Diophantine sets

TL;DR

This paper explores the concept of Diophantine sets, their generalizations to other rings, and proves a key surjectivity property for finitely presented schemes over rings, contributing to algebraic geometry and number theory.

Contribution

It compares various definitions of Diophantine sets across different rings and establishes a surjectivity result for finitely presented schemes over rings.

Findings

Comparison of Diophantine set definitions in different rings

Existence of affine schemes with surjective rational points maps

Advancement in understanding Diophantine properties in algebraic geometry

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 $R'$.