{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,6,11]],"date-time":"2022-06-11T00:50:47Z","timestamp":1654908647583},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8046,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1992,3]]},"abstract":"<jats:p>In this paper we study differential fields of characteristic 0 (with perhaps additional structure) whose theory is superstable. Our main result is that such a differential field has no proper strongly normal extensions in the sense of Kolchin [K1]. This is an approximation to the conjecture that a superstable differential field is differentially closed (although we believe the full conjecture to be false). Our result improves earlier work of Michaux [Mi] who proved that a (plain) differential field with quantifier elimination has no proper Picard-Vessiot extension. Our result is a generalisation of Michaux's, due to the fact that any plain differential field <jats:italic>K<\/jats:italic> with quantifier elimination is \u03c9-stable. (Any quantifier free type over <jats:italic>K<\/jats:italic> defines a unique type over <jats:italic>K<\/jats:italic> in the sense of dc(<jats:italic>k<\/jats:italic>), the differential closure of <jats:italic>K<\/jats:italic>, and as we mention below the theory of differentially closed fields is \u03c9-stable.) The proof of our main result depends on (i) Kolchin's theory [K3] which states that any strongly normal extension <jats:italic>L<\/jats:italic> of an algebraically closed differential field <jats:italic>K<\/jats:italic> is generated over <jats:italic>K<\/jats:italic> by an element \u03b7 of some algebraic group <jats:italic>G<\/jats:italic> defined over <jats:italic>C<jats:sub>K<\/jats:sub><\/jats:italic>, the constants of <jats:italic>K<\/jats:italic>, where \u03b7 satisfies some specific differential equations over <jats:italic>K<\/jats:italic> related to invariant differential forms on <jats:italic>G<\/jats:italic> (\u03b7 is \u201c<jats:italic>G<\/jats:italic>-primitive\u201d over <jats:italic>K<\/jats:italic>), and (ii) the fact that a superstable field has a unique generic type which is semiregular.<\/jats:p>","DOI":"10.2307\/2275178","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:43:45Z","timestamp":1146955425000},"page":"97-108","source":"Crossref","is-referenced-by-count":5,"title":["Superstable differential fields"],"prefix":"10.1017","volume":"57","author":[{"given":"A.","family":"Pillay","sequence":"first","affiliation":[]},{"given":"\u017d.","family":"Sokolovi\u0107","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200023197_ref014","volume-title":"Groupes stables","author":"Poizat","year":"1987"},{"key":"S0022481200023197_ref012","first-page":"1151","volume":"48","author":"Poizat","year":"1983","journal-title":"Une th\u00e9orie de Galois imaginaire"},{"key":"S0022481200023197_ref010","first-page":"287","article-title":"Sur l'\u00e9limination des quantificateurs dans les anneaux diff\u00e9rentials","volume":"302","author":"Michaux","year":"1986","journal-title":"Comptes Rendus des S\u00e9ances de l'Acad\u00e9mie des Sciences, S\u00e9rie I: Math\u00e9matique"},{"key":"S0022481200023197_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(83)90055-5"},{"key":"S0022481200023197_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757234"},{"key":"S0022481200023197_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/2372550"},{"key":"S0022481200023197_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(86)90035-7"},{"key":"S0022481200023197_ref013","first-page":"339","volume":"48","author":"Poizat","year":"1983","journal-title":"Groupes stables, avec types g\u00e9n\u00e9riques r\u00e9guliers"},{"key":"S0022481200023197_ref015","first-page":"43","article-title":"A note on derivations and differentials in algebraic groups","volume":"16","author":"Rosenlicht","year":"1957","journal-title":"Portugaliae Mathematica"},{"key":"S0022481200023197_ref002","first-page":"177","volume-title":"Model theory of groups","author":"Bouscaren","year":"1989"},{"key":"S0022481200023197_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(80)90006-6"},{"key":"S0022481200023197_ref005","doi-asserted-by":"publisher","DOI":"10.2307\/2372603"},{"key":"S0022481200023197_ref011","volume-title":"Cours de th\u00e9orie des mod\u00e8les","author":"Poizat","year":"1985"},{"key":"S0022481200023197_ref008","doi-asserted-by":"publisher","DOI":"10.4064\/fm-70-3-253-270"},{"key":"S0022481200023197_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/2372939"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200023197","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T22:32:22Z","timestamp":1558045942000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200023197\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,3]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1992,3]]}},"alternative-id":["S0022481200023197"],"URL":"https:\/\/doi.org\/10.2307\/2275178","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,3]]}}}