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Oscillation of analytic curves

Author: Y. Yomdin
Journal: Proc. Amer. Math. Soc. 126 (1998), 357-364
MSC (1991): Primary 30B10, 34A20, 30C55, 34A25, 34C15
MathSciNet review: 1443861
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Abstract: The number of zeroes of the restriction of a given polynomial to the trajectory of a polynomial vector field in $(\mathbb{C}^n,0)$, in a neighborhood of the origin, is bounded in terms of the degrees of the polynomials involved. In fact, we bound the number of zeroes, in a neighborhood of the origin, of the restriction to the given analytic curve in $(\mathbb{C}^n,0)$ of an analytic function, linearly depending on parameters, through the stabilization time of the sequence of zero subspaces of Taylor coefficients of the composed series (which are linear forms in the parameters). Then a recent result of Gabrielov on multiplicities of the restrictions of polynomials to the trajectories of polynomial vector fields is used to bound the above stabilization moment.

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Additional Information

Y. Yomdin
Affiliation: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel

Received by editor(s): January 4, 1996
Additional Notes: This research was partially supported by the Israel Science Foundation, Grant No. 101/95-1, and by the Minerva Foundation
Communicated by: Hal L. Smith
Article copyright: © Copyright 1998 American Mathematical Society

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