On multiple and non-multiple solutions of algebraic ODEs

Here we consider formal solutions to an ordinary differential equation (ODE) of a polynomial form. These solutions are Taylor or Laurent series with a finite main part. At that, we look at the ODE as an algebraic equation (AE) of $n+2$ variables. It allows to avoid a notion of a formal variation in proves of lemmas. Further we give a notion of a multiple solution to AE and show that each AE with non-multiple solution reduces to special form AE by means of some transformation. At last, we return to special form ODE. For these equations earlier we formulated and proved theorem on sufficient condition of the convergence of its formal solutions.