# Restricted analytic functions

Our next goal is to give a different construction method for o-minimal structures. The main example discussed for this construction method is the expansion $\Ran$ of the real field by all restricted analytic functions, although we will mention some other expansions to which this same method applies. To achieve this, we will again work in an axiomatic setting.

All of these examples involve working with power series: for $X = (X_1, \dots, X_n)$, we let $\Ps{R}{X}$ be the set of all power series $$F(X) = \sum_{\alpha \in \NN^n} a_\alpha X^\alpha,$$ where we abbreviate $X^\alpha = X_1^{\alpha_1} \cdots X_n^{\alpha_n}$. We add and multiply such series in the usual way, which makes $\Ps{R}{X}$ a ring containing the ring of polynomials $\RR[X]$ as a subring.

For a polyradius $r = (r_1, \dots, r_n) \in (0,\infty)^n$ and $F = \sum a_\alpha X^\alpha \in \Ps{R}{X}$, we set $$\|F\|_r:= \sum \left|a_\alpha\right| \left|r^\alpha\right| \in [0,\infty]$$ and $$\Pc{R}{X}_r := \set{F \in \Ps{R}{X}:\ \|F\|_r \lt \infty}.$$ For polyradii $r,s \in (0,\infty)^n$, we write $r \lt s$ if and only if $r_i \lt s_i$ for each $i$.

Exercise
Show that each $\Pc{R}{X}_r$ is a complete, normed ring with norm $\| \cdot \|_r$ containing $\RR[X]$ and, for polyradii $r,s \in (0,\infty)^n$, that $r \lt s$ implies $\Pc{R}{X}_s \subseteq \Pc{R}{X}_r$.

Next, we set $$\Pc{R}{X} := \bigcup_{r \in (0,\infty)^n} \Pc{R}{X}_r$$ and, for a polyradius $r$, $$\Pc{R}{X}_{>r}:= \bigcup_{s \gt r} \Pc{R}{X}_s.$$

Exercise
Show that $\Pc{R}{X}$ is a subring of $\Ps{R}{X}$ with maximal ideal generated by the monomials $X_1, \dots, X_n$ and, for a polyradius $r$, that $\Pc{R}{X}_{\gt r}$ is a normed subring of $\Ps{R}{X}$ with norm $\|\cdot\|_r$ and maximal ideal generated by the monomials $X_1, \dots, X_n$.

For a polyradius $r \in (0,\infty)^n$, we set $$B(r):= (-r_1,r_1) \times \cdots \times (-r_n,r_n)$$ and $$\bar B(r):= [-r_1,r_1] \times \cdots \times [-r_n,r_n].$$ For $F = \sum a_\alpha X^\alpha \in \Pc{R}{X}_{r}$ and $x \in \bar B(r)$, we denote by $F(x)$ the sum of the absolutely convergent series of real numbers $\sum_{\alpha} a_\alpha x^\alpha$.

Definition
For $F \in \Pc{R}{X}_{\gt 1}$, where we write $1 = (1, \dots, 1)$, we define the associated restricted analytic function $f_F:\RR^n \into \RR$ by $$f_F(x):= \begin{cases} F(x) &\text{if } x \in [-1,1]^n, \\ 0 &\text{otherwise.} \end{cases}$$ We let $\Ran$ be the expansion of the real field by all restricted analytic functions $f:\RR^n \into \RR$ with $n \in \NN$.

Our goal is to prove:

Theorem
The structure $\Ran$ is model complete and o-minimal.

What are the quantifier-free definable sets in $\Ran$ (also called globally semianalytic sets)? To describe them, note first that, for a polyradius $r \in (0,\infty)^n$ and $F \in \Pc{R}{X}_r$, there exists $\epsilon \gt 0$ such that $F(\epsilon X) \in \Pc{R}{X}_{\gt 1}$. The following sets are quantifier-free definable sets in $\Ran$:

Definition
A basic $\Pc{R}{X}$-set is a set of the form $$\set{x \in B(r):\ F(x) = 0, G_1(x) \gt 0, \dots, G_k(x) \gt 0},$$ where $r$ is a polyradius and $F,G_1, \dots, G_k \in \Pc{R}{X}_{\gt r}$. An $\Pc{R}{X}$-set is a finite union of basic $\Pc{R}{X}$-sets.

However, these are not all the quantifier-free definable sets, since $\Ran$ expands the real field; in particular, preimages of basic $\Pc{R}{X}$-sets under polynomial maps are also quantifier-free definable.

The next definition is due to Lojasiewicz:

Definition
Let $S \subseteq \RR^n$.

1. $S$ is semianalytic at $a \in \RR^n$ if there exists a polyradius $r$ such that $(S-a) \cap B(r)$ is an $\Pc{R}{X}$-set.
2. $S$ is semianalytic if $S$ is semianalytic at every $a \in \RR^n$.

Exercise
Show that every bounded semianalytic set is quantifier-free definable in $\Ran$.

Our strategy for proving the above theorem is to establish a version of Gabrielov’s Theorem of the complement for all existentially definable sets in $\Ran$ (also called globally subanalytic sets).

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