One-Dimensional Fibers of Rigid Subanalytic Sets

Let K be an algebraically closed eld of any characteristic, complete with respect to the non-trivial ultrametric absolute value jj : K ! R +. By R denote the valuation ring of K, and by } its maximal ideal. We work within the class of subanalytic sets deened in L2], but our results here also hold for the strongly subanalytic sets introduced in S] as well as for those subanalytic sets considered in LR2]. Let X R 1 be subanalytic. In LR1], we showed that there is a decomposition of X as a union of a nite 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 L D an (see L2] and LR2]) is C-minimal in the sense of HM] and MS]. The analogous uniformity result in the p-adic case was recently proved in DHM]. 1.1 Deenition. (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 nite union of discs.