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Mathematics > Algebraic Geometry

arXiv:1807.10258 (math)
[Submitted on 26 Jul 2018 (v1), last revised 7 Jul 2020 (this version, v3)]

Title:Moment Varieties of Measures on Polytopes

Authors:Kathlén Kohn, Boris Shapiro, Bernd Sturmfels
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Abstract:The uniform probability measure on a convex polytope induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex coordinates. We study projective varieties that are parametrized by finite collections of such rational functions. Our focus lies on determining the prime ideals of these moment varieties. Special cases include Hankel determinantal ideals for polytopal splines on line segments, and the relations among multisymmetric functions given by the cumulants of a simplex. In general, our moment varieties are more complicated than in these two special cases. They offer challenges for both numerical and symbolic computing in algebraic geometry.
Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO); Probability (math.PR)
MSC classes: 13P25, 52B11, 14Q15, 52B11, 62H05
Cite as: arXiv:1807.10258 [math.AG]
  (or arXiv:1807.10258v3 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1807.10258

Submission history

From: Kathlén Kohn [view email]
[v1] Thu, 26 Jul 2018 17:37:41 UTC (285 KB)
[v2] Tue, 9 Apr 2019 01:56:25 UTC (287 KB)
[v3] Tue, 7 Jul 2020 10:53:24 UTC (288 KB)
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