Oscillation of analytic curves
Author:
Y. Yomdin
Journal:
Proc. Amer. Math. Soc. 126 (1998), 357364
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 , 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.
 1.
N.
N. Bautin, On the number of limit cycles which appear with the
variation of coefficients from an equilibrium position of focus or center
type, American Math. Soc. Translation 1954 (1954),
no. 100, 19. MR 0059426
(15,527h)
 2.
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
(89h:58139), http://dx.doi.org/10.1090/S00029947198909300752
 3.
J.P.
Françoise and C.
C. Pugh, Keeping track of limit cycles, J. Differential
Equations 65 (1986), no. 2, 139–157. MR 861513
(88a:58162), http://dx.doi.org/10.1016/00220396(86)900306
 4.
J.P. Francoise and Y. Yomdin, Bernstein inequality and applications to analytic geometry and differential equations, to appear.
 5.
A.
M. Gabrièlov, Projections of semianalytic sets,
Funkcional. Anal. i Priložen. 2 (1968), no. 4,
18–30 (Russian). MR 0245831
(39 #7137)
 6.
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
(97c:14055), http://dx.doi.org/10.4310/MRL.1995.v2.n4.a5
 7.
A.
Gabrièlov, Multiplicities of Pfaffian intersections, and the
Łojasiewicz inequality, Selecta Math. (N.S.) 1
(1995), no. 1, 113–127. MR 1327229
(96d:32007), http://dx.doi.org/10.1007/BF01614074
 8.
W.
K. Hayman, Differential inequalities and local valency,
Pacific J. Math. 44 (1973), 117–137. MR 0316693
(47 #5240)
 9.
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
(97a:34010), http://dx.doi.org/10.1006/jdeq.1996.0045
 10.
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
(92h:14039)
 11.
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
(90d:11085)
 12.
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
(90b:58227), http://dx.doi.org/10.1007/BFb0083072
 13.
N. Roytvarf and Y. Yomdin, Bernstein classes, to appear.
 1.
 N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Transl. AMS, Series I, 5 (1962), 396413. MR 15:527h
 2.
 C. Chicone and M. Jacobs, Bifurcations of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312, N2 (1989), 433486. MR 89h:58139
 3.
 J.P. Francoise and C. C. Pugh, Keeping track of limit cycles, J. Diff. Equations 65 (1986), 139157. MR 88a:58162
 4.
 J.P. Francoise and Y. Yomdin, Bernstein inequality and applications to analytic geometry and differential equations, to appear.
 5.
 A. Gabrielov, Projections of semianalytic sets, Funct. Anal. Appl. 2 (4) (1968), 282291. MR 39:7137
 6.
 A. Gabrielov, Multiplicities of zeroes of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy, Math. Res. Lett. 2 (1995), 437451. MR 97c:14055
 7.
 A. Gabrielov, Multiplicities of Pfaffian intersections and the Lojasiewicz inequality, Selecta Matematica, New Series 1 (1) (1995), 113127. MR 96d:32007
 8.
 W. K. Hayman, Differential inequalities and local valency, Pacific J. of Math. 44 (1) (1973), 117137. MR 47:5240
 9.
 Yu. Il'yashenko and S. Yakovenko, Counting real zeroes of analytic functions, satisfying linear ordinary differential equations, J. Diff. Equations 126 (1) (1996), 87105. MR 97a:34010
 10.
 A. G. Khovanski, Fewnomials, AMS Publ., Providence, RI, 1991. MR 92h:14039
 11.
 Yu. V. Nesterenko, Estimates for the number of zeroes of certain functions, in Transcendence Theory, Alan Baker Ed., Cambridge Univ. Press (1988). MR 90d:11085
 12.
 R. Roussarie, A note on finite cyclicity and Hilbert's problem, LNM 1331, Springer, New York and Berlin (1988), 161168. MR 90b:58227
 13.
 N. Roytvarf and Y. Yomdin, Bernstein classes, to appear.
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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:
http://dx.doi.org/10.1090/S0002993998042658
PII:
S 00029939(98)042658
Received by editor(s):
January 4, 1996
Additional Notes:
This research was partially supported by the Israel Science Foundation, Grant No. 101/951, and by the Minerva Foundation
Communicated by:
Hal L. Smith
Article copyright:
© Copyright 1998 American Mathematical Society
