Hilbert’s tenth problem via additive combinatorics
Abstract
For all infinite rings R R that are finitely generated over Z \mathbb {Z} , we show that Hilbert’s tenth problem has a negative answer. This is accomplished by constructing elliptic curves E E without rank growth in certain quadratic extensions L / K L/K . To achieve such a result unconditionally, our key innovation is to combine techniques from additive combinatorics with 2 2 -descent.
Fields of Study & Subjects
Graph Labeling and Dimension Problems
Advanced Mathematical Theories and Applications
graph theory and CDMA systems
Publication details
- Journal / source
- Journal of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Volume
- 40
- Issue
- 1
- Pages
- 195–234
- Publication date
- 10 August 2026
- Language
- EN