The generalized symmetry condition means that trajectories lying in x ≥ 0 can be mapped onto trajectories in x ≤ 0, with the points on x = 0 being fixed. If we know that the flow encircles the origin, then trajectories sufficiently close to the origin must be closed. Thus if the critical point is known to be of focal type, this generalized symmetry is enough to imply the existence of a center. 6, and consider h(x) = 2 A(x) − A(0) = x2 + O(x3 ). A (0) Clearly F and G are also polynomials of this polynomial, so that F = L(h(x)) and G = M (h(x)) for some polynomials L and M .

Cn are constants of k which are linearly independent over Q and u1 , . . un and v are in K. Then u1 , . . , un ∈ k and v = ct + d, with c a constant of k and d ∈ k. In our applications, we shall take k = C to ensure that k is algebraically closed. In particular, all elements of k are constants. The application we need of this result could probably be obtained by an application of complex variables, but we use the above result to emphasize the algebraic nature of the computations. We consider the system x˙ = y, y˙ = P0 (x) + P1 (x)y + P2 (x)y 2 .

3) has a non-degenerate center at the point x = p if and only g(p) = 0, g (p) > 0 and F and G are both polynomials of a polynomial A which satisfies A (p) = 0 with A (p) = 0. Proof. If we shift the x-axis to bring x = p to the origin, then it is clear that the new F and G calculated will differ from the original ones only by a constant. 6. 8. 3) into itself, reversing the directions. Thus the origin has a generalized symmetry. 3. 10) after a scaling. 3). Proof. 2. The generalized symmetry condition means that trajectories lying in x ≥ 0 can be mapped onto trajectories in x ≤ 0, with the points on x = 0 being fixed.