Multiplicity of the solutions of a differential polynomial Original Research Article

We propose two notions of multiplicity for the solutions (Puiseux series) of differential polynomials. The first one is based on the Newton-Puiseux Fine polygon and it is of a combinatorial nature. The second one is of a differential nature and extends the standard notion of algebraic multiplicity. We show that both concepts, defined independently, are equivalent. We also establish the limit of the multiplicity for a given differential polynomial, and we give a precise procedure to reduce the multiplicity. Finally, we shall present the problem of computability of these entities, which still remains open.