# 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 \((3,3)\).