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Mosc. Math. J., 2010, Volume 10, Number 2, Pages 343–375 (Mi mmj384)  

This article is cited in 9 scientific papers (total in 9 papers)

Mixed volume and an extension of intersection theory of divisors

Kiumars Kaveha, A. G. Khovanskiibca

a Department of Mathematics, University of Toronto, Toronto, Canada
b Institute for Systems Analysis, Russian Academy of Sciences
c Moscow Independent Univarsity

Abstract: Let $\mathbf K_\mathrm{rat}(X)$ be the collection of all non-zero finite dimensional subspaces of rational functions on an $n$-dimensional irreducible variety $X$. For any $n$-tuple $L_1,…,L_n\in\mathbf K_\mathrm{rat}(X)$, we define an intersection index $[L_1,…,L_n]$ as the number of solutions in $X$ of a system of equations $f_1=…=f_n=0$ where each $f_i$ is a generic function from the space $L_i$. In counting the solutions, we neglect the solutions $x$ at which all the functions in some space $L_i$ vanish as well as the solutions at which at least one function from some subspace $L_i$ has a pole. The collection $\mathbf K_\mathrm{rat}(X)$ is a commutative semigroup with respect to a natural multiplication. The intersection index $[L_1,…,L_n]$ can be extended to the Grothendieck group of $\mathbf K_\mathrm{rat}(X)$. This gives an extension of the intersection theory of divisors. The extended theory is applicable even to non-complete varieties. We show that this intersection index enjoys all the main properties of the mixed volume of convex bodies. Our paper is inspired by the Bernstein–Kushnirenko theorem from the Newton polytope theory.

Key words and phrases: system of algebraic equations, mixed volume of convex bodies, Bernstein–Kushirenko theorem, linear system on a variety, Cartier divisor, intersection index.


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MSC: 14C20, 52A39
Received: August 10, 2009

Citation: Kiumars Kaveh, A. G. Khovanskii, “Mixed volume and an extension of intersection theory of divisors”, Mosc. Math. J., 10:2 (2010), 343–375

Citation in format AMSBIB
\by Kiumars~Kaveh, A.~G.~Khovanskii
\paper Mixed volume and an extension of intersection theory of divisors
\jour Mosc. Math.~J.
\yr 2010
\vol 10
\issue 2
\pages 343--375

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    This publication is cited in the following articles:
    1. Kaveh K., Khovanskii A.G., “Moment polytopes, semigroup of representations and Kazarnovskii's theorem”, J. Fixed Point Theory Appl., 7:2 (2010), 401–417  crossref  mathscinet  zmath  isi  elib  scopus
    2. Kiumars Kaveh, A. G. Khovanskii, “Newton polytopes for horospherical spaces”, Mosc. Math. J., 11:2 (2011), 265–283  mathnet  mathscinet
    3. A. G. Khovanskii, “Intersection theory and Hilbert function”, Funct. Anal. Appl., 45:4 (2011), 305–315  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. Kaveh K. Khovanskii A.G., “Newton-Okounkov Bodies, Semigroups of Integral Points, Graded Algebras and Intersection Theory”, Ann. Math., 176:2 (2012), 925–978  crossref  mathscinet  zmath  isi  elib  scopus
    5. Kaveh K. Khovanskii A., “Algebraic Equations and Convex Bodies”, Perspectives in Analysis, Geometry, and Topology: on the Occasion of the 60th Birthday of Oleg Viro, Progress in Mathematics, 296, ed. Itenberg I. Joricke B. Passare M., Birkhauser Verlag Ag, 2012, 263–282  crossref  mathscinet  zmath  isi  scopus
    6. Kiumars Kaveh, Askold G. Khovanskii, “Convex bodies associated to actions of reductive groups”, Mosc. Math. J., 12:2 (2012), 369–396  mathnet  mathscinet  zmath
    7. Malajovich G., “On the Expected Number of Zeros of Nonlinear Equations”, Found. Comput. Math., 13:6 (2013), 867–884  crossref  mathscinet  zmath  isi  elib  scopus
    8. Kaveh K., Khovanskii A.G., “Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov'S Cartier B-Divisors”, Can. Math. Bul.-Bul. Can. Math., 57:3 (2014), 562–572  crossref  mathscinet  zmath  isi  scopus
    9. Harada M. Kaveh K., “Integrable Systems, Toric Degenerations and Okounkov Bodies”, Invent. Math., 202:3 (2015), 927–985  crossref  mathscinet  zmath  isi  elib  scopus
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