Soit un polynôme réel de deux variables et une forme différentielle quelconque à coefficients polynomiaux réels de degré . Nous montrons que le nombre des ovales (c’est-à-dire les composantes compactes connexes) des courbes de niveau , telles que l’intégrale de la forme s’annule, est au plus quand , où ne dépend que du polynôme . 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’un opérateur irréductible, a au plus zéros réels isolés, quand .
We show that for a generic polynomial and an arbitrary differential 1-form with polynomial coefficients of degree , the number of ovals of the foliation , which yield the zero value of the complete Abelian integral , grows at most as as , where depends only on . 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 , , be a fundamental system of real solutions to a linear ordinary differential equation with rational coefficients and without singularities on the interval . If the differential operator is irreducible, then any real function representable in the form with polynomial coefficients of degree less or equal to , may have at most real isolated zeros on as .
@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/
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