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) no. 4, pp. 897-927.

Soit $H=H(x,y)$ un polynôme réel de deux variables et $\omega =P(x,y)dx+Q(x,y)dy$ 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 $H=\mathrm{const}$, telles que l’intégrale de la forme s’annule, est au plus $exp{O}_H\left(d\right)$ quand $d\to \infty$, où $O_H\left(d\right)$ 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 $expO\left(d\right)$ zéros réels isolés, quand $d\to \infty$.

We show that for a generic polynomial $H=H(x,y)$ and an arbitrary differential 1-form $\omega =P(x,y)dx+Q(x,y)dy$ with polynomial coefficients of degree $\le d$, the number of ovals of the foliation $H=\mathrm{const}$, which yield the zero value of the complete Abelian integral $I\left(t\right)={\oint }_{H=t}\omega$, grows at most as $exp{O}_H\left(d\right)$ as $d\to \infty$, where $O_H\left(d\right)$ 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 $f_1\left(t\right),\cdots ,f_n\left(t\right)$, $t\in K⋐ℝ$, 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 $\sum _{j,k=1}^n{p}_{jk}\left(t\right)f_j^{(k-1)}\left(t\right)$ with polynomial coefficients $p_{jk}\in ℂ\left[t\right]$ of degree less or equal to $d$, may have at most $exp{O}_{L,K}\left(d\right)$ real isolated zeros on $K$ as $d\to \infty$.

[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