{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20193"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20193","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Polynomially bounded o-minimal structures","abstract":"O-minimal expansions of ordered fields are investigated, with particular emphasis on polynomially bounded o-minimal expansions of $\\overline\\IR := (\\IR, <, +, -, \\cdot, 0,1).$","abstract_html":"O-minimal expansions of ordered fields are investigated, with particular emphasis on polynomially bounded o-minimal expansions of $\\overline\\IR := (\\IR, &lt;, +, -, \\cdot, 0,1).$","abstract_has_math":true,"creators":["Miller, Christopher Lee"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Benson, C. Ward"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:31:48Z","date_published":"2011-05-07T12:31:48Z","updated_at":"2026-07-22T22:25:15Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1994 Miller, Christopher Lee"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9512487","(UMI)AAI9512487"],"render_values":[{"text":"AAI9512487","href":null,"code":true},{"text":"(UMI)AAI9512487","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20193","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Benson, C. 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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.","Made available in DSpace on 2011-05-07T12:31:48Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9512487.pdf: 2006991 bytes, checksum: 6e07a5c91b6a6a093d3456afe2d977cc (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:42:15Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:18:21-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Polynomially bounded o-minimal structures"]}]}],"canonical_facts":{"dc:contributor":["Benson, C. Ward"],"dc:creator":["Miller, Christopher Lee"],"dc:date":["2011-05-07T12:31:48Z","10000-01-01","1994"],"dc:description":["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.","Made available in DSpace on 2011-05-07T12:31:48Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9512487.pdf: 2006991 bytes, checksum: 6e07a5c91b6a6a093d3456afe2d977cc (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:42:15Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:18:21-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9512487","(UMI)AAI9512487","http://hdl.handle.net/2142/20193"],"dc:language":["eng"],"dc:rights":["Copyright 1994 Miller, Christopher Lee"],"dc:subject":["Mathematics"],"dc:title":["Polynomially bounded o-minimal structures"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:15Z"}