Hilbert's tenth problem for finitely generated rings
Peter Koymans, Carlo Pagano
TL;DR
This article surveys Hilbert's tenth problem for finitely generated rings, tracing the negative answer over $\mathbb{Z}$ via the MRDP theorem and outlining how Poonen's elliptic-curve criterion reduces the problem to showing $\mathbb{Z}$ is Diophantine over maximal orders. It explains how descent and Selmer groups provide a mechanism to produce rank-1 elliptic curves in families, enabling unconditional (through arithmetic combinatorics) undecidability results for broad classes of rings, and discusses BSD-based conditional implications. The recent breakthroughs by KP and ABHSZywina, together with ideas from Green–Tao-type prime results, establish undecidability for all infinite finitely generated rings, while leaving the case of $\mathbb{Q}$ open as a major outstanding problem. The work highlights a deep interplay between Diophantine geometry, descent methods, and additive combinatorics in extending classical undecidability to a wide algebraic-number-theoretic setting with significant implications for decision problems in algebra and number theory.
Abstract
This expository article covers the recent developments surrounding Hilbert's tenth problem for finitely generated rings. We start by recounting the history of Hilbert's tenth problem over the integers, which was resolved negatively by Matiyasevich--Robinson--Davis--Putnam in 1970. In order to pass from $\mathbb{Z}$ to the finitely generated setting, we explain a criterion of Poonen that connects this to a problem in the theory of elliptic curves. Finally, we outline the main ideas behind the recent resolution of this elliptic curve problem by the authors.