The tropical discriminant of a polynomial map on a plane

2022/02/10 by Boulos El Hilany, Hilany, Boulos El
Computer Science · Mathematics · #58K15 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Polynomial and algebraic computation #Primary 14D06 #Secondary: 14T20

paper · pdf · doi:10.48550/arxiv.2202.05052

openalex publication_date 2022/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The discriminant of a polynomial map is central to problems from affine geometry and singularity theory. Standard methods for characterizing it rely on elimination techniques that can often be ineffective. This paper concerns polynomial maps on the two-dimensional torus defined over a field of Puiseux series. We present a combinatorial procedure for computing the tropical curve of the discriminant of maps determined by generic polynomials with given supports. Our results enable one to compute the Newton polytope of the discriminant of complex polynomial maps on the plane.

Citations

• Brief introduction to tropical geometry
• A Field of Generalised Puiseux Series for Tropical Geometry
• Enumerative tropical algebraic geometry in R2

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