Polynomially bounded o-minimal structures

Abstract

O-minimal expansions of ordered fields are investigated, with particular emphasis on polynomially bounded o-minimal expansions of $\overline\IR := (\IR, <, +, -, \cdot, 0,1).$

Growth dichotomy. Let $\Re$ be an o-minimal expansion of $\overline\IR$. If $\Re$ is not polynomially bounded, then the real exponential function $x\mapsto e\sp{x}: \IR\to \IR$ is 0-definable in $\Re$. If $\Re$ is polynomially bounded, then for every $\Re$-definable function $f: \IR\to\IR$, not ultimately identically 0, there exist $c,r\in\IR, c\ne 0$, such that the real power function $x\mapsto x\sp{r}:(0, + \infty)\to\IR$ is definable in $\Re$ and $f(x) = cx\sp{r} + o(x\sp{r})$ as $x\to +\infty$.

Piecewise uniform asymptotics. Let $\Re$ be a polynomially bounded o-minimal expansion of $\overline\IR$. Let $f : A\times\IR\to\IR$ be definable, $A\subseteq\IR\sp{m}$, such that for all $a\in A$, the function $x\mapsto f(a,x)$ is ultimately nonzero. Then there exist $r\sb1,\... ,r\sb{l}\in \IR$ and a definable function $g:A\to\IR\\\{0\}$ such that for all $a\in A, f(a,x) = g(a)x\sp{r\sb{i}}+o(x\sp{r\sb{i}})$ for some $i\in\{1,\...,l\}$.

The notions of exponential and power functions are extended to o-minimal expansions of arbitrary ordered fields, and the notion of "power bounded" is introduced as a generalization of "polynomially bounded". Versions of the above two results are established in this more general setting.

For any fixed subfield K of $\IR$, the expansion of $\overline\IR$ by all restricted analytic functions and all real power functions with exponents from K admits elimination of quantifiers and has a universal axiomatization. From this is derived that every function of one variable definable in this structure, not ultimately identically 0, is asymptotic at +$\infty$ to a real function of the form $x \mapsto cx\sp{r}$, $c\ne 0$ and $r\in K$. The proof generalizes to yield various model completeness results, and a method for expanding a given polynomially bounded o-minimal expansion $\Re$ of $\overline\IR$ by a set of power functions $\{x\sp{r}: r\in S\}$, $S\subseteq\IR$, preserving o-minimality and polynomial bounds, provided that the expansion of $\Re$ by the set of restrictions $\{x\sp{r}\ \vert\ \lbrack 1,2\rbrack : r\in S\}$ is o-minimal and polynomially bounded.