Hilbert's tenth problem for finitely generated rings

TL;DR

This paper reviews recent progress on Hilbert's tenth problem for finitely generated rings, highlighting the connection to elliptic curves and the authors' recent resolution of related problems.

Contribution

It explains a criterion linking Hilbert's tenth problem for finitely generated rings to elliptic curve theory and outlines the authors' recent solution to the key elliptic curve problem.

Findings

Resolution of Hilbert's tenth problem for certain finitely generated rings.

Connection established between Hilbert's tenth problem and elliptic curve theory.

Authors' recent proof of the elliptic curve problem related to finitely generated rings.

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.