MULTIPLICITY OF THE SOLUTIONS OF A DIFFERENTIAL POLYNOMIAL

We propose two notions of multiplicity for the solutions (Puiseux seri es) of differential polynomials. The first one is based on the Newton- Puiseux-Fine polygon and it is of a combinatorial nature. The second o ne is of a differential nature and extends the standard notion of alge braic 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 red uce the multiplicity. Finally, we shall present the problem of computa bility of these entities, which still remains open.