{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/110462"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/110462","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Derivations on o-minimal fields","abstract":"\"Let $K$ be an o-minimal expansion of a real closed ordered field and let $T$ be the theory of $K$. In this thesis, we study derivations $\\der$ on $K$. We require that these derivations be compatible with the $\\mathcal{C}^1$-functions definable in $K$. For example, if $K$ defines an exponential function, then we require that $\\der\\exp(a) = \\exp(a)\\der a$ for all $a \\in K$. We capture this compatibility with the notion of a $T$-derivation. Let $T^\\der$ be the theory of structures $(K,\\der)$, where $K\\models T$ and $\\der$ is a $T$-derivation on $K$. We show that $T^\\der$ has a model completion $T^\\der_{\\mathcal{G}}$, in which derivation behaves \"\"generically.\"\" The theory $T^\\der_{\\mathcal{G}}$ is model theoretically quite tame; it is distal, it has o-minimal open core, and it eliminates imaginaries. Following our investigation of $T^\\der_{\\mathcal{G}}$, we turn our attention to $T$-convex $T$-differential fields. These are models $K\\models T$ equipped with a $T$-derivation which is continuous with respect to a $T$-convex valuation ring of $K$, as defined by van den Dries and Lewenberg. We show that if $K$ is a $T$-convex $T$-differential field, then under certain conditions (including the necessary condition of power boundedness), $K$ has an immediate $T$-convex $T$-differential field extension which is spherically complete. In the penultimate chapter, we consider $T$-convex $T$-differential fields which are also $H$-fields, as defined by Aschenbrenner and van den Dries. We call these structures $H_T$-fields, and we show that if $T$ is power bounded, then every $H_T$-field $K$ has either exactly one or exactly two minimal Liouville closed $H_T$-field extensions up to $K$-isomorphism. We end with two theorems when $T= T_{\\operatorname{re}}$, the theory of the real field expanded by restricted elementary functions. First, we prove a model completeness result for the expansion of the ordered valued differential field $\\mathbb{T}$ of logarithmic-exponential transseries by its natural restricted elementary functions. We then use this result to prove that the theory of $H_{T_{\\operatorname{re}}}$-fields has a model companion.\"","abstract_html":"&quot;Let $K$ be an o-minimal expansion of a real closed ordered field and let $T$ be the theory of $K$. In this thesis, we study derivations $\\der$ on $K$. We require that these derivations be compatible with the <span class=\"etd-inline-math\">\\mathcal{C}<sup>1</sup></span>-functions definable in $K$. For example, if $K$ defines an exponential function, then we require that $\\der\\exp(a) = \\exp(a)\\der a$ for all $a \\in K$. We capture this compatibility with the notion of a $T$-derivation. Let <span class=\"etd-inline-math\">T<sup>\\</sup>der</span> be the theory of structures $(K,\\der)$, where $K\\models T$ and $\\der$ is a $T$-derivation on $K$. We show that <span class=\"etd-inline-math\">T<sup>\\</sup>der</span> has a model completion <span class=\"etd-inline-math\">T<sup>\\</sup>der<sub>\\mathcal{G}</sub></span>, in which derivation behaves &quot;&quot;generically.&quot;&quot; The theory <span class=\"etd-inline-math\">T<sup>\\</sup>der<sub>\\mathcal{G}</sub></span> is model theoretically quite tame; it is distal, it has o-minimal open core, and it eliminates imaginaries. Following our investigation of <span class=\"etd-inline-math\">T<sup>\\</sup>der<sub>\\mathcal{G}</sub></span>, we turn our attention to $T$-convex $T$-differential fields. These are models $K\\models T$ equipped with a $T$-derivation which is continuous with respect to a $T$-convex valuation ring of $K$, as defined by van den Dries and Lewenberg. We show that if $K$ is a $T$-convex $T$-differential field, then under certain conditions (including the necessary condition of power boundedness), $K$ has an immediate $T$-convex $T$-differential field extension which is spherically complete. In the penultimate chapter, we consider $T$-convex $T$-differential fields which are also $H$-fields, as defined by Aschenbrenner and van