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Mixed multiplicities of ideals versus mixed volumes of polytopes

Author(s): Ngo Viet Trung; Jugal Verma
Journal: Trans. Amer. Math. Soc. 359 (2007), 4711-4727.
MSC (2000): Primary 52B20, 13D40; Secondary 13H15, 05E99
Posted: May 1, 2007
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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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Additional Information:

Ngo Viet Trung
Affiliation: Institute of Mathematics, Viên Toán Hoc, 18 Hoàng Quôc Viêt, 10307 Hanoi, Vietnam
Email: nvtrung@math.ac.vn

Jugal Verma
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India 400076
Email: jkv@math.iitb.ac.in

DOI: 10.1090/S0002-9947-07-04054-8
PII: S 0002-9947(07)04054-8
Keywords: Mixed volume, mixed multiplicities, multigraded Rees algebra, diagonal algebra, toric rings, Hilbert functions of multigraded algebras
Received by editor(s): March 1, 2005
Received by editor(s) in revised form: March 30, 2005
Posted: May 1, 2007
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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