Josephine Yu

THE HYPERDETERMINANT AND TRIANGULATIONS OF THE 4-CUBE

PETER HUGGINS, BERND STURMFELS,JOSEPHINE YU AND DEBBIE S. YUSTERAbstract. The hyperdeterminant of format 2×2×2×2 is a polynomialof degree 24 in 16 unknowns which has 2894276 terms. We compute theNewton polytope of this polynomial and the secondary polytope of the 4-cube. The 87959448 regular triangulations of the 4-cube are classified into25448 D-equivalence classes, one for each vertex of the Newton polytope.The 4-cube has 80876 coarsest regular subdivisions, one for each facet of thesecondary polytope, but only 268 of them come from the hyperdeterminant.

Tropical Polytopes and Cellular Resolutions

Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This viewpoint gives rise to a family of cellular resolutions of monomial ideals that generalize the hull complex of Bayer and Sturmfels [Bayer and Sturmfels 98], instances of which improve upon the hull resolution in the sense of being smaller. We also suggest a new definition of a face of a tropical polytope, which has nicer properties than previous definitions; we give examples and provide many conjectures and directions for further research in this area.

An implicitization challenge for binary factor analysis

We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant in (Drton et al., 2009, Ch. VI, Problem 7.7). The model is obtained from the undirected graphical model of the complete bipartite graph K2,4 by marginalizing two of the six binary random variables. We present algorithms for computing the Newton polytope of its defining equation by parallel walks along the polytope and its normal fan. In this way we compute vertices of the polytope. Finally, we also compute and certify its facets by studying tangent cones of the polytope at the symmetry classes of vertices. The Newton polytope has 17214912 vertices in 44938 symmetry classes and 70646 facets in 246 symmetry classes.

Tropical Implicitization and Mixed Fiber Polytopes

The software TrIm offers implementations of tropical implicitization and tropical elimination, as developed by Tevelev and the authors. Given a polynomial map with generic coefficients, TrIm computes the tropical variety of the image. When the image is a hypersurface, the output is the Newton polytope of the defining polynomial. TrIm can thus be used to compute mixed fiber polytopes, including secondary polytopes.

Computing tropical resultants

Abstract We fix the supports A = ( A 1 , … , A k ) of a list of tropical polynomials and define the tropical resultant TR ( A ) to be the set of choices of coefficients such that the tropical polynomials have a common solution. We prove that TR ( A ) is the tropicalization of the algebraic variety of solvable systems and that its dimension can be computed in polynomial time. The tropical resultant inherits a fan structure from the secondary fan of the Cayley configuration of A , and we present algorithms for the traversal of TR ( A ) in this structure. We also present a new algorithm for recovering a Newton polytope from the support of its tropical hypersurface. We use this to compute the Newton polytope of the sparse resultant polynomial in the case when TR ( A ) is of codimension 1. Finally we consider the more general setting of specialized tropical resultants and report on experiments with our implementations.

Stable intersections of tropical varieties

We give several characterizations of stable intersections of tropical cycles and establish their fundamental properties. We prove that the stable intersection of two tropical varieties is the tropicalization of the intersection of the classical varieties after a generic rescaling. A proof of Bernstein’s theorem follows from this. We prove that the tropical intersection ring of tropical cycle fans is isomorphic to McMullen’s polytope algebra. It follows that every tropical cycle fan is a linear combination of pure powers of tropical hypersurfaces, which are always realizable. We prove that every stable intersection of constant coefficient tropical varieties defined by prime ideals is connected through codimension one. We also give an example of a realizable tropical variety that is connected through codimension one but whose stable intersection with a hyperplane is not.

The space of tropically collinear points is shellable

AbstractThe spaceTd;n ofn tropically collinear points in a fixed tropical projective spacen

Generalized Permutohedra from Probabilistic Graphical Models

mathbb{T}mathbb{P}^{d - 1}