Configurations of limit cycles and planar polynomial vector fields. (Q1428643)

A configuration of limit cycles is a finite set \(\{C_1,\ldots,C_n\}\) of disjoint simple closed curves of the plane such that \(C_i\cap C_j=\emptyset\) for all \(i\neq j.\) To a given configuration of limit cycles \(C=\{C_1,\dots,C_n\}\) the curve \(C_i\) is called primary if there is no curve \(C_j\) of \(C\) contained into the bounded region limited by \(C_i.\) It is said that a planar polynomial vector field \(X\) realizes a configuration of limit cycles \(C\) if the set of all limit cycles of \(X\) is equivalent (via an homeomorphism) to \(C.\) The main result of this paper is the following theorem: Let \(C\) be a configuration of limit cycles, and let \(r\) be its number of primary curves. Then the configuration \(C\) is realizable as algebraic limit cycles by a polynomial vector field of degree less or equal than \(2(n+r)-1.\) The vector field exhibiting the given configuration \(C\) is constructed in such a way that the limit cycles are circles and that it admits an integrating factor of the form \(1/V(x,y).\) The function \(V(x,y)\) is a polynomial such that its zero level set contains all circles and \(r\) isolated points, each one of them surrounded by a primary curve.