On the 16th Hilbert Problem

For a polynomial planar vector field of degree $n\geq 3$ with $S$ ($S\geq 2$) invariant nonsingular algebraic curves of degree greater than or equal to two, we proved that the maximal number of algebraic limit cycles is $n-1$. We use the Pontryagin method to analyze the problem of the maximal number of limit cycles for Lienard's equation.