On ordinary differentially large fields

TL;DR

This paper characterizes differentially large fields across all characteristics, providing axioms and expansion results, and explores their model-theoretic properties, including completeness and prime extensions.

Contribution

It introduces a unified axiomatic characterization of differentially large fields and demonstrates their expansion and model-theoretic properties in various settings.

Findings

Characterization of differentially large fields in arbitrary characteristic.

Existential closure in differential algebraic Laurent series rings.

No real closed differential field has a prime model extension unless it is closed ordered.

Abstract

We provide a characterisation of differentially large fields in arbitrary characteristic and a single derivation in the spirit of Blum axioms for differentially closed fields. In the case of characteristic zero, we use these axioms to characterise differential largeness in terms of being existentially closed in the differential algebraic Laurent series ring, and we prove that any large field of infinite transcendence degree can be expanded to a differentially large field even under certain prescribed constant fields. As an application, we show that the theory of proper dense pairs of models of a complete and modelcomplete theory of large fields, is a complete theory. As a further consequence of the expansion result we show that there is no real closed and differential field that has a prime model extension in closed ordered differential fields, unless it is itself a closed ordered differential field.