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 $$(3,3)$$.