This article is cited in 5 papers

Newton polytopes for horospherical spaces

Kiumars Kaveh^a, A. G. Khovanskii<sup>bcd

a</sup> 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 $f_1(x)=\dots=f_n(x)=0$ 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.

MSC: 14M17, 14M25

Received: July 14, 2010; in revised form October 18, 2010

Language: English

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