The hyperdeterminant and triangulations of the 4-cube

Abstract

The hyperdeterminant of format 2 × 2 × 2 × 2 2\times 2 \times 2 \times 2 is a polynomial of degree 24 24 in 16 16 unknowns which has 2894276 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4 4 -cube. The 87959448 87959448 regular triangulations of the 4 4 -cube are classified into 25448 25448 D D -equivalence classes, one for each vertex of the Newton polytope. The 4 4 -cube has 80876 80876 coarsest regular subdivisions, one for each facet of the secondary polytope, but only 268 268 of them come from the hyperdeterminant.