den Dries. We call these structures <span class=\"etd-inline-math\">H<sub>T</sub></span>-fields, and we show that if $T$ is power bounded, then every <span class=\"etd-inline-math\">H<sub>T</sub></span>-field $K$ has either exactly one or exactly two minimal Liouville closed <span class=\"etd-inline-math\">H<sub>T</sub></span>-field extensions up to $K$-isomorphism. We end with two theorems when <span class=\"etd-inline-math\">T= T<sub>\\operatorname{re}</sub></span>, the theory of the real field expanded by restricted elementary functions. First, we prove a model completeness result for the expansion of the ordered valued differential field $\\mathbb{T}$ of logarithmic-exponential transseries by its natural restricted elementary functions. We then use this result to prove that the theory of <span class=\"etd-inline-math\">H<sub>T<sub>\\operatorname{re}</sub></sub></span>-fields has a model companion.&quot;","abstract_has_math":true,"creators":["Kaplan, Elliot Alexander"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["van den Dries, Lou","Hieronymi, Philipp","Tserunyan, Anush","Chen, Ruiyuan"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-09-17T01:10:47Z","date_published":"2021-09-17T01:10:47Z","updated_at":"2026-07-22T22:24:50Z","subjects":["model theory","o-minimality","differential algebra","valued fields"],"languages":["en"],"rights":["Copyright 2021 Elliot Alexander Kaplan"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/110462","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["van den Dries, Lou","Hieronymi, Philipp","Tserunyan, Anush","Chen, Ruiyuan"]},{"key":"dc:creator","label":"Author","values":["Kaplan, Elliot Alexander"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2021-09-17T01:10:47Z","2021-04-13","2021-05"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["model theory","o-minimality","differential algebra","valued fields"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2021 Elliot Alexander Kaplan"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/110462"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"Let $K$ be an o-minimal expansion of a real closed ordered field and let $T$ be the theory of $K$. In this thesis, we study derivations $\\der$ on $K$. We require that these derivations be compatible with the $\\mathcal{C}^1$-functions definable in $K$. For example, if $K$ defines an exponential function, then we require that $\\der\\exp(a) = \\exp(a)\\der a$ for all $a \\in K$. We capture this compatibility with the notion of a $T$-derivation. Let $T^\\der$ be the theory of structures $(K,\\der)$, where $K\\models T$ and $\\der$ is a $T$-derivation on $K$. We show that $T^\\der$ has a model completion $T^\\der_{\\mathcal{G}}$, in which derivation behaves \"\"generically.\"\" The theory $T^\\der_{\\mathcal{G}}$ is model theoretically quite tame; it is distal, it has o-minimal open core, and it eliminates imaginaries. Following our investigation of $T^\\der_{\\mathcal{G}}$, we turn our attention to $T$-convex $T$-differential fields. These are models $K\\models T$ equipped with a $T$-derivation which is continuous with respect to a $T$-convex valuation ring of $K$, as defined by van den Dries and Lewenberg. We show that if $K$ is a $T$-convex $T$-differential field, then under certain conditions (including the necessary condition of power boundedness), $K$ has an immediate $T$-convex $T$-differential field extension which is spherically complete. In the penultimate chapter, we consider $T$-convex $T$-differential fields which are also $H$-fields, as defined by Aschenbrenner and van den Dries. We call these structures $H_T$-fields, and we show that if $T$ is power bounded, then every $H_T$-field $K$ has either exactly one or exactly two minimal Liouville closed $H_T$-field extensions up to $K$-isomorphism. We end with two theorems when $T= T_{\\operatorname{re}}$, the theory of the real field expanded by restricted elementary functions. First, we prove a model completeness result for the expansion of the ordered valued differential field $\\mathbb{T}$ of logarithmic-exponential transseries by its natural restricted elementary functions. We then use this result to prove that the theory of $H_{T_{\\operatorname{re}}}$-fields has a model companion.