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- W1788404431 abstract "Consider a planar algebraic curve $C$ over the field $CC {{t}}$ of Puiseux series. Suppose $C$ passes through the point $(1,1)$ with multiplicity $m$. The Newton polygon of $C$ is provided with a subdivision dual to the tropicalization $mathrm{Trop}(C)$ of the curve $C$. I describe properties of such a subdivision. Two of them are the following: a) if the point $(0,0)$ is on an edge of $mathrm{Trop}(C)$ then the subdivision contains a collection of faces with sum of areas at least $frac{1}{2}m^2$, these faces correspond to the vertices of the curve $mathrm{Trop}(C)$ which lie on the maximal long edge passing through the point $(0,0)$; b) if the point $(0,0)$ is a vertex of $mathrm{Trop}(C)$ then the corresponding sum of areas is at least $frac{3}{8}m^2$. Also the properties of the subdivision lead to a definition of tropical singularities in pure tropical terms. These results can be formulated for complex algebraic curves with prescribed asymptotics of coefficients of their equations and can be generalized to positive characteristic greater than $m$." @default.
- W1788404431 created "2016-06-24" @default.
- W1788404431 creator A5073288292 @default.
- W1788404431 date "2013-06-19" @default.
- W1788404431 modified "2023-09-27" @default.
- W1788404431 title "Newton polygon of a planar singular curve" @default.
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