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

Résumé

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 $H={\rm const}$, telles que l’intégrale de la forme s’annule, est au plus $\exp O_H(d)$ quand $d\rightarrow \infty $, 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 $\exp O(d)$ zéros réels isolés, quand $d\rightarrow \infty $.

Abstract

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 $H={\rm const}$, which yield the zero value of the complete Abelian integral $I(t)=\oint _{H=t}\omega $, grows at most as $\exp O_H(d)$ as $d\rightarrow \infty $, 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 $f_1(t),\dots ,f_n(t)$, $t\in K\Subset {\Bbb R}$, 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 $\sum \limits _{j,k=1}^n p_{jk}(t)\,f_j^{(k-1)}(t)$ with polynomial coefficients $p_{jk}\in {\Bbb C}[t]$ of degree less or equal to $d$, may have at most $\exp O_{L,K}(d)$ real isolated zeros on $K$ as $d\rightarrow \infty $.

Détail

Publié le : 1995-01-01

MR 97b:14053 | Zbl 0832.58028 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1478

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