A proof of a Dumortier-Roussarie's conjecture

<p style='text-indent:20px;'>Dumortier and Roussarie proposed a conjecture in their paper (2009, Discrete Con. Dyn. Sys., 2,723-781): For any <inline-formula><tex-math id="M1">\begin{document}$ q\in {\mathbb{N}} $\end{document}</tex-math></inline-formula>, the Abelian integrals id="M2">\begin{document}$ J_{2j+1}(h) = \int_{\gamma_h}x^{2j-1}\,\mathrm dy id="M3">\begin{document}$ j 0, 1, 2, \cdots, q form strict Chebyshev system on intervals id="M4">\begin{document}$ h\in (0, \frac{1}{2}] where id="M5">\begin{document}$ \gamma_h \{(x, y)| \mathrm e^{-2y}(y+\frac{1}{2}-x^2) h\} $\end{document}</tex-math></inline-formula>. If this holds, then they obtain precise upper bound of number limit cycles that appear near slow-fast Hopf point codimension. In present we develop method to estimate zeros prove conjecture.</p>