The maximum likelihood degree of toric varieties

Varieties with maximum likelihood degree one

We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under monomial maps with finite fibers. The maximum likelihood estimator corresponding to such a variety is Kapranov’s Horn uniformization. This extends Kapranov’s characterization of A-discriminantal hypersurfaces to varieties of arbitrary codimension.

The Maximum Likelihood Degree

Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement. Under suitable hypotheses...

Secant Varieties of Toric Varieties

If X is a smooth projective toric variety of dimension n we give explicit conditions on characters of the torus giving an embedding X →֒ Pr that guarantee dimSecX = 2n+ 1. We also give necessary and sufficient conditions for a general point of SecX to lie on a unique secant line when X is embedded into Pr using a complete linear system. For X of dimension 2 and 3 we give formulas for deg SecX in...

Toric Varieties

4.1.5. The weighted projective space P(q0, . . . ,qn), gcd(q0, . . . ,qn) = 1, is built from a fan in N = Z/Z(q0, . . . ,qn). Let ui ∈ N be the image of ei ∈ Z . The dual lattice is M = {(a0, . . . ,an) ∈ Z n+1 | a0q0+ · · ·+anqn = 0}. Also assume that gcd(q0, . . . , q̂i, . . . ,qn) = 1 for i = 0, . . . ,n. (a) Prove that the ui are the primitive ray generators of the fan giving P(q0, . . . ,qn...

The Geometry of Toric Varieties