\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2021-09-16 without embargo terms","The student, Elliot Kaplan, accepted the attached license on 2021-04-13 at 09:22.","The student, Elliot Kaplan, submitted this Dissertation for approval on 2021-04-13 at 09:42.","This Dissertation was approved for publication on 2021-04-13 at 15:03.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16296 on 2021-09-16 at 16:41:03","Made available in DSpace on 2021-09-17T01:10:47Z (GMT). No. of bitstreams: 2 KAPLAN-DISSERTATION-2021.pdf: 1109897 bytes, checksum: 10d6d08a3d36ae66cff245ce232d2e7b (MD5) LICENSE.txt: 4210 bytes, checksum: f79957140c99a248fec14c39eaaedcd8 (MD5) Previous issue date: 2021-04-13"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Derivations on o-minimal fields"]}]}],"canonical_facts":{"dc:contributor":["van den Dries, Lou","Hieronymi, Philipp","Tserunyan, Anush","Chen, Ruiyuan"],"dc:creator":["Kaplan, Elliot Alexander"],"dc:date":["2021-09-17T01:10:47Z","2021-04-13","2021-05"],"dc:description":["\"Let $K$ be an o-minimal expansion of a real closed ordered field and let $T$ be the theory of $K$. In this thesis, we study derivations $\\der$ on $K$. We require that these derivations be compatible with the $\\mathcal{C}^1$-functions definable in $K$. For example, if $K$ defines an exponential function, then we require that $\\der\\exp(a) = \\exp(a)\\der a$ for all $a \\in K$. We capture this compatibility with the notion of a $T$-derivation. Let $T^\\der$ be the theory of structures $(K,\\der)$, where $K\\models T$ and $\\der$ is a $T$-derivation on $K$. We show that $T^\\der$ has a model completion $T^\\der_{\\mathcal{G}}$, in which derivation behaves \"\"generically.\"\" The theory $T^\\der_{\\mathcal{G}}$ is model theoretically quite tame; it is distal, it has o-minimal open core, and it eliminates imaginaries. Following our investigation of $T^\\der_{\\mathcal{G}}$, we turn our attention to $T$-convex $T$-differential fields. These are models $K\\models T$ equipped with a $T$-derivation which is continuous with respect to a $T$-convex valuation ring of $K$, as defined by van den Dries and Lewenberg. We show that if $K$ is a $T$-convex $T$-differential field, then under certain conditions (including the necessary condition of power boundedness), $K$ has an immediate $T$-convex $T$-differential field extension which is spherically complete. In the penultimate chapter, we consider $T$-convex $T$-differential fields which are also $H$-fields, as defined by Aschenbrenner and van den Dries. We call these structures $H_T$-fields, and we show that if $T$ is power bounded, then every $H_T$-field $K$ has either exactly one or exactly two minimal Liouville closed $H_T$-field extensions up to $K$-isomorphism. We end with two theorems when $T= T_{\\operatorname{re}}$, the theory of the real field expanded by restricted elementary functions. First, we prove a model completeness result for the expansion of the ordered valued differential field $\\mathbb{T}$ of logarithmic-exponential transseries by its natural restricted elementary functions. We then use this result to prove that the theory of $H_{T_{\\operatorname{re}}}$-fields has a model companion.\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2021-09-16 without embargo terms","The student, Elliot Kaplan, accepted the attached license on 2021-04-13 at 09:22.","The student, Elliot Kaplan, submitted this Dissertation for approval on 2021-04-13 at 09:42.","This Dissertation was approved for publication on 2021-04-13 at 15:03.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16296 on 2021-09-16 at 16:41:03","Made available in DSpace on 2021-09-17T01:10:47Z (GMT). No. of bitstreams: 2 KAPLAN-DISSERTATION-2021.pdf: 1109897 bytes, checksum: 10d6d08a3d36ae66cff245ce232d2e7b (MD5) LICENSE.txt: 4210 bytes, checksum: f79957140c99a248fec14c39eaaedcd8 (MD5) Previous issue date: 2021-04-13"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/110462"],"dc:language":["en"],"dc:rights":["Copyright 2021 Elliot Alexander Kaplan"],"dc:subject":["model theory","o-minimality","differential algebra","valued fields"],"dc:title":["Derivations on o-minimal fields"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:50Z"}