Strongly minimal reducts of algebraically closed valued fields
In this thesis we prove the following restricted version of Zilber's Trichotomy: Let $K=(K,+,\cdot,v,\Gamma)$ be an algebraically closed valued field and let $(G,\+)$ be a K$-definable group that is either the multiplicative group or contains a finite index subgroup that is $ K$-definably isomo...
- Autores:
-
Pinzón Palacios, Santiago Iván
- Tipo de recurso:
- Doctoral thesis
- Fecha de publicación:
- 2023
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/69392
- Acceso en línea:
- http://hdl.handle.net/1992/69392
- Palabra clave:
- ACVF
Zilber's Trichotomy
Valued Fields
Strongly Minimal
Matemáticas
- Rights
- openAccess
- License
- Atribución 4.0 Internacional
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Strongly minimal reducts of algebraically closed valued fields |
| title |
Strongly minimal reducts of algebraically closed valued fields |
| spellingShingle |
Strongly minimal reducts of algebraically closed valued fields ACVF Zilber's Trichotomy Valued Fields Strongly Minimal Matemáticas |
| title_short |
Strongly minimal reducts of algebraically closed valued fields |
| title_full |
Strongly minimal reducts of algebraically closed valued fields |
| title_fullStr |
Strongly minimal reducts of algebraically closed valued fields |
| title_full_unstemmed |
Strongly minimal reducts of algebraically closed valued fields |
| title_sort |
Strongly minimal reducts of algebraically closed valued fields |
| dc.creator.fl_str_mv |
Pinzón Palacios, Santiago Iván |
| dc.contributor.advisor.none.fl_str_mv |
Hasson, Assaf Onshuus Niño, Alf |
| dc.contributor.author.none.fl_str_mv |
Pinzón Palacios, Santiago Iván |
| dc.contributor.jury.none.fl_str_mv |
Cubides Kovacsics, Pablo Kowalski, Piotr Peterzil, Kobi |
| dc.subject.keyword.none.fl_str_mv |
ACVF Zilber's Trichotomy Valued Fields Strongly Minimal |
| topic |
ACVF Zilber's Trichotomy Valued Fields Strongly Minimal Matemáticas |
| dc.subject.themes.es_CO.fl_str_mv |
Matemáticas |
| description |
In this thesis we prove the following restricted version of Zilber's Trichotomy: Let $K=(K,+,\cdot,v,\Gamma)$ be an algebraically closed valued field and let $(G,\+)$ be a K$-definable group that is either the multiplicative group or contains a finite index subgroup that is $ K$-definably isomorphic to a $K$-definable subgroup of $(K,+)$. Then if $\mathcal G=(G,\+,\ldots)$ is a strongly minimal non locally modular structure definable in $ K$ and expanding $(G,\oplus)$, it interprets an infinite field. |
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2023 |
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2023-08-08T17:06:01Z |
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2023-08-08T17:06:01Z |
| dc.date.issued.none.fl_str_mv |
2023-08-01 |
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Trabajo de grado - Doctorado |
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http://hdl.handle.net/1992/69392 |
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10.57784/1992/69392 |
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eng |
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eng |
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Shreeram Shankar Abhyankar. Local analytic geometry, volume 14. World Scientific, 2001. Juan Pablo Acosta. One dimensional commutative groups definable in algebraically closed valued fields and in the pseudo-local fields. arXiv preprint arXiv:2112.00430, 2021. Toni Annala. B´ezout's theorem. Helsingfors universitet, 2016. Emil Artin. Geometric algebra. Interscience Publishers, Inc., 1957. Elisabeth Bouscaren, A Nesin, and A Pillay. The group configuration-after e. hrushovski. The model theory of groups, pages 199-209, 1989. Benjamin Castle. Restricted trichotomy in characteristic zero. arXiv preprint arXiv:2209.00730, 2022. Benjamin Castle and Assaf Hasson. Very ampleness in strongly minimal sets. arXiv preprint arXiv:2212.03774, 2022. Pantelis E Eleftheriou, Assaf Hasson, and Ya'acov Peterzil. Strongly minimal groups in o-minimal structures. Journal of the European Mathematical Society, 23(10):3351-3418, 2021. Robin Hartshorne. Algebraic geometry, volume 52. Springer Science & Business Media, 2013. Deirdre Haskell, Ehud Hrushovski, and Dugald Macpherson. Stable domination and independence in algebraically closed valued fields. arXiv preprint math/0511310, 2005. Jan E Holly. Canonical forms for definable subsets of algebraically closed and real closed valued fields. The Journal of Symbolic Logic, 60(3):843-860, 1995. Ehud Hrushovski. Contributions to stable model theory. University of California, Berkeley, 1986. Ehud Hrushovski. A new strongly minimal set. Annals of pure and applied logic, 62(2):147-166, 1993. Assaf Hasson and Dmitry Sustretov. Incidence systems on cartesian powers of algebraic curves. arXiv preprint arXiv:1702.05554, 2017. Ehud Hrushovski and Boris Zilber. Zariski geometries. Journal of the American mathematical society, pages 1-56, 1996. Piotr Kowalski and Serge Randriambololona. Strongly minimal reducts of valued fields. The Journal of Symbolic Logic, 81(2):510-523, 2016. Juan Pablo Acosta L´opez. One dimensional groups definable in the p-adic numbers. The Journal of Symbolic Logic, 86(2):801-816, 2021. David Marker. Model theory: an introduction, volume 217. Springer Science & Business Media, 2006. David Marker. Strongly minimal sets and geometry. In Colloquium'95 (Haifa), pages 191-213, 2017. Samaria Montenegro, Alf Onshuus, and Pierre Simon. Stabilizers,-generics, and prc fields. Journal of the Institute of Mathematics of Jussieu, 19(3):821-853, 2020. Evgenia D Rabinovich. Definability of a field in sufficiently rich incidence systems, volume 14. School of Mathematical Sciences, Queen Mary and Westfield College, 1993. Stan Wagon. The Banach-Tarski Paradox. Number 24. Cambridge University Press, 1993. |
