Differentially large fields Article Swipe
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· 2024
· Open Access
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· DOI: https://doi.org/10.2140/ant.2024.18.249
· OA: W3021508062
We introduce the notion of differential largeness for fields equipped with\nseveral commuting derivations (as an analogue to largeness of fields). We lay\nout the foundations of this new class of "tame" differential fields. We state\nseveral characterizations and exhibit plenty of examples and applications. Our\nresults strongly indicate that differentially large fields will play a key role\nin differential field arithmetic. For instance, we characterise differential\nlargeness in terms of being existentially closed in their power series field\n(furnished with natural derivations), we give explicit constructions of\ndifferentially large fields in terms of iterated powers series, we prove that\nthe class of differentially large fields is elementary, and we show that\ndifferential largeness is preserved under algebraic extensions, therefore\nshowing that their algebraic closure is differentially closed.\n
