Flat expansions and their applications

We consider polynomial ODEs at degenerate singularities. We study families of solutions to an ODE which are exponentially close to a solution represented by a formal power series. We demonstrate that for plane polynomial systems all solutions of such a family are uniquely determined as series of flat functions. At present, such (flat) expansions are poorly understood. Power series involved in flat expansions may converge or diverge. We give some examples of computation of flat expansions and consider their applications. We compute a flat expansion of the solution to the Blasius problem at infinity and demonstrate that this asymptotic expansion can be matched with the Blasius power series expansion at the origin.