  We present some results on the existence and nonexistence of centers for
polynomial first order ordinary differential equations with complex
coefficients. In particular, we show that binomial differential equations
without linear terms do not complex centers. Classes of polynomial differential
equations, with more than two terms, are presented that do not have complex
centers. We also study the relation between complex centers and the Pugh
problem. An algorithm is described to solve the Pugh problem for equations
without complex centers. The method of proof involves phase plane analysis of
the polar equations an a local study of periodic solutions.
