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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Mixed multiplicities of ideals versus mixed volumes of polytopes


Authors: Ngo Viet Trung and Jugal Verma
Journal: Trans. Amer. Math. Soc. 359 (2007), 4711-4727
MSC (2000): Primary 52B20, 13D40; Secondary 13H15, 05E99
Published electronically: May 1, 2007
MathSciNet review: 2320648
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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: http://dx.doi.org/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
Published electronically: May 1, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.