The Pila-Wilkie theorem and analytic Ax-Kochen-Ersov theory

Bhardwaj, Neer

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https://hdl.handle.net/2142/116173 Copy

Description

Title

The Pila-Wilkie theorem and analytic Ax-Kochen-Ersov theory

Author(s)

Bhardwaj, Neer

Issue Date

2022-06-30

Director of Research (if dissertation) or Advisor (if thesis)

van den Dries, Lou

Doctoral Committee Chair(s)

Hieronymi, Philipp

Committee Member(s)

• Heller, Jeremiah
• Freitag, James

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Logic
• Model Theory
• Square-free integers
• O-minimality
• Pila-Wilkie counting theorem
• Valued fields with analytic structure
• Ax-Kochen-Ersov principle

Abstract

In Part I, we consider the four structures $(\mathbb{Z}; \SF)$, $(\mathbb{Z}; <, \SF)$, $(\mathbb{Q}; \SQ)$, and $(\mathbb{Q}; <, \SQ)$ where $\mathbb{Z}$ is the additive group of integers, $\SF$ is the set of $a \in \mathbb{Z}$ such that $v_{p}(a) < 2$ for every prime $p$ and corresponding $p$-adic valuation $v_{p}$, $\mathbb{Q}$ and $\SQ$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences. Part II gives an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We also include complete treatments of a result due to Pila and Bombieri and of the o-minimal Yomdin-Gromov theorem that are used in this proof. For the latter we follow Binyamini and Novikov. Part III develops an extension theory for analytic valuation rings in order to establish Ax-Kochen-Ersov type results for these structures. New is that we can add in salient cases lifts of the residue field and the value group and show that the induced structure on the lifted residue field is just its field structure, and on the lifted value group is just its ordered abelian group structure. This restores an analogy with the non-analytic AKE-setting that was missing in earlier treatments of analytic AKE-theory.

Graduation Semester

2022-08

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Neer Bhardwaj

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