Differentially Large Fields

TL;DR

This paper introduces the concept of differentially large fields, establishing foundational properties, characterizations, and examples, and demonstrating their significance in differential field arithmetic.

Contribution

It defines differential largeness for fields with multiple derivations and explores its fundamental properties, characterizations, and applications, advancing the understanding of tame differential fields.

Findings

Differential largeness characterized by existential closure in power series fields

Explicit constructions of differentially large fields via iterated power series

Differential largeness preserved under algebraic extensions

Abstract

We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of "tame" differential fields. We state several characterizations and exhibit plenty of examples and applications. Our results strongly indicate that differentially large fields will play a key role in differential field arithmetic. For instance, we characterise differential largeness in terms of being existentially closed in their power series field (furnished with natural derivations), we give explicit constructions of differentially large fields in terms of iterated powers series, we prove that the class of differentially large fields is elementary, and we show that differential largeness is preserved under algebraic extensions, therefore showing that their algebraic closure is differentially closed.