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Moscow Mathematical Journal, 2011, Volume 11, Number 2, Pages 265–283 (Mi mmj421)  

This article is cited in 5 scientific papers (total in 5 papers)

Newton polytopes for horospherical spaces

Kiumars Kaveha, A. G. Khovanskiibcd

a Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA
b Institute for Systems Analysis, Russian Academy of Sciences
c Independent University of Moscow
d Department of Mathematics, University of Toronto, Toronto, Canada
Full-text PDF Citations (5)
References:
Abstract: A subgroup $H$ of a reductive group $G$ is horospherical if it contains a maximal unipotent subgroup. We describe the Grothendieck semigroup of invariant subspaces of regular functions on $G/H$ as a semigroup of convex polytopes. From this we obtain a formula for the number of solutions of a system of equations $f_1(x)=\dots=f_n(x)=0$ on $G/H$, where $n=\dim(G/H)$ and each $f_i$ is a generic element from an invariant subspace $L_i$ of regular functions on $G/H$. The answer is in terms of the mixed volume of polytopes associated to the $L_i$. This generalizes the Bernstein–Kushnirenko theorem from toric geometry. We also obtain similar results for the intersection numbers of invariant linear systems on $G/H$.
Key words and phrases: reductive group, moment polytope, Newton polytope, horospherical variety, Bernstein–Kushnirenko theorem, Grothendieck group.
Received: July 14, 2010; in revised form October 18, 2010
Bibliographic databases:
Document Type: Article
MSC: 14M17, 14M25
Language: English
Citation: Kiumars Kaveh, A. G. Khovanskii, “Newton polytopes for horospherical spaces”, Mosc. Math. J., 11:2 (2011), 265–283
Citation in format AMSBIB
\Bibitem{KavKho11}
\by Kiumars~Kaveh, A.~G.~Khovanskii
\paper Newton polytopes for horospherical spaces
\jour Mosc. Math.~J.
\yr 2011
\vol 11
\issue 2
\pages 265--283
\mathnet{http://mi.mathnet.ru/mmj421}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2859237}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000288967100005}
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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