Oscillation of analytic curves
HTML articles powered by AMS MathViewer
- by Y. Yomdin PDF
- Proc. Amer. Math. Soc. 126 (1998), 357-364 Request permission
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.References
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- Carmen Chicone and Marc Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989), no. 2, 433–486. MR 930075, DOI 10.1090/S0002-9947-1989-0930075-2
- J.-P. Françoise and C. C. Pugh, Keeping track of limit cycles, J. Differential Equations 65 (1986), no. 2, 139–157. MR 861513, DOI 10.1016/0022-0396(86)90030-6
- J.-P. Francoise and Y. Yomdin, Bernstein inequality and applications to analytic geometry and differential equations, to appear.
- A. M. Gabrièlov, Projections of semianalytic sets, Funkcional. Anal. i Priložen. 2 (1968), no. 4, 18–30 (Russian). MR 0245831
- Andrei Gabrielov, Multiplicities of zeroes of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy, Math. Res. Lett. 2 (1995), no. 4, 437–451. MR 1355706, DOI 10.4310/MRL.1995.v2.n4.a5
- A. Gabrièlov, Multiplicities of Pfaffian intersections, and the Łojasiewicz inequality, Selecta Math. (N.S.) 1 (1995), no. 1, 113–127. MR 1327229, DOI 10.1007/BF01614074
- W. K. Hayman, Differential inequalities and local valency, Pacific J. Math. 44 (1973), 117–137. MR 316693, DOI 10.2140/pjm.1973.44.117
- Yuliĭ Il′yashenko and Sergeĭ Yakovenko, Counting real zeros of analytic functions satisfying linear ordinary differential equations, J. Differential Equations 126 (1996), no. 1, 87–105. MR 1382058, DOI 10.1006/jdeq.1996.0045
- A. G. Khovanskiĭ, Fewnomials, Translations of Mathematical Monographs, vol. 88, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Smilka Zdravkovska. MR 1108621, DOI 10.1090/mmono/088
- Yu. V. Nesterenko, Estimates for the number of zeros of certain functions, New advances in transcendence theory (Durham, 1986) Cambridge Univ. Press, Cambridge, 1988, pp. 263–269. MR 972005
- R. Roussarie, A note on finite cyclicity property and Hilbert’s 16th problem, Dynamical systems, Valparaiso 1986, Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, pp. 161–168. MR 961099, DOI 10.1007/BFb0083072
- N. Roytvarf and Y. Yomdin, Bernstein classes, to appear.
Additional Information
- Y. Yomdin
- Affiliation: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
- MR Author ID: 185690
- Email: yomdin@wisdom.weizmann.ac.il
- 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
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 357-364
- MSC (1991): Primary 30B10, 34A20, 30C55, 34A25, 34C15
- DOI: https://doi.org/10.1090/S0002-9939-98-04265-8
- MathSciNet review: 1443861