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

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), tK, 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 [t] of degree less or equal to d, may have at most expO L,K (d) real isolated zeros on K as d.

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.

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {45},
     number = {4},
     year = {1995},
     pages = {897-927},
     doi = {10.5802/aif.1478},
     zbl = {0832.58028},
     mrnumber = {97b:14053},
     language = {en},
     url = {http://www.numdam.org/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, Volume 45 (1995) no. 4, pp. 897-927. doi : 10.5802/aif.1478. http://www.numdam.org/item/AIF_1995__45_4_897_0/

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