{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/107953"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/107953","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On asymptotic valued differential fields with small derivation","abstract":"This thesis is a contribution to the algebra and model theory of certain valued differential fields and ordered valued differential fields. We focus on those with small derivation, which is a strong form of continuity of the derivation with respect to the valuation topology, and especially on those that are also asymptotic, which is a weak valuation-theoretic analogue of l'Hôpital's Rule. The first component of this thesis concerns three conjectures for valued differential fields $K$ with small derivation and linearly surjective differential residue field: the uniqueness of maximal immediate extensions of $K$, the equivalence of differential-algebraic maximality and differential-henselianity for asymptotic $K$, and the existence and uniqueness of differential-henselizations of asymptotic $K$. First, we show that any two maximal immediate extensions of $K$ are isomorphic over $K$ whenever the value group of $K$ has only finitely many convex subgroups. More significantly, we also establish this conjecture when $K$ is asymptotic. Next, we show that if $K$ is asymptotic and differential-henselian, then it is differential-algebraically maximal; this is optimal, as Aschenbrenner, van den Dries, and van der Hoeven have shown that the asymptoticity assumption is necessary. They have also shown that if $K$ is differential-algebraically maximal, then it is differential-henselian, so this establishes the equivalence of differential-algebraic maximality and differential-henselianity for asymptotic $K$. Finally, we use this equivalence to show that if $K$ is asymptotic, then it has a differential-henselization, and that differential-henselizations are unique. The second component of this thesis builds on the first to study the model theory of pre-$H$-fields with gap 0, which are certain asymptotic ordered valued differential fields with small derivation that are transexponential in some sense. We show that the theory $T^*$ of differential-henselian, real closed pre-$H$-fields that have exponential integration and closed ordered differential residue field (such pre-$H$-fields necessarily have gap 0) has quantifier elimination in the language $\\{+, -, \\cdot, 0, 1, \\leqslant, \\preccurlyeq, \\der\\}$. From quantifier elimination, we deduce that this theory is complete and is the model completion of the theory of pre-$H$-fields with gap 0 (equivalently, it axiomatizes the class of existentially closed pre-$H$-fields with gap 0). Moreover, we show that it is combinatorially tame in the sense that it is distal, and hence has NIP. Finally, we consider a two-sorted structure with one sort for a model of $T^*$ and one sort for its residue field in a language $\\mathcal{L}_{\\res}$ expanding the language $\\{+, -, \\cdot, 0, 1, \\leqslant, \\der\\}$ of ordered differential rings, and show that the theory of this two-sorted structure is model complete when the theory of the residue field is model complete in $\\mathcal{L}_{\\res}$.","abstract_html":"This thesis is a contribution to the algebra and model theory of certain valued differential fields and ordered valued differential fields. We focus on those with small derivation, which is a strong form of continuity of the derivation with respect to the valuation topology, and especially on those that are also asymptotic, which is a weak valuation-theoretic analogue of l&#x27;Hôpital&#x27;s Rule. The first component of this thesis concerns three conjectures for valued differential fields $K$ with small derivation and linearly surjective differential residue field: the uniqueness of maximal immediate extensions of $K$, the equivalence of differential-algebraic maximality and differential-henselianity for asymptotic $K$, and the existence and uniqueness of differential-henselizations of asymptotic $K$. First, we show that any two maximal immediate extensions of $K$ are isomorphic over $K$ whenever the value group of $K$ has only finitely many convex subgroups. More significantly, we also establish this conjecture when $K$ is asymptotic. Next, we show that if $K$ is asymptotic and differential-henselian, then it is differential-algebraically maximal; this is optimal, as Aschenbrenner, van den Dries, and van der Hoeven have shown that the asymptoticity assumption is necessary. They have also shown that if $K$ is differential-algebraically maximal, then it is differential-henselian, so this establishes the equivalence of differential-algebraic maximality and differential-henselianity for asymptotic $K$. Finally, we use this equivalence to show that if $K$ is asymptotic, then it has a differential-henselization, and that differential-henselizations are unique. The second component of this thesis builds on the first to study the model theory of pre-$H$-fields with gap 0, which are certain asymptotic ordered valued differential fields with small derivation that are transexponential in some sense. We show that the theory <span class=\"etd-inline-math\">T<sup>*</sup></span> of differential-henselian, real closed pre-$H$-fields that have exponential