The hyperdeterminant and triangulations of the 4-cube

Abstract

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

High-dimensional microbiome interactions shape host fitness

“…The gut microbiome of the fruit fly Drosophila melanogaster is an effective combinatorial model because as few as five species of bacteria consistently inhabit the gut of wild and laboratory flies, [20][21][22] yielding 2 5 possible combinations of species. Here, we isolated the five core fly gut bacteria in culture, constructed germ-free flies by bleaching the embryos, and reinoculated the newly emerged adult flies via continuous feeding with defined flora using established protocols.…”

“…In general, however, it is an open problem to determine all circuits in an n-cube, resp. in an n-genotype system, as the number increases dramatically with n, see [5] where the 4-dimensional case has been considered. Moreover, there is not a clear biological interpretation of all possible circuits of the n-cube.…”

“…Microbiome impacts on host health can shape key aspects of fitness, such as development, 1 fecundity, 2 and lifespan, 3,4 while the host in turn can shape the microbiome. 5 However, complex interactions between microbes can make downstream effects unpredictable, such as when toxin-producing Clostridium species cause pathogenesis after antibiotics reduce gut diversity. 6 A pressing need exists to deconstruct the effects of gut diversity on host health, and new mathematical approaches are needed to quantify the complex patterns of interactions.…”

Faithful tropicalisation and torus actions

“…4. This is the spherical complex of the normal fan of the bipyramid over a tetrahedron from [14,Section 2]. The matrix A sends τ = (ρ, δ, ν) ∈ R 6 to the 2 × 2 × 2-array with entries (ρ i + δ j + ν k ) i jk .…”

The 3 × 3 × 3 Hyperdeterminant as a Polynomial in the Fundamental Invariants for $${{SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})}}$$ S L 3 ( C ) × S L 3 ( C ) × S L 3 ( C )

“…In theory Schläfli's method may be used to compute the 3×3×3 hyperdeterminant, see Gelfand et al [15] (1994); this computation turns out to require a large amount of memory, but it may still be useful for evaluating the hyperdeterminant and was used by one of us [25] (2012). In general, these computations become very large very quickly; see Huggins et al [16] (2008) for explicit results on the 2 × 2 × 2 × 2 hyperdeterminant. Domokos and Drensky [12, Remark 2(ii)] (2012) computed the single defining relation of the algebra of invariants for the action of SL(3, C) × SL(3, C) on triples of 3 × 3 matrices, and provided an alternative proof of Vinberg's result.…”