Moment Varieties of Measures on Polytopes
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.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 DOI:
 10.48550/arXiv.1807.10258
 arXiv:
 arXiv:1807.10258
 Bibcode:
 2018arXiv180710258K
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 Mathematics  Probability;
 13P25;
 52B11;
 14Q15;
 52B11;
 62H05