# Ilyashenko algebras based on definable monomials

This post generalizes the construction of quasianalytic Ilyashenko algebras based on log monomials to certain other definable monomials. This construction is joint work with my student Zeinab Galal.

Recall that $\H = \Hanexp$ is the Hardy field of germs at $+\infty$ of univariate functions definable in $\Ranexp$, and that $\I$ is the set of all infinitely increasing $f \in \H$, that is, all $f \in \H$ such that $\lim_{x \to +\infty} f(x) = +\infty$. We denote by $v:\H \setminus \{0\} \into \Gamma$ the standard valuation on $\H$, extended to $0$ by setting $v(0):= \infty$. Note that $\Gamma$ is a divisible, ordered abelian group and an $\RR$-vector space.

For $f_1, \dots, f_k \in \H$, we set $$\frac1\exp \circ \left(f_1, \dots, f_k\right) := \left(\frac1{\exp \circ f_1}, \dots, \frac1{\exp \circ f_k}\right),$$ and we denote by $G(f_1, \dots, f_k)$ the multiplicative $\RR$-vector subspace of $\H$ generated by $\left\{\frac1{\exp \circ f_1}, \dots, \frac1{\exp \circ f_k}\right\}$.

In this post, we constructed an increasing sequence of qaa fields $\left(\F^0_k, G(\log_0, \dots, \log_k), T^0_k\right)$, for $k \in \NN$. The goal of this post is to generalize this construction to more general $f_1, \dots, f_k \in \I$ satisfying at least the following properties:

(I1)

$x = f_0 > f_1 > \cdots > f_k$;

(I2)

the set of values $\set{v(f_0), \dots, v(f_k)}$ is linearly independent in $\Gamma$.

In this situation, we hope to obtain qaa fields $\left(\F(f_0, \dots, f_k), G(f_0, \dots, f_k), T(f_0, \dots, f_k)\right)$ such that, for each $i < k$ and strictly increasing $\iota:\{1, \dots, i\} \into \{1, \dots, k\}$, we have $$\tag{I}\F\left(f_0, f_{\iota(1)}, \dots, f_{\iota(i)}\right) \subseteq \F(f_0, \dots, f_k)$$ and $$\tag{II}T\left(f_0, f_{\iota(1)}, \dots, f_{\iota(i)}\right) = T(f_0, \dots, f_k)\rest{\F\left(f_0, f_{\iota(1)}, \dots, f_{\iota(i)}\right)}.$$ The idea is to mimic the inductive construction of $\F^0_k$ as follows: consider the germs $x = g_0 > g_1 > \cdots g_{k-1}$, where $$g_i:= f_i \circ f_1^{-1}, \quad\text{for } i=1, \dots, k-1.$$ These $g_i$ also satisfy conditions (I1) and (I2) above and, assuming we have constructed qaa fields $\left(\F(g_0, \dots, g_{k-1}), G(g_0, \dots, g_{k-1}), T(g_0, \dots, g_{k-1})\right)$ such that properties $\text{(I)}$ and $\text{(II)}$ hold we let, in analogy with this construction, $\A(f_0, \dots, f_k)$ be the set of all germs at $+\infty$ of functions $f:\RR \into \RR$ that have a bounded, holomorphic extension to (the closure of) some standard quadratic domain and for which there exist real numbers $0 \le \nu_0 < \nu_1 < \cdots$ and $a_i \in \F(g_0, \dots, g_{k-1})$ such that $\lim_{n \to \infty} \nu_n = +\infty$ and, for all $N \in \NN$, we have $$a_N(f_1(x)) = O\left(e^{\nu_Nx}\right) \quad\text{and}\quad f(x) – \sum_{n=0}^N a_n(f_1(x)) e^{-\nu_n x} = o\left(e^{-\nu_Nx}\right) \quad\text{as}\quad x \to +\infty.$$

#### Remark

Condition (I2) guarantees that there is at most one such asymptotic expansion.

In this situation, we set
$$T(f):= \sum_{n=0}^\infty (a_n\circ f_1) e^{-\nu_nx};$$ by the Phragmén-Lindelöf Principle, the map $T$ is injective. The same then holds for the fraction field $\F(f_0, \dots, f_k)$ of $\A(f_0, \dots, f_k)$, and we define $T(f_0, \dots, f_k):\F(f_0, \dots, f_k) \into \RR\big(\big(G(f_0, \dots, f_k)\big)\big)$ by $$T(f_0, \dots, f_k)(f):= \sum_{n=0}^\infty (T(g_0, \dots, g_{k-1})(a_n) \circ f_1) e^{-\nu_n x},$$ where $T(f) = \sum_{n=0}^\infty (a_n \circ f_1) e^{-\nu_n x}$. This then completes the construction.

### The first problem

…is that, if some of the monomials $\frac1{\exp} \circ f_i$ do not have bounded holomorphic extensions to some standard quadratic domain, then the field $\F(f_0, \dots, f_k)$ will not contain these monomials, which implies that its image under $T(f_0, \dots, f_k)$ is not truncation closed; in particular, $(\F(f_0, \dots, f_k), G(f_0, \dots, f_k),T(f_0, \dots, f_k))$ is not a qaa field.

Indeed, the problem is even more subtle: we also need to know that, for $z$ in some standard quadratic domain, the modulus $\left|\frac1{\exp} \circ f_i(z)\right|$ has a lower bound of the form $\frac1{\exp} \circ f_i\left(k\sqrt{|z|}\right)$ for some $k>0$ (actually, any positive power of $|z|$ would do).

Both these conditions are met if, in addition to (I1) and (I2) above, we also assume that

(I3′)

each $f_i$ has a holomorphic extension $\f_i:\Omega_i \into \Delta_i$, for some standard quadratic domains $\Omega_i$ and $\Delta_i$.

### The second problem

…is that, in order for the inductive construction to work, condition (I3′) also has to hold for $g_0, \dots, g_{k-1}$ in place of $f_0, \dots, f_k$. This leads to the additional assumption

(I3”)

for $i > 1$, each $f_i \circ f_1^{-1}$ has a holomorphic extension $\g_i:\Omega_{i,1} \into \Delta_{i,1}$, for some standard quadratic domains $\Omega_{i,1}$ and $\Delta_{i,1}$.

Note that both conditions (I3′) and (I3”) are implied by the stronger condition

(I3)

for $0 \le i < j \le k$, the germ $f_j \circ f_i^{-1}$ has a holomorphic extension $\f_{ij}:\Omega_{ij} \into \Delta_{ij}$, for some standard quadratic domains $\Omega_{ij}$ and $\Delta_{ij}$.

Moreover, condition (I3) implies the same condition holds with $k-1$ and $g_0, \dots, g_{k-1}$ in place of $k$ and $f_0, \dots, f_k$.

We then obtain:

## Theorem (with Zeinab Galal)

Assume that $f_0, \dots, f_k$ satisfy conditions (I1)(I3). Then $(\F(f_0, \dots, f_k), G(f_0, \dots, f_k), T(f_0, \dots, f_k))$ is a qaa field that contains the monomials $(\exp \circ f_i)^{-1}$, for $i=0, \dots, k$, and has properties $\text{(I)}$ and $\text{(II)}$.

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