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 $\text{(II)}$.