2010/07/24 by Kiumars Kaveh, Kaveh, Kiumars, Askold Khovanskiĭ +2 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #math.AG #msc:14M17 #msc:14M25
paper · pdf · doi:10.48550/arxiv.1007.4270
17 pages
arxiv created 2010/07/24 · arxiv updated 2010/07/27
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 generic system of equations on G/H in terms of mixed volume of polytopes. This generalizes Bernstein-Kushnirenko theorem from toric geometry.