Differentially large fields

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 characterize 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. 1. Introduction 249 2. Preliminaries 252 3. The Taylor morphism 257 4. Differentially large fields and algebraic characterizations 261 5. Fundamental properties, constructions and applications 266 6. Algebraic-geometric axioms 275 References 279 León Sánchez was partially supported by EPSRC grant EP/V03619X/1.