A refinement of the Kushnirenko-Bernstein estimate

Details A refinement of the Kushnirenko-Bernstein estimate A refinement of the Kushnirenko-Bernstein estimate

2007-09-20

arxiv.org/abs/0709.3306v2

Mathematics

Algebraic Geometry

45 pp., 6 figures

Scientific paper

A theorem of Kushnirenko and Bernstein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry.

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