Note on convergence and orders of vanishing along curves.(English)Zbl 0711.32002

Let X be a complex space and C a family of curves passing through a point x in X. Using a theorem due to Gabrielov, the author gives conditions under which a formal function germ at x is convergent (or, respectively, has a high order of vanishing) when this property holds along all the curves in the family C.
Reviewer: A.Dimca

MSC:

 32A05 Power series, series of functions of several complex variables 32B05 Analytic algebras and generalizations, preparation theorems 13J05 Power series rings
Full Text:

References:

 [1] Alexander, H.:Projective capacity: in Recent developments in several complex variables (Ann. Math. Studies 100), 3–27. Princeton: Priceton Univ. 1981 [2] Becker, J.:Exposé on a conjecture of Tougeron. Ann. Inst. Fourier (Grenoble)27, 9–27 (1977) · Zbl 0337.14002 [3] Bedford, E., Taylor, B. A.:A new capacity for plurisubharmonic functions, Acta Math.149, 1–40 (1982) · Zbl 0547.32012 [4] Bloom, T., Risler, J.-J.:Familles de courbes sur les germs d’espaces analytiques Bull. Soc. Math. France105, 261–280 (1977) · Zbl 0365.32003 [5] Gabrièlov, A. M.:Formal relations between analytic functions. Izv. Akad. Nauk. SSSR.37, 1056–1088 (1973) [6] Grauert, H. Remmert, R.:Coherent analytic sheaves (GMW 265), Berl in Heidelberg: Springer 1984 [7] Hironaka, H.:Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. Ann. Math.79 109–326 (1964) · Zbl 0122.38603 [8] Izumi, S.:A measure of integrity for local analytic algebras. Publ. RIMS, Kyoto Univ.21, 719–735 (1985) · Zbl 0587.32016 [9] –:The rank condition and convergence of formal functions. Duke Math. J.59, 241–264 (1989) · Zbl 0688.32008 [10] –:Propagation of convergence along Moisezon subspace: in Technical Report Series in Math. #12. Sapporo: Hokkaido Univ. Press 1989 [11] Lejeune-Jalabert, M., Teissier, B.:Clôture integral des idéaux et équisingularite (chapt. 1) Univ. Sc. & Medicale de Gremoble 1974 · Zbl 0298.14002 [12] Levenberg, N., Molzon, R. E.:Convergence set of a formal power series. Math. Z.197, 411–420 (1988) · Zbl 0617.32001 [13] Łojasiewicz, S.:Sur le problème de la division. Studia Math.18, 87–136 (1959) · Zbl 0115.10203 [14] Matsumura, H.:Commutative ring theory (Cambridge Studies in Adv. Math.. 8). Cambridge: Cambridge Univ. Press 1986 [15] Nagata, M.:Local rings (Interscience Tracts 13), New York London: Interscience P. 1962 [16] Rees, D.:Lectures on the asymptotic theory of ideals (London Math. Soc. LNS 113). Cambridge: Cambridge Univ. Press 1988 [17] Siciak, J.: Extremal plurisubharmonic functions and capacities in $$\mathbb{C}$$ n (Sophia Kokyuroku in Math. 14). Tokyo 1982 · Zbl 0579.32025 [18] Tougeron, J-Cl.:Courbes analytiques sur un germ d’espace analytique et applications. Ann. Inst Fourier Grenoble26, 117–131 (1976) · Zbl 0318.32005 [19] Tougeron, J-Cl.:Sur les racines d’un polynome a coefficients series formelles. preprint 1988 [20] Whitney, H.:Complex analytic varieties Reading, Massachusetts: Addison-Wesley 1972 · Zbl 0265.32008 [21] Wiegerinck, J.:Convergence of formal power series and analytic extension: in Complex analysis II (LNM 1276), 313–320. Berlin Heidelberg: Springer 1987
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.