{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,28]],"date-time":"2024-03-28T14:50:29Z","timestamp":1711637429552},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8593,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,9]]},"abstract":"<jats:p>We point out that a group first order definable in a differentially closed field <jats:italic>K<\/jats:italic> of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over <jats:italic>K<\/jats:italic>. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).<\/jats:p><jats:p>I restrict myself here to showing (Theorem 20) how one can find a large \u201cdifferentially algebraic group chunk\u201d inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].<\/jats:p><jats:p>What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1\u20134 below are known.<\/jats:p><jats:p><jats:italic>Fact<\/jats:italic> 1. <jats:italic>The theory of differentially closed fields of characteristic<\/jats:italic> 0 <jats:italic>is complete and has quantifier elimination in the language of differential fields<\/jats:italic> (+, \u00b7,0,1, <jats:sup>\u22121<\/jats:sup>,<jats:italic>d<\/jats:italic>).<\/jats:p><jats:p><jats:italic>Fact<\/jats:italic> 2. <jats:italic>Affine n-space over a differentially closed field is a <jats:bold>Noetherian<\/jats:bold> space when equipped with the differential Zariski topology<\/jats:italic>.<\/jats:p><jats:p><jats:italic>Fact<\/jats:italic> 3. <jats:italic>If K is a differentially closed field, k<\/jats:italic> \u2286 <jats:italic>K a differential field, and a and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025615_inline1\" \/> are in k, then a is in the definable closure of k<\/jats:italic> \u25e1 <jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025615_inline1\" \/> iff a<\/jats:italic> \u2208 \u2039<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025615_inline1\" \/>\u203a (<jats:italic>where k<\/jats:italic> \u2039<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025615_inline1\" \/>\u203a <jats:italic>denotes the differential field generated by k and<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025615_inline1\" \/>).<\/jats:p><jats:p><jats:italic>Fact<\/jats:italic> 4. <jats:italic>The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, <jats:bold>stable<\/jats:bold>)<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2274479","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:36:50Z","timestamp":1146955010000},"page":"1138-1142","source":"Crossref","is-referenced-by-count":6,"title":["Differentially algebraic group chunks"],"prefix":"10.1017","volume":"55","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025615_ref007","doi-asserted-by":"publisher","DOI":"10.2307\/2372535"},{"key":"S0022481200025615_ref003","first-page":"1350","volume":"49","author":"Pillay","year":"1984","journal-title":"Closed sets and chain conditions in stable theories"},{"key":"S0022481200025615_ref001","first-page":"177","volume-title":"The model theory of groups (stable group seminar, Notre Dame, Indiana, 1985\u20131987","volume":"11","author":"Bouscaren","year":"1989"},{"key":"S0022481200025615_ref005","volume-title":"Saturated model theory","author":"Sacks","year":"1972"},{"key":"S0022481200025615_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(88)90053-X"},{"key":"S0022481200025615_ref004","first-page":"339","volume":"48","author":"Poizat","year":"1983","journal-title":"Groupes stables, avec types g\u00e9n\u00e9riques r\u00e9guliers"},{"key":"S0022481200025615_ref002","volume-title":"Differentially algebraic groups","author":"Kolchin","year":"1985"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025615","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T20:25:14Z","timestamp":1558211114000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025615\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0022481200025615"],"URL":"https:\/\/doi.org\/10.2307\/2274479","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}