The Reduced Bautin Index of Planar Vector Fields

the aij and bij being real or complex. This vector field is a deformation of the vector field x∂y −y∂x whose trajectories are concentric circles around 0. We prove in this paper a precise version of the following assertion: for any compact K in the space of the (aij ,bij ), there exist a number p(q) and a neighborhood U(q,K) of 0 such that for (a,b) ∈ K , either 0 is again a center of Wa,b (i.e., 0 is an elliptic nondegenerate singular point of W , and W is integrable near 0) or Wa,b has at most p(q) limit cycles in U(q,K). The local sixteenth Hilbert problem consists of finding explicit expressions for U(q,K) and p(q). This problem is solved only for q = 2 by the socalled Bautin theorem (see [B], [Ya]). Bautin considered the Poincaré first return map around the origin restricted to a line with coordinate X as a series Fz(X) in X with coefficients depending on the parameters z = (aij ,bij ). The limit cycles correspond to the zeroes of Fz(X)−X. Given a series