $f$ is admissible. In particular, if each $f_i \in \U$ and $f_0 \succ \cdots \succ f_k$, then $f$ is admissible.

We denote by $\langle f \rangle$ the additive real vector space generated by $\{f_0, \dots, f_k\}$. Note that the set of monomials generated by $e^x \circ f$ is then $e^x \circ \langle f \rangle$.

### Corollary 2

For each $f \in \U^{k+1}$ such that $f_0 \succ \cdots \succ f_k$, we obtain a qaa field $(\K_f,e^x \circ \langle f \rangle, T_f)$. Moreover, if $g = (g_0, \dots, g_l) \in \U^{k+1}$ is such that $g_0 \succ \cdots \succ g_l$ and $\{f_0, \dots, f_k\} \subseteq \{g_0, \dots, g_l\}$, then the qaa field $(\K_g, e^x \circ \langle g \rangle, T_g)$ extends $(\K_f,e^x \circ \langle f \rangle, T_f)$.

This means that the system of all qaa fields $(\K_f,e^x \circ \langle f \rangle, T_f)$, with $f \in \U^{k+1}$ such that $f_0 \succ \cdots \succ f_k,$ is a directed system. Note that the corresponding direct limit of all monomial space $e^x \circ \langle f \rangle$ is $\la$.

### Definition

We let $(\K,\la,T)$ be the direct limit of this system of qaa fields.

This last qaa field has the following additional properties:

### Theorem 3

1. The field $\K$ is a Hardy field and $T$ preserves differentiation.
2. $(\K,\la,T)$ extends both $(\F,L,T)$ and $(\H,\la,T)$.

### Remark

The Hardy field $\K$ is not closed under exponentiation. To see this, consider a germ $f \in \K$ with divergent Dulac series $T(f) = \sum a_\alpha e^{-\alpha x}$. (There are such almost regular germs, and all almost regular germs are contained in $\K$.) Then $f \cdot \exp_2 \in \K$ as well, and the transseries $T(f \cdot \exp_2) = T(f) \cdot \exp_2$ is divergent and purely infinite. So $\exp \circ T(f \cdot \exp_2)$ is a transmonomial that is not convergent, hence is not in the image $T(\K)$. It follows that $\exp \circ (f \cdot \exp_2) \notin \K$.

### Open questions

1. Is the Liouville closure of $\K$ also a qaa field? One possible way to answer this may be by giving a positive answer to the next question.
2. Is the image $T(\K)$ a transserial Hardy field in the sense of Van der Hoeven?

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