The Newton polytope of the discriminant of a quaternary cubic form

2019/09/19 by Lars Kastner, Kastner, Lars, Robert Loewe +1 · 1 citation
Mathematics · #14Q10 #52B55 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1909.08910

openalex publication_date 2019/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Twenty-Seven Questions about the Cubic Surface

Related

• Newton Polytopes Of Nondegenerate Quadratic Forms
• Determinantal representations of the cubic discriminant
• B-facets in dimension 4
• Regular ternary polygonal forms
• Root polytopes, triangulations, and the subdivision algebra, I