caa a22 4500 388039507 CHVBK 20180307125017.0 cr unu---uuuuu 161130e199803 xx s 000 0 eng 10.2307/2586589 doi S0022481200015346 pii (NATIONALLICENCE)cambridge-10.2307/2586589 One-dimensional fibers of rigid subanalytic sets [Elektronische Daten] Let K be an algebraically closed field of any characteristic, complete with respect to the non-trivial ultrametric absolute value ∣·∣: K → ℝ+. By R denote the valuation ring of K, and by ℘ 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 X ⊂ R 1 be subanalytic. In [8], we showed that there is a decomposition of X as a union of a finite number of special sets U ⊂ R 1 (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 K in the language (see [5] and [6]) is C-minimal in the sense of [3] and [9]. The analogous uniformity result in the p-adic case was recently proved in [12]. Definition 1.1. (i) A disc in R 1 is a set of one of the two following forms: A special set in R 1 is a disc minus a finite union of discs. (ii) R-domains u ⊂ R m , and their associated rings of analytic functions, , are defined inductively as follows. Rm is an R-domain and , the ring of strictly convergent power series in X 1, , X m over K. If u is an R-domain with associated ring , (where K 〈X, Y〉 〚ρ〛 S is a ring of separated power series, see [5, §2] and [1, §1]) and f, have no common zero on u and ◸ ϵ {<, ≤}, then is an R-domain and where J is the ideal generated by I and f − gZ (Z is a new variable) if ◸ is ≤, and where J is the ideal generated by I and f − gτ (τ a new variable) if ◸ is <. (See [8, Definition 2.2].) R-domains generalize the rational domains of [2, §7.2.3]. It is true, but not easy to prove, that only depends on u as a point set, and is independent of the particular representation of u. Copyright © Association for Symbolic Logic 1998 Lipshitz L. Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA, E-mail: lipshitz@math.purdue.edu Robinson Z. Department of Mathematics, University of East Carolina, Greenville, NC 27858, USA, E-mail: robinson@math.ecu.edu The Journal of Symbolic Logic Cambridge University Press 63/1(1998-03), 83-88 0022-4812 63:1<83 1998 63 JSL https://doi.org/10.2307/2586589 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586589 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Lipshitz L. Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA, E-mail: lipshitz@math.purdue.edu NATIONALLICENCE 700 1- Robinson Z. Department of Mathematics, University of East Carolina, Greenville, NC 27858, USA, E-mail: robinson@math.ecu.edu NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/1(1998-03), 83-88 0022-4812 63:1<83 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge