no. 4
Simple exponential estimate for the number of real zeros of complete abelian integrals
Annales de l'Institut Fourier, Tome 45 (1995) no. 4, pp. 897-927.

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), t∈K⋐ℝ, 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 ∈ℂ[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},
     pages = {897--927},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {4},
     year = {1995},
     doi = {10.5802/aif.1478},
     zbl = {0832.58028},
     mrnumber = {97b:14053},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1478/}
}
TY  - JOUR
AU  - Novikov, Dmitri
AU  - Yakovenko, Sergei
TI  - Simple exponential estimate for the number of real zeros of complete abelian integrals
JO  - Annales de l'Institut Fourier
PY  - 1995
SP  - 897
EP  - 927
VL  - 45
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1478/
DO  - 10.5802/aif.1478
LA  - en
ID  - AIF_1995__45_4_897_0
ER  - 
%0 Journal Article
%A Novikov, Dmitri
%A Yakovenko, Sergei
%T Simple exponential estimate for the number of real zeros of complete abelian integrals
%J Annales de l'Institut Fourier
%D 1995
%P 897-927
%V 45
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1478/
%R 10.5802/aif.1478
%G en
%F 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) no. 4, pp. 897-927. doi : 10.5802/aif.1478. https://aif.centre-mersenne.org/articles/10.5802/aif.1478/

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

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

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

[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

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

[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 | Zbl

[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

[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 | Zbl

[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

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

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

Cité par Sources :