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 \((3,3)\).