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) = 0. Valuations of the coefficients of F define a subdivision of the Newton polygon ∆ 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 ∆. Namely, we find a distinguished collection of faces of the above subdivision, with total area at least 3 8 m2 . In a sense, the union of these faces is “the region of influence” of the singular point p on the subdivision of ∆. Also, we discuss three different definitions of a tropical point of multiplicity m.

• Type: Journal
• Year: 2016
• Where: JCT
• Authors: Nikita Kalinin
• Keywords: JCT 2016