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# One-dimensional fibers of rigid subanalytic sets

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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 XR1 be subanalytic. In [8], we showed that there is a decomposition of X as a union of a finite number of special sets UR1 (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 R1 is a set of one of the two following forms:

A special set in R1 is a disc minus a finite union of discs.

(ii) R-domains uRm, 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 X1,…, Xm over K. If u is an R-domain with associated ring , (where KX, 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 fgZ (Z is a new variable) if ◸ is ≤, and

where J is the ideal generated by I and fgτ (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.

## References

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[1] Bartenwerfer, W., Die Beschränktheit der Stückzahl der Fasenk-analytischer abbildungen, Journal für die Reine und Angewandte Mathematik, vol. 416 (1991), pp. 4970.
[2] Bosch, S., Güntzer, U., and Remmert, R., Non-Archimedean analysis, Springer-Verlag, 1984.
[3] Haskell, D. and Macpherson, H., Cell decompositions of C-minimal structures, Annals of Pure and Applied Logic, vol. 66 (1994), pp. 113162.
[4] Lipshitz, L., Isolated points on fibers of ajfinoid varieties, Journal für die Reine und Angewandte Mathematik, vol. 384 (1988), pp. 208220.
[5] Lipshitz, L.et al., Rigid subanalytic sets, American Journal of Mathematics, vol. 115 (1993), pp. 77108.
[6] Lipshitz, L. and Robinson, Z., Rigid subanalytic sets II, submitted.
[7] Lipshitz, L. and Robinson, Z., Rings of separated power series, preprint.
[8] Lipshitz, L. and Robinson, Z., Rigid subanalytic subsets of the line and the plane, American Journal of Mathematics, vol. 118 (1996), pp. 493527.
[9] Macpherson, H. and Steinhorn, C., On variants of 0-minimality, to appear in Annals of Pure and Applied Logic.
[10] Matsumura, H., Commutative ring theory, Cambridge University Press, 1989.
[11] Schoutens, H., Rigid subanalytic sets, Compositio Mathematica, vol. 94 (1994), pp. 269295.
[12] van den Dries, L., Haskell, D., and Macpherson, D., One dimensional p-adic subanalytic sets, preprint.

# One-dimensional fibers of rigid subanalytic sets

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