A refinement of the Kushnirenko-Bernstein estimate
Algebraic Geometry
2007-12-06 v2 Combinatorics
Abstract
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.
Keywords
Cite
@article{arxiv.0709.3306,
title = {A refinement of the Kushnirenko-Bernstein estimate},
author = {Patrice Philippon and Martin Sombra},
journal= {arXiv preprint arXiv:0709.3306},
year = {2007}
}
Comments
45 pp., 6 figures