Mixed multiplicities of ideals versus mixed volumes of polytopes

Trung, Ngo; Verma, Jugal

doi:10.1090/S0002-9947-07-04054-8

Mixed multiplicities of ideals versus mixed volumes of polytopes
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by Ngo Viet Trung and Jugal Verma PDF

Trans. Amer. Math. Soc. 359 (2007), 4711-4727 Request permission

Abstract:

The main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as Samuel’s multiplicities. In particular, we can give a purely algebraic proof of Bernstein’s theorem which asserts that the number of common zeros of a system of Laurent polynomial equations in the torus is bounded above by the mixed volume of their Newton polytopes.

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