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 Mosc. Math. J., 2011, Volume 11, Number 2, Pages 265–283 (Mi mmj421)

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

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 $f_1(x)=…=f_n(x)=0$ 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.

Full text: http://www.ams.org/.../abst11-2-2011.html
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Bibliographic databases:
MSC: 14M17, 14M25
Received: July 14, 2010; in revised form October 18, 2010
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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://www.ams.org/mathscinet-getitem?mr=2859237} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000288967100005} 

• http://mi.mathnet.ru/eng/mmj421
• http://mi.mathnet.ru/eng/mmj/v11/i2/p265

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. G. Khovanskii, “Intersection theory and Hilbert function”, Funct. Anal. Appl., 45:4 (2011), 305–315
2. Kaveh K., Khovanskii A.G., “Newton-Okounkov Bodies, Semigroups of Integral Points, Graded Algebras and Intersection Theory”, Ann. Math., 176:2 (2012), 925–978
3. 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, eds. Itenberg I., Joricke B., Passare M., Birkhauser Verlag Ag, 2012, 263–282
4. K. Kaveh, A. G. Khovanskii, “Complete intersections in spherical varieties”, Sel. Math.-New Ser., 22:4, SI (2016), 2099–2141