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Atribución 4.0 Internacional |
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85 páginas |
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Universidad de los Andes |
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Doctorado en Matemáticas |
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Facultad de Ciencias |
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Departamento de Matemáticas |
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Atribución 4.0 Internacionalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Hasson, AssafOnshuus Niño, Alfvirtual::20915-1Pinzón Palacios, Santiago Iván26597600Cubides Kovacsics, PabloKowalski, PiotrPeterzil, Kobi2023-08-08T17:06:01Z2023-08-08T17:06:01Z2023-08-01http://hdl.handle.net/1992/6939210.57784/1992/69392instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/In this thesis we prove the following restricted version of Zilber's Trichotomy: Let $K=(K,+,\cdot,v,\Gamma)$ be an algebraically closed valued field and let $(G,\+)$ be a K$-definable group that is either the multiplicative group or contains a finite index subgroup that is $ K$-definably isomorphic to a $K$-definable subgroup of $(K,+)$. Then if $\mathcal G=(G,\+,\ldots)$ is a strongly minimal non locally modular structure definable in $ K$ and expanding $(G,\oplus)$, it interprets an infinite field.Doctor en MatemáticasDoctoradoModel TheoryZilbers Trichotomy85 páginasapplication/pdfengUniversidad de los AndesDoctorado en MatemáticasFacultad de CienciasDepartamento de MatemáticasStrongly minimal reducts of algebraically closed valued fieldsTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttps://purl.org/redcol/resource_type/TDACVFZilber's TrichotomyValued FieldsStrongly MinimalMatemáticasShreeram Shankar Abhyankar. Local analytic geometry, volume 14. World Scientific, 2001.Juan Pablo Acosta. One dimensional commutative groups definable in algebraically closed valued fields and in the pseudo-local fields. arXiv preprint arXiv:2112.00430, 2021.Toni Annala. B´ezout's theorem. Helsingfors universitet, 2016.Emil Artin. Geometric algebra. Interscience Publishers, Inc., 1957.Elisabeth Bouscaren, A Nesin, and A Pillay. The group configuration-after e. hrushovski. The model theory of groups, pages 199-209, 1989.Benjamin Castle. Restricted trichotomy in characteristic zero. arXiv preprint arXiv:2209.00730, 2022.Benjamin Castle and Assaf Hasson. Very ampleness in strongly minimal sets. arXiv preprint arXiv:2212.03774, 2022.Pantelis E Eleftheriou, Assaf Hasson, and Ya'acov Peterzil. Strongly minimal groups in o-minimal structures. Journal of the European Mathematical Society, 23(10):3351-3418, 2021.Robin Hartshorne. Algebraic geometry, volume 52. Springer Science & Business Media, 2013.Deirdre Haskell, Ehud Hrushovski, and Dugald Macpherson. Stable domination and independence in algebraically closed valued fields. arXiv preprint math/0511310, 2005.Jan E Holly. Canonical forms for definable subsets of algebraically closed and real closed valued fields. The Journal of Symbolic Logic, 60(3):843-860, 1995.Ehud Hrushovski. Contributions to stable model theory. University of California, Berkeley, 1986.Ehud Hrushovski. A new strongly minimal set. Annals of pure and applied logic, 62(2):147-166, 1993.Assaf Hasson and Dmitry Sustretov. Incidence systems on cartesian powers of algebraic curves. arXiv preprint arXiv:1702.05554, 2017.Ehud Hrushovski and Boris Zilber. Zariski geometries. Journal of the American mathematical society, pages 1-56, 1996.Piotr Kowalski and Serge Randriambololona. Strongly minimal reducts of valued fields. The Journal of Symbolic Logic, 81(2):510-523, 2016.Juan Pablo Acosta L´opez. One dimensional groups definable in the p-adic numbers. The Journal of Symbolic Logic, 86(2):801-816, 2021.David Marker. Model theory: an introduction, volume 217. Springer Science & Business Media, 2006.David Marker. Strongly minimal sets and geometry. In Colloquium'95 (Haifa), pages 191-213, 2017.Samaria Montenegro, Alf Onshuus, and Pierre Simon. Stabilizers,-generics, and prc fields. Journal of the Institute of Mathematics of Jussieu, 19(3):821-853, 2020.Evgenia D Rabinovich. Definability of a field in sufficiently rich incidence systems, volume 14. School of Mathematical Sciences, Queen Mary and Westfield College, 1993.Stan Wagon. The Banach-Tarski Paradox. Number 24. 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