A proof of a Dumortier-Roussarie's conjecture

Dumortier and Roussarie proposed a conjecture in their paper (2009, Discrete Con. Dyn. Sys., 2,723-781): For any \begin{document}$ q\in {\mathbb{N}} $\end{document} , the Abelian integrals \begin{document}$ J_{2j+1}(h) = \int_{\gamma_h}x^{2j-1}\,\mathrm dy $\end{document} , \begin{document}$ j = 0, 1, 2, \cdots, q $\end{document} , form a strict Chebyshev system on intervals \begin{document}$ h\in (0, \frac{1}{2}] $\end{document} , where \begin{document}$ \gamma_h = \{(x, y)| \mathrm e^{-2y}(y+\frac{1}{2}-x^2) = h\} $\end{document} . If this conjecture holds, then they obtain the precise upper bound of the number of limit cycles that appear near a slow-fast Hopf point of any codimension. In the present paper we develop a method to estimate the number of zeros of Abelian integrals and prove this conjecture.