La logique continue des corps globalement valu\'es
Antoine Chambert-Loir

TL;DR
This paper explores the model-theoretic properties of globally valued fields, including algebraic numbers, using continuous logic, and proves their existential closure in the context of diophantine equations and heights.
Contribution
It demonstrates that the field of algebraic numbers is existentially closed as a globally valued field, extending previous work to number fields using Arakelov geometry.
Findings
Algebraic numbers form an existentially closed globally valued field.
The proof adapts methods from function fields to number fields.
Connections between continuous logic and diophantine geometry are established.
Abstract
The continuous logic of globally valued fields -- A globally valued field is a field endowed with a family of absolute values that satisfy a product formula. Number fields and function fields in one variable give classical and fundamental examples; Nevanlinna theory also gives rise to such structures on the field of meromorphic functions on . These globally valued fields can be studied in the context of continuous logic (for which the predicates are real valued), and such a study has been undertaken some 10 years ago by Ben Yaacov and Hrushovski, thus providing a model-theoretic framework for the diophantine theory of heights. One of the first fundamental results in the tehory states the the field of algebraic numbers, with its essentially unique structure of a globally valued field, is existentially closed: every system involving polynomial equalities and inequalities, as…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsFrench Urban and Social Studies
