{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,7]],"date-time":"2023-05-07T10:40:01Z","timestamp":1683456001442},"reference-count":16,"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":7132,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that Diophantine equivalence of two suitably presented countable rings implies that the existential polynomial languages of the two rings have the same \u201cexpressive power\u201d and that their Diophantine sets are in some sense the same. We also show that a Diophantine class of countable rings is contained completely within a relative enumeration class and demonstrate that one consequence of this fact is the existence of infinitely many Diophantine classes containing holomorphy rings of \u211a.<\/jats:p>","DOI":"10.2307\/2275929","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:52:43Z","timestamp":1146955963000},"page":"1068-1095","source":"Crossref","is-referenced-by-count":5,"title":["Diophantine equivalence and countable rings"],"prefix":"10.1017","volume":"59","author":[{"given":"Alexandra","family":"Shlapentokh","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200019721_ref014","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160440706"},{"key":"S0022481200019721_ref010","first-page":"341\u2013360","article-title":"Computable algebra, general theory and theory of countable fields","volume":"95","author":"Rabin","year":"1960","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200019721_ref012","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160420805"},{"key":"S0022481200019721_ref009","first-page":"611\u2013620","article-title":"Hilbert's tenth problem for a class of rings of algebraic integers","volume":"104","author":"Pheidas","year":"1988","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481200019721_ref007","first-page":"242\u2013247","volume-title":"Proceedings of the Asian Mathematical Conference, 1990","author":"Kim","year":"1992"},{"key":"S0022481200019721_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-07216-5"},{"key":"S0022481200019721_ref004","first-page":"227\u2013236","article-title":"Diophantine sets of algebraic integers","volume":"257","author":"Denef","year":"1980","journal-title":"II, Transactions of the American Mathematical Society"},{"key":"S0022481200019721_ref003","first-page":"214\u2013220","article-title":"Hilbert's tenth problem for quadratic rings","volume":"48","author":"Denef","year":"1975","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481200019721_ref001","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1973.11993265"},{"key":"S0022481200019721_ref016","unstructured":"Shlapentokh A. , Diophantine classes of the holomorphy rings of the global fields(to appear)."},{"key":"S0022481200019721_ref005","first-page":"385\u2013391","article-title":"Diophantine sets over some rings of algebraic integers","volume":"18","author":"Denef","year":"1978","journal-title":"Journal of the London Mathematical Society"},{"key":"S0022481200019721_ref015","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093634397"},{"key":"S0022481200019721_ref011","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200019721_ref002","first-page":"323\u2013378","volume-title":"Proceedings of Symposia in Pure Mathematics","volume":"28","author":"Davis","year":"1976"},{"key":"S0022481200019721_ref013","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160420703"},{"key":"S0022481200019721_ref008","doi-asserted-by":"crossref","first-page":"35\u201345","DOI":"10.1080\/10586458.1992.10504244","article-title":"The topology of rational points","volume":"1","author":"Mazur","year":"1992","journal-title":"Experimental Mathematics"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200019721","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,7]],"date-time":"2023-05-07T10:10:06Z","timestamp":1683454206000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200019721\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,9]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1994,9]]}},"alternative-id":["S0022481200019721"],"URL":"https:\/\/doi.org\/10.2307\/2275929","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,9]]}}}