A survey of local-global methods for Hilbert's Tenth Problem

TL;DR

This survey reviews recent advances in local-global methods applied to Hilbert's Tenth Problem across various rings and fields, highlighting key results and future research directions in logic and number theory.

Contribution

It provides a comprehensive overview of how local-global principles are used to study Hilbert's Tenth Problem in different algebraic settings, emphasizing recent progress and open problems.

Findings

H10 is unsolvable for integers, but remains open for number fields and Q.

Recent work uses local-global principles for quadratic forms and central simple algebras.

The survey outlines future research directions inspired by model theory and arithmetic geometry.

Abstract

Hilbert's Tenth Problem (H10) for a ring R asks for an algorithm to decide correctly, for each $f\in\mathbb{Z}[X_{1},\dots,X_{n}]$, whether the diophantine equation $f(X_{1},...,X_{n})=0$ has a solution in R. The celebrated `Davis-Putnam-Robinson-Matiyasevich theorem' shows that {\bf H10} for $\mathbb{Z}$ is unsolvable, i.e.~there is no such algorithm. Since then, Hilbert's Tenth Problem has been studied in a wide range of rings and fields. Most importantly, for {number fields and in particular for $\mathbb{Q}$}, H10 is still an unsolved problem. Recent work of Eisentr\"ager, Poonen, Koenigsmann, Park, Dittmann, Daans, and others, has dramatically pushed forward what is known in this area, and has made essential use of local-global principles for quadratic forms, and for central simple algebras. We give a concise survey and introduction to this particular rich area of interaction between logic and number theory, without assuming a detailed background of either subject. We also sketch two further directions of future research, one inspired by model theory and one by arithmetic geometry.