The Newton polytope of the discriminant of a quaternary cubic form

TL;DR

This paper explicitly determines all extremal monomials of the discriminant of a quaternary cubic form, linking them to regular triangulations of a dilated tetrahedron and developing efficient computational methods.

Contribution

It provides a complete enumeration of extremal monomials of the discriminant and introduces algorithms for computing associated triangulations and equivalence classes.

Findings

Identified 166,104 extremal monomials of the discriminant.

Established a bijection with D-equivalence classes of triangulations.

Developed efficient algorithms for triangulation and class computation.

Abstract

We determine the $166\,104$ extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with $D$-equivalence classes of regular triangulations of the $3$-dilated tetrahedron. We describe how to compute these triangulations and their $D$-equivalence classes in order to arrive at our main result. The computation poses several challenges, such as dealing with the sheer amount of triangulations effectively, as well as devising a suitably fast algorithm for computation of a $D$-equivalence class.