{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,30]],"date-time":"2024-03-30T22:40:45Z","timestamp":1711838445740},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1197,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in<jats:italic>T<jats:sub>A<\/jats:sub><\/jats:italic>. The thesis concerns minimal formulae of the form<jats:italic>x<\/jats:italic>\u0404<jats:italic>A<\/jats:italic>\u2227 \u03c3(<jats:italic>x<\/jats:italic>) =<jats:italic>f(x)<\/jats:italic>for an algebraic curve<jats:italic>A<\/jats:italic>and a dominant rational function<jats:italic>f: A<\/jats:italic>\u2192 \u03c3(<jats:italic>A<\/jats:italic>). These are shown to be uniform in the Zilber trichotomy, and the pairs<jats:italic>(A, f)<\/jats:italic>that fall into each of the three cases are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to<jats:italic>T<jats:sub>A<\/jats:sub><\/jats:italic>, the model companion of the class of models of an arbitrary totally-transcendental theory<jats:italic>T<\/jats:italic>with an injective endomorphism, if this model-companion exists. A<jats:italic>T<jats:sub>A<\/jats:sub><\/jats:italic>analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry.<\/jats:p>","DOI":"10.2178\/jsl\/1286198157","type":"journal-article","created":{"date-parts":[[2010,10,4]],"date-time":"2010-10-04T13:16:13Z","timestamp":1286198173000},"page":"1462-1488","source":"Crossref","is-referenced-by-count":0,"title":["Grouplike minimal sets in ACFA AND in<i>T<sub>A<\/sub><\/i>"],"prefix":"10.1017","volume":"75","author":[{"given":"Alice","family":"Medvedev","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200002346_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-09494-6"},{"key":"S0022481200002346_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-69904-2"},{"key":"S0022481200002346_ref014","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534372.001.0001","volume-title":"Geometric stability theory","volume":"32","author":"Pillay","year":"1996"},{"key":"S0022481200002346_ref012","unstructured":"Medvedev Alice and Scanlon Thomas , Polynomial dynamics, 2009."},{"key":"S0022481200002346_ref009","unstructured":"Hrushovski Ehud , The elementary theory of the Frobenius automorphisms, 2004."},{"key":"S0022481200002346_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(01)00096-3"},{"key":"S0022481200002346_ref006","volume-title":"Difference algebra","author":"Cohn","year":"1965"},{"key":"S0022481200002346_ref011","unstructured":"Medvedev Alice , Around latt\u00e8s functions, in preparation."},{"key":"S0022481200002346_ref004","doi-asserted-by":"publisher","DOI":"10.1112\/S0024611502013576"},{"key":"S0022481200002346_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-99-02498-8"},{"key":"S0022481200002346_ref002","unstructured":"Blossier Thomas , Wagner Frank O. , and Martin-Pizarro Amador . G\u00e9om\u00e9tries relatives, 2009."},{"key":"S0022481200002346_ref001","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1063372196"},{"key":"S0022481200002346_ref013","doi-asserted-by":"publisher","DOI":"10.4171\/011-1\/1"},{"key":"S0022481200002346_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-03-03264-1"},{"key":"S0022481200002346_ref010","unstructured":"Medvedev Alice , Minimal sets in ACFA, Ph.D. thesis, 2007."},{"key":"S0022481200002346_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(98)00021-9"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200002346","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,30]],"date-time":"2024-03-30T22:24:27Z","timestamp":1711837467000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200002346\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,12]]},"references-count":16,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,12]]}},"alternative-id":["S0022481200002346"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1286198157","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,12]]}}}