2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74838The 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.19 pages, to appear in Trans. Amer. Math. SocCommutative AlgebraAlgebraic Geometry13D40; 52B20text