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Title:  Polynomially bounded ominimal structures 
Author(s):  Miller, Christopher Lee 
Doctoral Committee Chair(s):  Benson, C. Ward 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  Ominimal expansions of ordered fields are investigated, with particular emphasis on polynomially bounded ominimal expansions of $\overline\IR := (\IR, <, +, , \cdot, 0,1).$ Growth dichotomy. Let $\Re$ be an ominimal expansion of $\overline\IR$. If $\Re$ is not polynomially bounded, then the real exponential function $x\mapsto e\sp{x}: \IR\to \IR$ is 0definable 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 ominimal 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 ominimal 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 ominimal expansion $\Re$ of $\overline\IR$ by a set of power functions $\{x\sp{r}: r\in S\}$, $S\subseteq\IR$, preserving ominimality 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 ominimal and polynomially bounded. 
Issue Date:  1994 
Type:  Text 
Language:  English 
URI:  http://hdl.handle.net/2142/20193 
Rights Information:  Copyright 1994 Miller, Christopher Lee 
Date Available in IDEALS:  20110507 
Identifier in Online Catalog:  AAI9512487 
OCLC Identifier:  (UMI)AAI9512487 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics
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