Simple exponential estimate for the number of real zeros of complete abelian integrals
Novikov, Dmitri ; Yakovenko, Sergei
Annales de l'Institut Fourier, Tome 45 (1995), p. 897-927 / Harvested from Numdam

Soit H=H(x,y) un polynôme réel de deux variables et ω=P(x,y)dx+Q(x,y)dy une forme différentielle quelconque à coefficients polynomiaux réels de degré d. Nous montrons que le nombre des ovales (c’est-à-dire les composantes compactes connexes) des courbes de niveau H= const , telles que l’intégrale de la forme s’annule, est au plus expO H (d) quand d, où O H (d) ne dépend que du polynôme H. En fait, on obtient ce résultat comme un corollaire du théorème plus général sur les zéros de fonctions dans les enveloppes polynomiales. Nous montrons que chaque fonction appartenant à l’enveloppe d’ordre d d’un opérateur irréductible, a au plus expO(d) zéros réels isolés, quand d.

We show that for a generic polynomial H=H(x,y) and an arbitrary differential 1-form ω=P(x,y)dx+Q(x,y)dy with polynomial coefficients of degree d, the number of ovals of the foliation H= const , which yield the zero value of the complete Abelian integral I(t)= H=t ω, grows at most as expO H (d) as d, where O H (d) depends only on H. The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let f 1 (t),,f n (t), tK, be a fundamental system of real solutions to a linear ordinary differential equation Lu=0 with rational coefficients and without singularities on the interval K. If the differential operator L is irreducible, then any real function representable in the form j,k=1 n p jk (t)f j (k-1) (t) with polynomial coefficients p jk [t] of degree less or equal to d, may have at most expO L,K (d) real isolated zeros on K as d.

@article{AIF_1995__45_4_897_0,
     author = {Novikov, Dmitri and Yakovenko, Sergei},
     title = {Simple exponential estimate for the number of real zeros of complete abelian integrals},
     journal = {Annales de l'Institut Fourier},
     volume = {45},
     year = {1995},
     pages = {897-927},
     doi = {10.5802/aif.1478},
     mrnumber = {97b:14053},
     zbl = {0832.58028},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1995__45_4_897_0}
}
Novikov, Dmitri; Yakovenko, Sergei. Simple exponential estimate for the number of real zeros of complete abelian integrals. Annales de l'Institut Fourier, Tome 45 (1995) pp. 897-927. doi : 10.5802/aif.1478. http://gdmltest.u-ga.fr/item/AIF_1995__45_4_897_0/

[AI]V. Arnold, Yu Il'Yashenko, Ordinary differential equations, Encyclopedia of mathematical sciences vol. 1 (Dynamical systems-I) Springer, Berlin, 1988. | Zbl 0718.34070

[F]G. Frobenius, Ueber die Determinante mehrerer Functionen Variablen, J. Reine Angew. Math., 7 (1874), 245-257. | JFM 06.0201.02

[H]P. Hartman, Ordinary Differential Equations, John Wiley, N. Y.-London-Sydney, 1964. | MR 30 #1270 | Zbl 0125.32102

[IY1]Yu. Il'Yashenko, S. Yakovenko, Double exponential estimate for the number of zeros of complete Abelian integrals and rational envelopes of linear ordinary differential equations with an irreductible monodromy group, Inventiones Mathematicae, 121, n° 3 (1995). | Zbl 0865.34007

[IY2]Yu. Il'Yashenko, S. Yakovenko, Counting real zeros of analytic functions satisfying linear ordinary differential equations, J. Diff. Equations, 1996 (to appear). | Zbl 0847.34010

[In]E. L. Ince, Ordinary Differential Equations, Dover Publ., 1956.

[M]P. Mardešić, An explicit bound for the multiplicity of zeros of generic Abelian integrals, Nonlinearity, 4 (1991), 845-852. | MR 92h:58163 | Zbl 0741.58043

[NY]D. Novikov, S. Yakovenko, Une borne simplement exponentielle pour le nombre de zéros réels isolés des intégrales complètes abéliennes, Comptes Rendus Acad. Sci. Paris, série I, 320 (1995), 853-858. | Zbl 0826.34032

[Pe]G. Petrov, Nonoscillation of elliptic integrals, Funkcional'nyĭ analiz i ego prilozheniya, 24-3 (1990), 45-50 (Russian); English translation, Functional Analysis and Applications. | MR 92c:33036 | Zbl 0738.33013

[Pó]G. Pólya, On the mean-value theorem corresponding to a given linear homogeneous differential equation, Trans. Amer. Math. Soc., 24 (1922), 312-324. | JFM 50.0299.02

[Sch]L. Schlesinger, Handbuch der Theorie der linearen Differentialgleichungen, Teubner, Leipzig, 1 (1895), 52, formula (14). | JFM 26.0329.01

[Y]S. Yakovenko, Complete Abelian Integrals as Rational Envelopes, Nonlinearity, 7 (1994), 1237-1250. | MR 95d:34049 | Zbl 0829.58035