An explicit linear estimate for the number of zeros of Abelian integrals (Q2895598)

Let \(H\) be a polynomial in two variables and let \(\omega\) be an algebraic form on \({\mathbb R}^2\). The infinitesimal Hilbert's 16th problem deals with the study of the number of zeroes of \(I_{H,\omega}(t)=\int_{\delta_t}\omega \) in terms of the degrees of \(H\) and \(\omega\), where \(\delta_t\subset\{H=t\}\) denotes a continuous family of real ovals.NEWLINENEWLINEIn this interesting paper, the authors are able to obtain an explicit bound \(N(\deg \omega,\deg H)\) of the number of zeroes of the above abelian integral which grows linearly with the degree of \(\omega\). This result extends and refines previous results of Petrov and Khovanskii and of Novikov, Yakovenko and the first author. The dependence on the degree of \(H\) involves five successive exponentiations. As the authors comment, the bound seems not to be realistic, but the appearance of at least two successive exponentiations can not be avoid with the used approach.