The tropical discriminant of a polynomial map on a plane

TL;DR

This paper introduces a combinatorial method to compute the tropical discriminant of polynomial maps on the plane, facilitating the calculation of the Newton polytope for complex polynomial discriminants.

Contribution

It presents a novel combinatorial procedure for computing the tropical curve of the discriminant for generic polynomial maps on the plane.

Findings

Provides an explicit method for computing the tropical discriminant

Enables calculation of the Newton polytope of the discriminant

Applies to polynomial maps over Puiseux series fields

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.