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

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.

DOI: https://doi.org/10.17323/1609-4514-2010-10-2-343-375

Full text: http://www.ams.org/.../abst10-2-2010.html
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MSC: 14C20, 52A39
Received: August 10, 2009
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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
\Bibitem{KavKho10} \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 \mathnet{http://mi.mathnet.ru/mmj384} \crossref{https://doi.org/10.17323/1609-4514-2010-10-2-343-375} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2722802} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000279342400005} 

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Citing articles on Google Scholar: Russian citations, English citations
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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
2. Kiumars Kaveh, A. G. Khovanskii, “Newton polytopes for horospherical spaces”, Mosc. Math. J., 11:2 (2011), 265–283
3. A. G. Khovanskii, “Intersection theory and Hilbert function”, Funct. Anal. Appl., 45:4 (2011), 305–315
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
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
6. Kiumars Kaveh, Askold G. Khovanskii, “Convex bodies associated to actions of reductive groups”, Mosc. Math. J., 12:2 (2012), 369–396
7. Malajovich G., “On the Expected Number of Zeros of Nonlinear Equations”, Found. Comput. Math., 13:6 (2013), 867–884
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
9. Harada M. Kaveh K., “Integrable Systems, Toric Degenerations and Okounkov Bodies”, Invent. Math., 202:3 (2015), 927–985
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