{"title":"Differentially large fields","authors":"Omar León Sánchez, Marcus Tressl","doi":"10.2140/ant.2024.18.249","DOIUrl":null,"url":null,"abstract":"<p>We introduce the notion of <span>differential largeness </span>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. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"236 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra & Number Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/ant.2024.18.249","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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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.
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