Analytic Reducibility of Nondegenerate Centers: Cherkas Systems

In this paper we study the center problem for polynomial differential systems and we prove that any center of an analytic differential system is analytically reducible. We also study the centers for the Cherkas polynomial differential systems ẋ = y, ẏ = P0(x) + P1(x)y + P2(x)y , where Pi(x) are polynomials of degree n, P0(0) = 0 and P ′ 0(0) < 0. Computing the focal values we find the center conditions for such systems for degree 3, and using modular arithmetics for degree 4. Finally we do a conjecture about the center conditions for Cherkas polynomial differential systems of degree n.