{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T10:27:51Z","timestamp":1772447271262,"version":"3.50.1"},"reference-count":12,"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":5855,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1998,3]]},"abstract":"<jats:p>Let <jats:italic>K<\/jats:italic> be an algebraically closed field of any characteristic, complete with respect to the non-trivial ultrametric absolute value \u2223\u00b7\u2223: <jats:italic>K<\/jats:italic> \u2192 \u211d<jats:sub>+<\/jats:sub>. By <jats:italic>R<\/jats:italic> denote the valuation ring of <jats:italic>K<\/jats:italic>, and by \u2118 its maximal ideal. We work within the class of subanalytic sets defined in [5], but our results here also hold for the strongly subanalytic sets introduced in [11] as well as for those subanalytic sets considered in [6]. Let <jats:italic>X<\/jats:italic> \u2282 <jats:italic>R<\/jats:italic><jats:sup>1<\/jats:sup> be subanalytic. In [8], we showed that there is a decomposition of <jats:italic>X<\/jats:italic> as a union of a finite number of special sets <jats:italic>U<\/jats:italic> \u2282 <jats:italic>R<\/jats:italic><jats:sup>1<\/jats:sup> (see below). In this note, in Theorem 1.6, we obtain a version of this result which is uniform in parameters, thereby answering a question brought to our attention by Angus Macintyre. It follows immediately from Theorem 1.6 that the theory of <jats:italic>K<\/jats:italic> in the language <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015346_inline1\"\/> (see [5] and [6]) is <jats:italic>C<\/jats:italic>-minimal in the sense of [3] and [9]. The analogous uniformity result in the <jats:italic>p<\/jats:italic>-adic case was recently proved in [12].<\/jats:p><jats:p>D<jats:sc>efinition<\/jats:sc> 1.1. (i) A <jats:italic>disc<\/jats:italic> in <jats:italic>R<\/jats:italic><jats:sup>1<\/jats:sup> is a set of one of the two following forms:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200015346_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>A <jats:italic>special set<\/jats:italic> in <jats:italic>R<\/jats:italic><jats:sup>1<\/jats:sup> is a disc minus a finite union of discs.<\/jats:p><jats:p>(ii) <jats:italic>R<\/jats:italic>-domains <jats:italic>u<\/jats:italic> \u2282 <jats:italic>R<\/jats:italic><jats:sup><jats:italic>m<\/jats:italic><\/jats:sup>, and their associated rings of analytic functions, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015346_inline2\"\/>, are defined inductively as follows. <jats:italic>R<jats:sup>m<\/jats:sup><\/jats:italic> is an <jats:italic>R<\/jats:italic>-domain and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015346_inline3\"\/>, the ring of strictly convergent power series in <jats:italic>X<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026, <jats:italic>X<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub> over <jats:italic>K<\/jats:italic>. If <jats:italic>u<\/jats:italic> is an <jats:italic>R<\/jats:italic>-domain with associated ring <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015346_inline4\"\/>, (where <jats:italic>K<\/jats:italic> \u3008<jats:italic>X, Y<\/jats:italic>\u3009 \u301a\u03c1\u301b<jats:sub><jats:italic>S<\/jats:italic><\/jats:sub> is a ring of separated power series, see [5, \u00a72] and [1, \u00a71]) and <jats:italic>f<\/jats:italic>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015346_inline5\"\/> have no common zero on <jats:italic>u<\/jats:italic> and \u25f8 \u03f5 {&lt;, \u2264}, then<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200015346_eqnU2\"\/><\/jats:disp-formula><\/jats:p><jats:p>is an <jats:italic>R<\/jats:italic>-domain and<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200015346_eqnU3\"\/><\/jats:disp-formula><\/jats:p><jats:p>where <jats:italic>J<\/jats:italic> is the ideal generated by <jats:italic>I<\/jats:italic> and <jats:italic>f<\/jats:italic> \u2212 <jats:italic>gZ<\/jats:italic> (<jats:italic>Z<\/jats:italic> is a new variable) if \u25f8 is \u2264, and<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200015346_eqnU4\"\/><\/jats:disp-formula><\/jats:p><jats:p>where <jats:italic>J<\/jats:italic> is the ideal generated by <jats:italic>I<\/jats:italic> and <jats:italic>f<\/jats:italic> \u2212 <jats:italic>g<\/jats:italic>\u03c4 (\u03c4 a new variable) if \u25f8 is &lt;. (See [8, Definition 2.2].) <jats:italic>R<\/jats:italic>-domains generalize the rational domains of [2, \u00a77.2.3]. It is true, but not easy to prove, that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015346_inline2\"\/> only depends on <jats:italic>u<\/jats:italic> as a point set, and is independent of the particular representation of <jats:italic>u<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2586589","type":"journal-article","created":{"date-parts":[[2006,4,18]],"date-time":"2006-04-18T18:43:03Z","timestamp":1145385783000},"page":"83-88","source":"Crossref","is-referenced-by-count":18,"title":["One-dimensional fibers of rigid subanalytic sets"],"prefix":"10.1017","volume":"63","author":[{"given":"L.","family":"Lipshitz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Z.","family":"Robinson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200015346_ref011","first-page":"269","article-title":"Rigid subanalytic sets","volume":"94","author":"Schoutens","year":"1994","journal-title":"Compositio Mathematica"},{"key":"S0022481200015346_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-52229-1"},{"key":"S0022481200015346_ref009","volume-title":"Annals of Pure and Applied Logic","author":"Macpherson"},{"key":"S0022481200015346_ref006","unstructured":"[6] Lipshitz L. and Robinson Z. , Rigid subanalytic sets II, submitted."},{"key":"S0022481200015346_ref005","doi-asserted-by":"publisher","DOI":"10.2307\/2374723"},{"key":"S0022481200015346_ref007","unstructured":"[7] Lipshitz L. and Robinson Z. , Rings of separated power series, preprint."},{"key":"S0022481200015346_ref001","first-page":"49","article-title":"Die Beschr\u00e4nktheit der St\u00fcckzahl der Fasenk-analytischer abbildungen","volume":"416","author":"Bartenwerfer","year":"1991","journal-title":"Journal f\u00fcr die Reine und Angewandte Mathematik"},{"key":"S0022481200015346_ref004","first-page":"208","article-title":"Isolated points on fibers of ajfinoid varieties","volume":"384","author":"Lipshitz","year":"1988","journal-title":"Journal f\u00fcr die Reine und Angewandte Mathematik"},{"key":"S0022481200015346_ref010","volume-title":"Commutative ring theory","author":"Matsumura","year":"1989"},{"key":"S0022481200015346_ref008","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1353\/ajm.1996.0027","article-title":"Rigid subanalytic subsets of the line and the plane","volume":"118","author":"Lipshitz","year":"1996","journal-title":"American Journal of Mathematics"},{"key":"S0022481200015346_ref012","unstructured":"[12] van den Dries L. , Haskell D. , and Macpherson D. , One dimensional p-adic subanalytic sets, preprint."},{"key":"S0022481200015346_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)90064-7"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200015346","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T19:04:16Z","timestamp":1557601456000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200015346\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,3]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1998,3]]}},"alternative-id":["S0022481200015346"],"URL":"https:\/\/doi.org\/10.2307\/2586589","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,3]]}}}