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
DOI:
https://doi.org/10.1090/S0002-9947-07-04054-8
Published electronically:
May 1, 2007
MathSciNet review:
2320648
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
MR Author ID:
207806
Email:
nvtrung@math.ac.vn
Jugal Verma
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India 400076
MR Author ID:
177990
Email:
jkv@math.iitb.ac.in
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.