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Algebraic points in tame expansions of fields
Thesis (MSc)--Stellenbosch University, 2021.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2021
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| _version_ | 1867613771661312000 |
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| access_status_str | Open Access |
| author | Harrison-Migochi, Andrew |
| author2 | Boxall, Gareth John |
| author_browse | Boxall, Gareth John Harrison-Migochi, Andrew |
| author_facet | Boxall, Gareth John Harrison-Migochi, Andrew |
| author_sort | Harrison-Migochi, Andrew |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2021. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/123974 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:41:26.849Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2021 |
| publishDateRange | 2021 |
| publishDateSort | 2021 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/123974 Algebraic points in tame expansions of fields Harrison-Migochi, Andrew Boxall, Gareth John Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Model theory Tame algebras p-Adic fields Pila-Wilkie results Tame expansions Schanuel property Dense pairs UCTD Thesis (MSc)--Stellenbosch University, 2021. ENGLISH ABSTRACT: We investigate the behaviour of algebraic points in several expansions of the real, complex and p-adic fields. We build off the work of Eleftheriou, Günaydin and Hieronymi in [17] and [18] to prove a Pila-Wilkie result for a p-adic subanalytic structure with a predicate for either a dense elementary substructure or a dense dcl-independent set. In the process we prove a structure theorem for p-minimal structures with a predicate for a dense independent set. We then prove quantifier reduction results for the complex field with a predicate for the singular moduli and the real field with an exponentially transcendental power function and a predicate for the algebraic numbers using a Schanuel property proved by Bays, Kirby and Wilkie [5]. Finally we adapt a theorem by Ax [2] about exponential fields, key to the proof of the Schanuel property for power functions, to power functions. AFRIKAANSE OPSOMMING: Ons ondersoek die gedrag van algebraïese punte in verskeie uitbreidings van die reële, komplekse en p-adiese liggame. Ons bou op die werk van Eleftheriou, Günaydin en Hieronymi in [17] en [18] om ‘n Pila-Wilkie resultaat vir ‘n p-adiese subanalitiese struktuur met ’n predikaat vir ‘n dig elementêre substruktuur of ‘n dig dcl-onafhanklike versameling te bewys. In die proses bewys ons ‘n struktuurstelling vir p-minimale strukture met ‘n predikaat vir ‘n dig onafhanklike versameling. Ons bewys dan kwantorreduksie resultate vir die komplekse liggame met ‘n predikaat vir die komplekse vermenigvuldigingspunte en die reële liggaam met ‘n eksponensieel-transendentale magsfunksie en ‘n predikaat vir die algebraïese getalle deur gebruik te maak van ‘n Schanuel-eienskap wat bewys is deur Bays, Kirby en Wilkie [5]. Uiteindelik pas ons ‘n stelling van Ax [2] aan oor exponensiële liggame, wat noodsaaklik is vir die bewys van die Schanuel-eienskap, tot magsfunksies. The financial assistance of the National Research Foundation(NRF) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at, are those of the author and are not necessarily to be attributed to the NRF. Masters 2021-12-07T09:39:19Z 2021-12-22T14:32:28Z 2021-12-07T09:39:19Z 2021-12-22T14:32:28Z 2021-12 Thesis http://hdl.handle.net/10019.1/123974 en_ZA Stellenbosch University viii, 88 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Model theory Tame algebras p-Adic fields Pila-Wilkie results Tame expansions Schanuel property Dense pairs UCTD Harrison-Migochi, Andrew Algebraic points in tame expansions of fields |
| title | Algebraic points in tame expansions of fields |
| title_full | Algebraic points in tame expansions of fields |
| title_fullStr | Algebraic points in tame expansions of fields |
| title_full_unstemmed | Algebraic points in tame expansions of fields |
| title_short | Algebraic points in tame expansions of fields |
| title_sort | algebraic points in tame expansions of fields |
| topic | Model theory Tame algebras p-Adic fields Pila-Wilkie results Tame expansions Schanuel property Dense pairs UCTD |
| url | http://hdl.handle.net/10019.1/123974 |
| work_keys_str_mv | AT harrisonmigochiandrew algebraicpointsintameexpansionsoffields |