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A short note on model theory of C((t))

Zhentao Zhang

TL;DR

The paper studies the model theory of $\mathbb{C}((t))$ in the language of valued rings, focusing on definable compactness and the topology of definable groups. It proves that a definable subset of $\mathbb{C}((t))^n$ is definably compact iff it is closed and bounded, using residue-field analysis and the transcendental type $p_{\mathrm{trans},M}$, and shows $\mathcal{O}$ is definably compact. It then builds a weak Lie-type structure on definable groups via a finite affine decomposition and a $\Gamma$-exhaustion, enabling a Peterzil–Steinhorn-type theorem for abelian groups and standard-part constructions. Finally, it proposes conjectures about $G^0 = G^{00}$ and reductions to algebraic groups, outlining a program to relate definable groups in $\mathbb{C}((t))$ to algebraic groups.

Abstract

In this short note, we study C((t)) in the language of valued rings. We show that a definable subset of C((t))n (or in monster model, Mn) is definably compact iff it is closed and unbounded. Then we give some comments on definable groups over C((t)).

A short note on model theory of C((t))

TL;DR

The paper studies the model theory of in the language of valued rings, focusing on definable compactness and the topology of definable groups. It proves that a definable subset of is definably compact iff it is closed and bounded, using residue-field analysis and the transcendental type , and shows is definably compact. It then builds a weak Lie-type structure on definable groups via a finite affine decomposition and a -exhaustion, enabling a Peterzil–Steinhorn-type theorem for abelian groups and standard-part constructions. Finally, it proposes conjectures about and reductions to algebraic groups, outlining a program to relate definable groups in to algebraic groups.

Abstract

In this short note, we study C((t)) in the language of valued rings. We show that a definable subset of C((t))n (or in monster model, Mn) is definably compact iff it is closed and unbounded. Then we give some comments on definable groups over C((t)).
Paper Structure (4 sections, 12 theorems, 3 equations)

This paper contains 4 sections, 12 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Definable compactness
  3. Topology on definable groups
  4. More results

Key Result

Lemma 2.1

Let $\mathcal{F}$ be a definable family in $M$ with F.I.P. (finite intersection property) and $\mathcal{F}\subset p\in S(M)$. Let $M\prec N$ and $\bar{p}\in S(N)$ be an heir of $p$. Then $\mathcal{F}(N)\subset \bar{p}$.

Theorems & Definitions (23)

  • Lemma 2.1: Without any assumption for theories
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 13 more