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 | References | Similar Articles | Additional Information

Abstract: The number of zeroes of the restriction of a given polynomial to the trajectory of a polynomial vector field in , 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 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

Email:
yomdin@wisdom.weizmann.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-98-04265-8

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