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 .
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 .
@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, Volume 45 (1995) no. 4, pp. 897-927. doi : 10.5802/aif.1478. https://aif.centre-mersenne.org/articles/10.5802/aif.1478/
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