The Newton polygon of a planar singular curve and its subdivision

Abstract

Let a planar algebraic curve $C$ be defined over a valuation field by an equation $F(x,y)=0$. Valuations of the coefficients of $F$ define a subdivision of the Newton polygon $\Delta$ of the curve $C$. If a given point $p$ is of multiplicity $m$ for $C$, then the coefficients of $F$ are subject to certain linear constraints. These constraints can be visualized on the above subdivision of $\Delta$. Namely, we find a distinguished collection of faces of the above subdivision, with total area at least $\frac{3}{8}m^2$. In a sense, the union of these faces in "the region of influence" of the singular point $p$ on the subdivision of $\Delta$. Also, we discuss three different definitions of a tropical point of multiplicity $m$.