The number of limit cycles from a smooth slow-fast Hopf bifurcation

Dumortier and Roussarie proposed a conjecture in their paper (2009, Discrete Con. Dyn. Sys., 2, 723-781): For any positive integer q, q Abelian integrals over some non-algebraic curve, form a strict Chebyshev system. If this conjecture holds, then one can obtain the precise upper bound of the number of limit cycles that appear near a slow-fast Hopf point. In this talk we develop a method to estimate the number of zeros of Abelian integrals and prove this conjecture. This is a joint work with Prof. Chengzhi Li.