Open Problems

Explicitly Stated Open Problems in Articles by Bernd Sturmfels

  • \(1704.01910\)
    Geometry of Log-Concave Density Estimation
    Problem 5.8: Which triangulations arise from log-concave MLE with unit weights?

  • \(1703.01660\)
    Sixty-Four Curves of Degree Six
    Conjecture 5.8: Real sextics in \(\mathbb{P}^2\) have between 12 and 306 real bitangents.

  • \(1510.08797\)
    Convexity in Tree Spaces
    Problem 8: Are geodesic triangles in orthant spaces always closed?

  • \(1612.01129\)
    Algebraic Identifiability of Gaussian Mixtures
    Conjecture 2: Moment varieties of bivariate Gaussian mixtures are identifiable.

  • \(1606.02253\)
    How to Flatten a Soccer Ball
    Problem 1: Map the unit 3-ball polynomially onto an arbitrary convex polygon.

  • \(1601.06574\)
    Real Rank Geometry of Ternary Forms
    Conjecture 4.3: The real rank boundary for ternary quartics has degree \(6+27+51\).

  • \(1504.08049\)
    Decomposing Tensors into Frames
    Conjecture 3.3: The fradeco variety for binary forms is determinantal.

  • \(1402.5651\)
    Tropicalization of Del Pezzo Surfaces
    Conjecture 5.3: The 270 trinomials generators are a tropical basis for the universal Cox ideal of cubic surfaces.

  • \(1412.6185\)
    Exponential Varieties
    Problem 7.5: Determine the exponential variety of inverse Hankel matrices.

  • \(1303.1132\)
    Tropicalization of classical moduli spaces
    Conjecture 3.5: The universal abelian surface in \(\mathbb{P}^3 \times \mathbb{P}^8\) is defined by 93 polynomials, nine of bidegree \((4,2)\) and 84 of bidegree \((3,3)\).