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Universally defining $\mathbb{Z}$ in $\mathbb{Q}$ with $10$ quantifiers

Nicolas Daans

Abstract

We show that for a global field $K$, every ring of $S$-integers has a universal first-order definition in $K$ with $10$ quantifiers. We also give a proof that every finite intersection of valuation rings of $K$ has an existential first-order definition in $K$ with $3$ quantifiers.

Universally defining $\mathbb{Z}$ in $\mathbb{Q}$ with $10$ quantifiers

Abstract

We show that for a global field , every ring of -integers has a universal first-order definition in with quantifiers. We also give a proof that every finite intersection of valuation rings of has an existential first-order definition in with quantifiers.
Paper Structure (6 sections, 29 theorems, 48 equations)

This paper contains 6 sections, 29 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Existentially definable subsets of fields and number of quantifiers
  3. Quaternion algebras over global and local fields
  4. Defining valuation rings, individually and uniformly
  5. Universally defining rings of $S$-integers
  6. Recursively enumerable subsets of $\mathbb{Q}$

Key Result

Theorem 1

Let $K$ be a global field, $S$ a finite set of valuations on $K$. There exists a polynomial $F \in K[X, Y_1, \ldots, Y_{10}]$ such that, for the ring of $S$-integers $\mathcal{O}_S$, we have

Theorems & Definitions (58)

  • Theorem : see \ref{['T:nicomainthmQO']}
  • Corollary : see \ref{['C:undecidable']}
  • Proposition : see \ref{['P:valE3']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof : Proof of \ref{['P:LringvsLfield']}
  • Corollary 2.4
  • proof
  • ...and 48 more