The Newton polygon of a planar singular curve and its subdivision

Let a planar algebraic curve C   be defined over a valuation field by an equation F(x,y)=0F(x,y)=0. Valuations of the coefficients of F define a subdivision of the Newton polygon Δ of the curve C.If a given point p is of multiplicity m on C, then the coefficients of F   are subject to certain linear constraints. These constraints can be visualized in the above subdivision of Δ. Namely, we find a distinguished collection of faces of the above subdivision, with total area at least 38m2. The union of these faces can be considered to be the “region of influence” of the singular point p in the subdivision of Δ. We also discuss three different definitions of a tropical point of multiplicity m.