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The maximum likelihood degree of a very affine variety

Part of: Local theory Polytopes and polyhedra Cycles and subschemes

Published online by Cambridge University Press:  24 May 2013

June Huh
June Huh*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA email junehuh@umich.edu
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Abstract

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.

Keywords

maximum likelihood degreelogarithmic differential formChern–Schwartz–MacPherson class

MSC classification

Secondary: 14B05: Singularities 14C17: Intersection theory, characteristic classes, intersection multiplicities 52B40: Matroids (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 8 , August 2013 , pp. 1245 - 1266
Copyright
© The Author(s) 2013 

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