The Newton polygon of a planar singular curve and its subdivision

Nikita Kalinin

doi:10.1016/j.jcta.2015.09.003

The Newton polygon of a planar singular curve and its subdivision

Nikita Kalinin^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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 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.

Original language	English
Pages (from-to)	226-256
Number of pages	31
Journal	Journal of Combinatorial Theory. Series A
Volume	137
DOIs	https://doi.org/10.1016/j.jcta.2015.09.003
State	Published - 1 Jan 2016
Externally published	Yes

Keywords

Extended newton polyhedron
Lattice width
M-Fold point
Tropical singular point

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10.1016/j.jcta.2015.09.003

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N2 - 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 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.

AB - 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 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.

KW - Extended newton polyhedron

KW - Lattice width

KW - M-Fold point

KW - Tropical singular point

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M3 - 文章

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SN - 0097-3165

VL - 137

SP - 226

EP - 256

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

ER -

The Newton polygon of a planar singular curve and its subdivision

Abstract

Keywords

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