integration and closed ordered differential residue field (such pre-$H$-fields necessarily have gap 0) has quantifier elimination in the language $\\{+, -, \\cdot, 0, 1, \\leqslant, \\preccurlyeq, \\der\\}$. From quantifier elimination, we deduce that this theory is complete and is the model completion of the theory of pre-$H$-fields with gap 0 (equivalently, it axiomatizes the class of existentially closed pre-$H$-fields with gap 0). Moreover, we show that it is combinatorially tame in the sense that it is distal, and hence has NIP. Finally, we consider a two-sorted structure with one sort for a model of <span class=\"etd-inline-math\">T<sup>*</sup></span> and one sort for its residue field in a language <span class=\"etd-inline-math\">\\mathcal{L}<sub>\\res</sub></span> expanding the language $\\{+, -, \\cdot, 0, 1, \\leqslant, \\der\\}$ of ordered differential rings, and show that the theory of this two-sorted structure is model complete when the theory of the residue field is model complete in <span class=\"etd-inline-math\">\\mathcal{L}<sub>\\res</sub></span>.","abstract_has_math":true,"creators":["Pynn-Coates, Nigel Adam Lucas"],"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","Freitag, James"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-08-26T21:54:41Z","date_published":"2020-08-26T21:54:41Z","updated_at":"2026-07-22T22:24:47Z","subjects":["algebra","valued differential fields","asymptotic fields","pre-H-fields","differential-henselianity","logic","quantifier elimination","model companion"],"languages":["en"],"rights":["Copyright 2020 Nigel Adam Lucas Pynn-Coates"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/107953","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","Freitag, James"]},{"key":"dc:creator","label":"Author","values":["Pynn-Coates, Nigel Adam Lucas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-08-26T21:54:41Z","2020-05-04","2020-05"]},{"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":["algebra","valued differential fields","asymptotic fields","pre-H-fields","differential-henselianity","logic","quantifier elimination","model companion"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2020 Nigel Adam Lucas Pynn-Coates"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/107953"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis is a contribution to the algebra and model theory of certain valued differential fields and ordered valued differential fields. We focus on those with small derivation, which is a strong form of continuity of the derivation with respect to the valuation topology, and especially on those that are also asymptotic, which is a weak valuation-theoretic analogue of l'Hôpital's Rule. The first component of this thesis concerns three conjectures for valued differential fields $K$ with small derivation and linearly surjective differential residue field: the uniqueness of maximal immediate extensions of $K$, the equivalence of differential-algebraic maximality and differential-henselianity for asymptotic $K$, and the existence and uniqueness of differential-henselizations of asymptotic $K$. First, we show that any two maximal immediate extensions of $K$ are isomorphic over $K$ whenever the value group of $K$ has only finitely many convex subgroups. More significantly, we also establish this conjecture when $K$ is asymptotic. Next, we show that if $K$ is asymptotic and differential-henselian, then it is differential-algebraically maximal; this is optimal, as Aschenbrenner, van den Dries, and van der Hoeven have shown that the asymptoticity assumption is necessary. They have also shown that if $K$ is differential-algebraically maximal, then it is differential-henselian, so this establishes the equivalence of differential-algebraic maximality and differential-henselianity for asymptotic $K$. Finally, we use this equivalence to show that if $K$ is asymptotic, then it has a differential-henselization, and that differential-henselizations are unique. The second component of this thesis builds on the first to study the model theory of pre-$H$-fields with gap 0, which are certain asymptotic ordered valued differential fields with small derivation that are transexponential in some sense. We show that the theory $T^*$ of differential-henselian, real closed pre-$H$-fields that have exponential integration and closed ordered differential residue field (such pre-$H$-fields necessarily have gap 0) has quantifier elimination in the language $\\{+, -, \\cdot, 0, 1, \\leqslant, \\preccurlyeq, \\der\\}$. From quantifier elimination, we deduce that this theory is complete and is the model completion of the theory of pre-$H$-fields with gap 0 (equivalently, it axiomatizes the class of existentially closed pre-$H$-fields with gap 0). Moreover, we show that it is combinatorially tame in the sense that it is distal, and hence has NIP. Finally, we consider a two-sorted structure with one sort for a model of $T^*$ and one sort for its residue field in a language $\\mathcal{L}_{\\res}$ expanding the language $\\{+, -, \\cdot, 0, 1, \\leqslant, \\der\\}$ of ordered differential rings, and show that the theory of this two-sorted structure is model complete when the theory of the residue field is model complete in $\\mathcal{L}_{\\res}$.