{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T00:48:28Z","timestamp":1773881308962,"version":"3.50.1"},"reference-count":18,"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":832,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2011,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let (<jats:italic>K, v<\/jats:italic>) be a henselian valued field of characteristic 0. Then <jats:italic>K<\/jats:italic> admits a definable partition on each piece of which the leading term of a polynomial in one variable can be computed as a definable function of the leading term of a linear map. The main step in obtaining this partition is an answer to the question, given a polynomial <jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>) \u2208 <jats:italic>K<\/jats:italic>[<jats:italic>x<\/jats:italic>], what is <jats:italic>v<\/jats:italic>(<jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>))?<\/jats:p><jats:p>Two applications are given: first, a constructive quantifier elimination relative to the leading terms, suggesting a relative decision procedure; second, a presentation of every definable subset of <jats:italic>K<\/jats:italic> as the pullback of a definable set in the leading terms subjected to a linear translation.<\/jats:p>","DOI":"10.2178\/jsl\/1318338847","type":"journal-article","created":{"date-parts":[[2011,10,11]],"date-time":"2011-10-11T14:32:16Z","timestamp":1318343536000},"page":"1240-1260","source":"Crossref","is-referenced-by-count":18,"title":["Relative decidability and definability in henselian valued fields"],"prefix":"10.1017","volume":"76","author":[{"given":"Joseph","family":"Flenner","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200001353_ref018","first-page":"655","volume":"74","author":"Yin","year":"2009","journal-title":"Henselianity and the Denef-Pas language"},{"key":"S0022481200001353_ref005","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160220202"},{"key":"S0022481200001353_ref008","volume-title":"Stable domination and independence in algebraically closed valued fields","volume":"30","author":"Haskell","year":"2008"},{"key":"S0022481200001353_ref004","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061307000664"},{"key":"S0022481200001353_ref016","first-page":"3","article-title":"Equivalent forms of Hensel's lemma","volume":"3","author":"Ribenboim","year":"1985","journal-title":"Expositiones Mathematicae"},{"key":"S0022481200001353_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2005.05.014"},{"key":"S0022481200001353_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/2373065"},{"key":"S0022481200001353_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/2373066"},{"key":"S0022481200001353_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BF02758645"},{"key":"S0022481200001353_ref003","doi-asserted-by":"publisher","DOI":"10.2307\/1970476"},{"key":"S0022481200001353_ref007","first-page":"175","article-title":"Definable sets in algebraically closed valued fields: elimination of imaginarles","volume":"597","author":"Haskell","year":"2006","journal-title":"Journal f\u00fcr die reine und angewandte Mathematik"},{"key":"S0022481200001353_ref009","first-page":"843","volume":"60","author":"Holly","year":"1995","journal-title":"Canonical forms for definable subsets of algebraically closed and real closed valued fields"},{"key":"S0022481200001353_ref010","first-page":"1093","volume":"62","author":"Holly","year":"1997","journal-title":"Prototypes for definable subsets of algebraically closed valued fields"},{"key":"S0022481200001353_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4532-8_4"},{"key":"S0022481200001353_ref012","unstructured":"Hrushovski Ehud and Martin Ben , Zeta functions from definable equivalence relations, 2006, Preprint http:\/\/arxiv.org\/abs\/matli\/0701011."},{"key":"S0022481200001353_ref014","first-page":"605","volume":"41","author":"Macintyre","year":"1976","journal-title":"On definable subsets of p-adic fields"},{"key":"S0022481200001353_ref017","volume-title":"Complete theories","author":"Robinson","year":"1956"},{"key":"S0022481200001353_ref006","first-page":"21","article-title":"On the elementary theory of maximal normed fields","volume":"165","author":"Er\u0161ov","year":"1965","journal-title":"Doklady Akademii Nauk SSSR"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200001353","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T01:55:14Z","timestamp":1556330114000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200001353\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,12]]},"references-count":18,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2011,12]]}},"alternative-id":["S0022481200001353"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1318338847","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,12]]}}}