Abstract. We consider polynomial vector fields of the formwhere Xm = Xm(x, y) and Ym = Ym(x, y) are homogenous polynomials of degree m. It is well-known that X has a center at the origin if and only if X has an analytic first integral of the formwhereThe classical center-focus problem already studied by H. Poincaré consists in distinguishing when the origin of X is either a center or a focus. In this paper we study the inverse center-focus problem. In particular for a given analytic function H defined in a neighborhood of the origin we want to determine the homogenous polynomials X m and Y m in such a way that H is a first integral of X and consequently the origin of X will be a center. Moreover, we study the case whenwhere Υ j is a convenient homogenous polynomial of degree j for j ≥ 1. The solution of the inverse center problem for polynomial differential systems with homogenous nonlinearities, provides a new mechanism to study the center problem, which is equivalent to Liapunov's Theorem and Reeb's criterion.

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Abstract. We prove the following two results.First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles.Second a planar polynomial vector field of degree S, admits at most S − 1 invariant circles which are algebraic limit cycles.In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S − 1 algebraic limit cycles given by circles, and this number is reached.

For planar polynomial vector fields of the form (−y + X(x, y)) ∂ ∂x + (x + Y (x, y)) ∂ ∂y , where X and Y start at least with terms of second order in the variables x and y, we determine necessary and sufficient conditions under which the origin is a center or a uniform isochronous centers. 2010 Mathematics Subject Classification. 34C07.

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