On the number of limit cycles surrounding a unique singular point for polynomial differential systems of arbitrary degree

We study the number of limit cycles that bifurcate from the periodic orbits of the center ẋ=−yR(x,y), ẏ=xR(x,y) where RR is a convenient polynomial of degree 2, when we perturb it inside the class of all polynomial differential systems of degree nn. We use averaging theory for computing this number. As a consequence of our study we provide the biggest number of limit cycles surrounding a unique singular point in terms of the degree of the system, known up to now.