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A subgroup H of a reductive group G is horospherical if it contains a maximalunipotent subgroup. We describe the Grothendieck semigroup of invariantsubspaces of regular functions on G/H as a semigroup of convex polytopes. Fromthis we obtain a formula for the number of solutions of a generic system ofequations on G/H in terms of mixed volume of polytopes. This generalizesBernstein-Kushnirenko theorem from toric geometry.

Newton polytopes for horospherical spaces