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Nigel Pynn-Coates, accepted the attached license on 2020-04-30 at 19:54.","The student, Nigel Pynn-Coates, submitted this Dissertation for approval on 2020-04-30 at 20:11.","This Dissertation was approved for publication on 2020-05-04 at 15:20.","DSpace SAF Submission Ingestion Package generated from Vireo submission #15138 on 2020-08-25 at 17:10:22","Made available in DSpace on 2020-08-26T21:54:41Z (GMT). No. of bitstreams: 3 PYNN-COATES-DISSERTATION-2020.pdf: 957343 bytes, checksum: 64ca765d06c11f49e6f7899a2015e113 (MD5) LICENSE.txt: 4214 bytes, checksum: 5b0e444c6d227c2aca26193b0ffe2f67 (MD5) PROQUEST_LICENSE.txt: 4560 bytes, checksum: a8814fe1ecd574f3e42c051d7b007c00 (MD5) Previous issue date: 2020-05-04"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["On asymptotic valued differential fields with small derivation"]}]}],"canonical_facts":{"dc:contributor":["van den Dries, Lou","Hieronymi, Philipp","Tserunyan, Anush","Freitag, James"],"dc:creator":["Pynn-Coates, Nigel Adam Lucas"],"dc:date":["2020-08-26T21:54:41Z","2020-05-04","2020-05"],"dc:description":["This thesis is a contribution to the algebra and model theory of certain valued differential fields and ordered valued differential fields. We focus on those with small derivation, which is a strong form of continuity of the derivation with respect to the valuation topology, and especially on those that are also asymptotic, which is a weak valuation-theoretic analogue of l'Hôpital's Rule. The first component of this thesis concerns three conjectures for valued differential fields $K$ with small derivation and linearly surjective differential residue field: the uniqueness of maximal immediate extensions of $K$, the equivalence of differential-algebraic maximality and differential-henselianity for asymptotic $K$, and the existence and uniqueness of differential-henselizations of asymptotic $K$. First, we show that any two maximal immediate extensions of $K$ are isomorphic over $K$ whenever the value group of $K$ has only finitely many convex subgroups. More significantly, we also establish this conjecture when $K$ is asymptotic. Next, we show that if $K$ is asymptotic and differential-henselian, then it is differential-algebraically maximal; this is optimal, as Aschenbrenner, van den Dries, and van der Hoeven have shown that the asymptoticity assumption is necessary. They have also shown that if $K$ is differential-algebraically maximal, then it is differential-henselian, so this establishes the equivalence of differential-algebraic maximality and differential-henselianity for asymptotic $K$. Finally, we use this equivalence to show that if $K$ is asymptotic, then it has a differential-henselization, and that differential-henselizations are unique. The second component of this thesis builds on the first to study the model theory of pre-$H$-fields with gap 0, which are certain asymptotic ordered valued differential fields with small derivation that are transexponential in some sense. We show that the theory $T^*$ of differential-henselian, real closed pre-$H$-fields that have exponential integration and closed ordered differential residue field (such pre-$H$-fields necessarily have gap 0) has quantifier elimination in the language $\\{+, -, \\cdot, 0, 1, \\leqslant, \\preccurlyeq, \\der\\}$. From quantifier elimination, we deduce that this theory is complete and is the model completion of the theory of pre-$H$-fields with gap 0 (equivalently, it axiomatizes the class of existentially closed pre-$H$-fields with gap 0). Moreover, we show that it is combinatorially tame in the sense that it is distal, and hence has NIP. Finally, we consider a two-sorted structure with one sort for a model of $T^*$ and one sort for its residue field in a language $\\mathcal{L}_{\\res}$ expanding the language $\\{+, -, \\cdot, 0, 1, \\leqslant, \\der\\}$ of ordered differential rings, and show that the theory of this two-sorted structure is model complete when the theory of the residue field is model complete in $\\mathcal{L}_{\\res}$.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Nigel Pynn-Coates, accepted the attached license on 2020-04-30 at 19:54.","The student, Nigel Pynn-Coates, submitted this Dissertation for approval on 2020-04-30 at 20:11.","This Dissertation was approved for publication on 2020-05-04 at 15:20.","DSpace SAF Submission Ingestion Package generated from Vireo submission #15138 on 2020-08-25 at 17:10:22","Made available in DSpace on 2020-08-26T21:54:41Z (GMT). No. of bitstreams: 3 PYNN-COATES-DISSERTATION-2020.pdf: 957343 bytes, checksum: 64ca765d06c11f49e6f7899a2015e113 (MD5) LICENSE.txt: 4214 bytes, checksum: 5b0e444c6d227c2aca26193b0ffe2f67 (MD5) PROQUEST_LICENSE.txt: 4560 bytes, checksum: a8814fe1ecd574f3e42c051d7b007c00 (MD5) Previous issue date: 2020-05-04"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/107953"],"dc:language":["en"],"dc:rights":["Copyright 2020 Nigel Adam Lucas Pynn-Coates"],"dc:subject":["algebra","valued differential fields","asymptotic fields","pre-H-fields","differential-henselianity","logic","quantifier elimination","model companion"],"dc:title":["On asymptotic valued differential fields with small derivation"],"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":"2026-07-22T22:24:47Z"}