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The Diophantine problem for addition and divisibility for rings of $S$-integers of quadratic imaginary extensions of $\mathbb{Q}$

Natalia Hormazábal, Carlos Martínez-Ranero

Abstract

Let $K$ be a quadratic imaginary extension of $\mathbb{Q}$, let $S$ be a finite nonempty set of non archimedean places, and let $\mathcal{O}_{K,S}$ denote the ring of $S$-integers of $K$. We show that there is no algorithm which solves the following problem. Given an arbitrary system of linear equations over the integers together with divisibility conditions on some of the variables, decide whether or not there exists a solution over $\mathcal{O}_{K,S}$. This contrasts with Lipshitz's result, which shows that such algorithm does exists for the ring of integers (i.e. $S=\emptyset).$

The Diophantine problem for addition and divisibility for rings of $S$-integers of quadratic imaginary extensions of $\mathbb{Q}$

Abstract

Let be a quadratic imaginary extension of , let be a finite nonempty set of non archimedean places, and let denote the ring of -integers of . We show that there is no algorithm which solves the following problem. Given an arbitrary system of linear equations over the integers together with divisibility conditions on some of the variables, decide whether or not there exists a solution over . This contrasts with Lipshitz's result, which shows that such algorithm does exists for the ring of integers (i.e.
Paper Structure (5 sections, 12 theorems, 38 equations)

This paper contains 5 sections, 12 theorems, 38 equations.

Table of Contents

  1. Introduction and main results
  2. Definability of "to be distinct"
  3. Defining the multiplication function restricted to units
  4. Abundance of large units
  5. Definability of the square function

Key Result

Theorem 1.1

( Theorem mainthm) Multiplication is positive-existentially definable over the $\mathcal{L}_{\rm div}$-structure $\mathcal{O}_{K,S}$. In particular, the positive-existential theory of this structure is undecidable.

Theorems & Definitions (16)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 4.1
  • Remark 4.2
  • ...and 6 more