Simultaneous Bifurcation of Limit Cycles from a Linear Center with Extra Singular Points

The period annuli of the planar vector field x′ = −yF (x, y), y′ = xF (x, y), where the set {F (x, y) = 0} consists of k different isolated points, is defined by k + 1 concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n. Additionally, we prove that the associated Abelian integral is piecewise rational and, when k = 1, the provided upper bound is reached. Finally, the case k = 2 is also treated.