Algebraic points in tame expansions of fields

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Stellenbosch : Stellenbosch University
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.
Thesis (MSc)--Stellenbosch University, 2021.
Model theory, Tame algebras, p-Adic fields, Pila-Wilkie results, Tame expansions, Schanuel property, Dense pairs, UCTD