Diophantine sets

TL;DR

This work analyzes how diophantine notions extend from $\mathbb{Z}$ to general rings $R$, introducing five frameworks—existential, positive-existential, many-polynomial-diophantine, one-polynomial-diophantine, and morphism-diophantine—and proving when they coincide. A key technical tool is an affine-covering theorem: any finitely presented $R$-scheme $Y$ admits an affine model $X\to Y$ with surjective $X(R)\to Y(R)$, enabling translation of morphism-diophantine problems into polynomial equations on affine spaces. The authors prove that many of these notions are equivalent under broad conditions, with the heart being that many-polynomial-diophantine and morphism-diophantine definitions align via the affine-covering construction. They also study the two-variable equation problem $g(x,y)=0$ having a unique solution at $(0,0)$, showing existence in various cases (notably when $\text{Frac}(R)$ is not algebraically closed) and outlining limitations when the fraction field is algebraically closed. Collectively, the results unify definability concepts across rings and provide criteria for when diophantine definitions preserve under embeddings and morphisms, offering a foundation for extending Hilbert's tenth problem-style analyses to broader algebraic contexts.

Abstract

Diophantine subsets of $\mathbb{Z}$ play a key role in the negative answer to Hilbert's tenth problem. The definition of diophantine set generalizes in several ways to other commutative rings. We compare these definitions. Along the way, we prove that for every finitely presented scheme $Y$ over a ring $R$, there exists an affine $R$-scheme $X$ with a finitely presented $R$-morphism $X \to Y$ such that $X(R') \to Y(R')$ is surjective for every $R$-algebra $R'$.

Theorems & Definitions (29)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Corollary 1.5
• proof
• Theorem 2.1
• Remark 2.2
• Remark 2.3
• proof : Proof of Theorem \ref{['T:covered by affine']}