
TL;DR
This paper introduces 'ez fields', a class of large fields where definable sets have well-behaved topological properties, unifying various theories of definable sets across different types of fields.
Contribution
The paper defines 'ez fields' as a new class of large fields with a generalized model completeness property, and develops their theory of definable sets, unifying multiple existing frameworks.
Findings
Existentially definable sets in perfect large fields are finite unions of étale open subsets.
Algebraically closed, real closed, p-adically closed, and bounded PAC fields are 'ez' fields.
Some Henselian and Frobenius fields are also 'ez'.
Abstract
Let be a field. The \'etale open topology on the -points of a -variety was introduced in our previous work. The \'etale open topology is non-discrete if and only if is large. If is separably, real, -adically closed then the \'etale open topology agrees with the Zariski, order, valuation topology, respectively. We show that existentially definable sets in perfect large fields behave well with respect to this topology: such sets are finite unions of \'etale open subsets of Zariski closed sets. This implies that existentially definable sets in arbitrary perfect large fields enjoy some of the well-known topological properties of definable sets in algebraically, real, and -adically closed fields. We introduce and study the class of \'ez fields: is \'ez if is large and every definable set is a finite union of \'etale open subsets of Zariski closed…
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
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms · Algebraic Geometry and Number Theory
