[Paper Review] Existential rank and essential dimension of diophantine sets

Nicolas Daans, Philip Dittmann+1 more|arXiv (Cornell University)|Feb 13, 2021

Algebraic Geometry and Number Theory40 references4 citations

TL;DR

This paper introduces and analyzes three complexity measures—existential rank, positive-existential rank, and essential fibre dimension—for Diophantine sets over fields, establishing connections between these measures and essential dimension of functors. A key contribution is a quantifier-saving bound: the existential rank of the intersection of two Diophantine sets over a finitely generated field is at most the sum of their ranks minus one, which provides a structural constraint on definability and offers new tools to investigate Hilbert’s tenth problem over number fields and function fields.

ABSTRACT

We study the minimal number of existential quantifiers needed to define a diophantine set over a field and relate this number to the essential dimension of the functor of points associated to such a definition.

Motivation & Objective

• To define and analyze new complexity measures—existential rank, positive-existential rank, and essential fibre dimension—for Diophantine sets over fields.
• To explore the interplay between these measures and the notion of essential dimension of functors, particularly in relation to algebraic geometry and arithmetic geometry.
• To provide structural constraints on Diophantine definability, especially in the context of Hilbert’s tenth problem over global fields.
• To investigate whether certain fundamental sets—like ℤ in ℚ or 𝔽ₚ[t] in 𝔽ₚ(t)—are Diophantine, using these complexity measures as tools.

Proposed method

• Introduces three complexity measures: existential rank, positive-existential rank, and essential fibre dimension, each capturing different aspects of Diophantine definitions.
• Uses the framework of essential dimension of functors, drawing on results from Karpenko, Merkurjev, and Totaro on quadrics and Severi-Brauer varieties.
• Applies arithmetic results from [Jeo97] to prove bounds in positive characteristic, particularly for intersection bounds.
• Employs model-theoretic techniques, including quantifier elimination and interpretations in the language of rings, to relate definability over a ring and its fraction field.
• Applies the theory to construct bounds on existential rank for sets like sums of squares and norm groups in Galois extensions.
• Uses a recursive construction via Lemma 8.20 and Corollary 5.11 to bound the existential rank of ℚ in terms of that of ℤ or ℤ≥₀.

Experimental results

Research questions

• RQ1 Can the existential rank of a Diophantine set be bounded in terms of its geometric and arithmetic structure, particularly via essential fibre dimension?
• RQ2 Is the existential rank of the intersection of two Diophantine sets over a finitely generated field always at most the sum of their individual ranks minus one?
• RQ3 Does the existential rank of ℚ or ℱₚ(t) lie strictly between 1 and ∞, or is it always 0, 1, or ∞?
• RQ4 Can the existential rank of sets such as sums of m squares or norm groups in Galois extensions be precisely determined?
• RQ5 Is it possible to prove that ℤ is not Diophantine in ℚ by showing that its existential rank is infinite?

Key findings

• The existential rank of the intersection of two Diophantine sets over a finitely generated field satisfies rk∃(D₁ ∩ D₂) ≤ rk∃(D₁) + rk∃(D₂) − 1, a saving of one quantifier over the trivial bound.
• For any field K of characteristic zero and any m > 0, there exists a field K such that the set of sums of m squares in K has existential rank exactly m.
• The essential fibre dimension of a Diophantine set controls the complexity of its existential definition, and this interplay underlies the main bounds.
• In positive characteristic, the existential rank of the set of p-th powers in 𝔽ₚ(t) is 1, despite being n in the completion 𝔽ₚ((t)), showing a key difference in definability.
• If the existential rank of ℚ is infinite and there exists an injective polynomial map from ℚ×ℚ to ℚ, then ℤ is not Diophantine in ℚ.
• The existential rank of ℚ is at least 2, and for all fields whose existential rank has been computed, it lies in {0, 1, ∞}, raising the question of whether any field has finite existential rank strictly greater than 1.