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First-order definability of affine Campana points in the projective line over a number field

Juan Pablo De Rasis

TL;DR

This work establishes a $\forall\exists$-definability framework for affine Campana points on $\mathbb{P}^1_K$, constructing a uniform, parameter-free description that realises a filtration between the number field $K$ and its rings of $S$-integers $\mathcal{O}_{K,S}$. The authors marry quaternion algebras, Hilbert symbols, and carefully engineered diophantine sets to encode local valuations and finite-place data, enabling a globally uniform description of Campana points with explicit quantifier and degree bounds. They prove that for any number field $K$, finite $S$ containing the archimedean places, and $n\ge 2$, the Campana set $C_{K,S,n}$ is $\forall\exists$-definable with a fixed, explicit quantifier structure and a polynomial degree bound, uniform in $S$. The approach extends to Campana points with arbitrary divisors, yielding a principled way to express $S$-integers and generalized Campana sets via first-order logic, with potential implications for the study of decidability and definability in number fields. The results provide a concrete, computable bridge between $K$ and $\mathcal{O}_{K,S}$ through a well-controlled logical framework and paves the way for further applications to Darmon-type points and related Diophantine constructions.

Abstract

We offer a $\forall\exists$-definition for (affine) Campana points over $\mathbb{P}^1_K$ (where $K$ is a number field), which constitute a set-theoretical filtration between $K$ and $\mathcal{O}_{K,S}$ ($S$-integers), which are well-known to be universally defined (Koenigsmann 2010, Park 2012, Eisentraeger & Morrison 2016). We also show that our formulas are uniform with respect to all possible $S$, are parameter-free as such, and we count the number of involved quantifiers and offer a bound for the degree of the defining polynomial.

First-order definability of affine Campana points in the projective line over a number field

TL;DR

This work establishes a -definability framework for affine Campana points on , constructing a uniform, parameter-free description that realises a filtration between the number field and its rings of -integers . The authors marry quaternion algebras, Hilbert symbols, and carefully engineered diophantine sets to encode local valuations and finite-place data, enabling a globally uniform description of Campana points with explicit quantifier and degree bounds. They prove that for any number field , finite containing the archimedean places, and , the Campana set is -definable with a fixed, explicit quantifier structure and a polynomial degree bound, uniform in . The approach extends to Campana points with arbitrary divisors, yielding a principled way to express -integers and generalized Campana sets via first-order logic, with potential implications for the study of decidability and definability in number fields. The results provide a concrete, computable bridge between and through a well-controlled logical framework and paves the way for further applications to Darmon-type points and related Diophantine constructions.

Abstract

We offer a -definition for (affine) Campana points over (where is a number field), which constitute a set-theoretical filtration between and (-integers), which are well-known to be universally defined (Koenigsmann 2010, Park 2012, Eisentraeger & Morrison 2016). We also show that our formulas are uniform with respect to all possible , are parameter-free as such, and we count the number of involved quantifiers and offer a bound for the degree of the defining polynomial.
Paper Structure (12 sections, 26 theorems, 28 equations)

This paper contains 12 sections, 26 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Campana points
  3. Definition of Campana Points
  4. Campana points over number fields
  5. Quaternion Algebras, Hilbert Symbol, and Diophantine sets
  6. Quaternion algebras
  7. Hilbert Symbols
  8. Relevant diophantine sets
  9. Parametrization of finite sets of non-archimedean places
  10. First-order description of Campana points
  11. Counting variables and bounding the degree
  12. Campana points with respect to arbitrary divisors

Key Result

Theorem 1.1

If $K$ is a number field, $n\in\mathbb{Z}_{\geq 2}$, and $S$ is any finite set of places containing the archimedean ones, then the set of Campana points is $\forall\exists$-definable in $K$. Moreover, the defining formula is uniform with respect to all possible $S$, and involves $838$ universal quantifiers, $558$ existential quantifiers, and a defining polynomial of degree at most $\max\left\{194n

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Definition 3.4
  • ...and 21 more