[Paper Review] Defining Z in Q

Jochen Koenigsmann|arXiv (Cornell University)|Nov 15, 2010

Algebraic Geometry and Number Theory13 citations

TL;DR

This paper establishes that the integers ℤ are definable in the rationals ℚ using a universal first-order formula, presenting an ∀∃-formula with only one universal quantifier. It also constructs new diophantine subsets of ℚ, including the complement of a norm image, and proves the non-existence of an existential definition for ℤ in ℚ under a strong variant of the Bombieri-Lang conjecture.

ABSTRACT

We show that ${\mathbb Z}$ is definable in ${\mathbb Q}$ by a universal first-order formula in the language of rings. We also present an $\forall\exists$-formula for ${\mathbb Z}$ in ${\mathbb Q}$ with just one universal quantifier. We exhibit new diophantine subsets of ${\mathbb Q}$ like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof of the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for ${\mathbb Z}$ in ${\mathbb Q}$, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over ${\mathbb Q}$ with many ${\mathbb Q}$-rational points.

Motivation & Objective

• To establish a definable characterization of the integers ℤ within the rationals ℚ using first-order logic.
• To construct explicit diophantine subsets of ℚ, including complements of norm images and the set of non-squares.
• To investigate the logical complexity of defining ℤ in ℚ, particularly whether an existential formula exists.
• To explore the implications of the Bombieri-Lang conjecture for the definability of ℤ in ℚ.

Proposed method

• Using model-theoretic techniques in the language of rings to define ℤ in ℚ via a universal first-order formula.
• Constructing an ∀∃-formula for ℤ in ℚ with only one universal quantifier, reducing logical complexity.
• Demonstrating that the complement of the image of a norm map under a quadratic extension is diophantine in ℚ.
• Providing an elementary proof that the set of non-squares in ℚ is diophantine.
• Applying a strong variant of the Bombieri-Lang conjecture to rule out the existence of an existential formula for ℤ in ℚ.

Experimental results

Research questions

• RQ1 Can ℤ be defined in ℚ using a universal first-order formula?
• RQ2 Is there an ∀∃-formula for ℤ in ℚ with only one universal quantifier?
• RQ3 Are there new diophantine subsets of ℚ, such as the complement of a norm image, that can be constructed?
• RQ4 Can the set of non-squares in ℚ be shown to be diophantine via elementary means?
• RQ5 Does the existence of an existential formula for ℤ in ℚ contradict a strong variant of the Bombieri-Lang conjecture?

Key findings

• The integers ℤ are definable in the rationals ℚ by a universal first-order formula in the language of rings.
• An ∀∃-formula for ℤ in ℚ is constructed with only one universal quantifier, improving logical efficiency.
• The complement of the image of the norm map under a quadratic extension is shown to be a diophantine subset of ℚ.
• The set of non-squares in ℚ is proven to be diophantine using an elementary argument.
• Under a strong variant of the Bombieri-Lang conjecture, no existential formula for ℤ in ℚ exists.