11 research outputs found
The maximum likelihood degree of a very affine variety
We show that the maximum likelihood degree of a smooth very affine variety is
equal to the signed topological Euler characteristic. This generalizes Orlik
and Terao's solution to Varchenko's conjecture on complements of hyperplane
arrangements to smooth very affine varieties. For very affine varieties
satisfying a genericity condition at infinity, the result is further
strengthened to relate the variety of critical points to the
Chern-Schwartz-MacPherson class. The strengthened version recovers the
geometric deletion-restriction formula of Denham et al. for arrangement
complements, and generalizes Kouchnirenko's theorem on the Newton polytope for
nondegenerate hypersurfaces.Comment: Improved readability. Final version, to appear in Compositio
Mathematic
A minimal set of generators for the canonical ideal of a non-degenerate curve
We give an explicit way of writing down a minimal set of generators for the
canonical ideal of a non-degenerate curve, or of a more general smooth
projective curve in a toric surface, in terms of its defining Laurent
polynomial.Comment: 14 pages, 6 figures, accepted for publication in Journal of the
Australian Mathematical Societ
Likelihood Geometry
We study the critical points of monomial functions over an algebraic subset
of the probability simplex. The number of critical points on the Zariski
closure is a topological invariant of that embedded projective variety, known
as its maximum likelihood degree. We present an introduction to this theory and
its statistical motivations. Many favorite objects from combinatorial algebraic
geometry are featured: toric varieties, A-discriminants, hyperplane
arrangements, Grassmannians, and determinantal varieties. Several new results
are included, especially on the likelihood correspondence and its bidegree.
These notes were written for the second author's lectures at the CIME-CIRM
summer course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.Comment: 45 pages; minor changes and addition