Bifurcation of limit cycles from a polynomial non-global center (Q957746)

The paper studies the number of limit cycles that bifurcate, when \(\varepsilon\) is small enough, from the period annulus of the system \[ \dot{x}=-yF(x,y) + \varepsilon P(x,y), \quad \dot{y}=xF(x,y) + \varepsilon Q(x,y), \] \(P(x,y)\) and \(Q(x,y)\) are arbitrary real polynomials of degree \(n\). The main subject is the situation when \({F(x,y)}=0\) is formed by \(k\) non-zero singular points. The main goal is to give lower and upper bounds for the zeros for corresponding Abelian integral in terms of \(k\) and \(n\). One of the key points is that the Abelian integral can be explicitly obtained as an application of the integral representation formula of harmonic functions through the Poisson kernel.