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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
50/2018

Moment Varieties of Measures on Polytopes

Kathlén Kohn, Bernd Sturmfels and Boris Shapiro

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.

Received:
Jul 27, 2018
Published:
Jul 27, 2018

Related publications

inJournal
2020 Repository Open Access
Kathlén Kohn, Boris Shapiro and Bernd Sturmfels

Moment varieties of measures on polytopes

In: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 21 (2020), pp. 739-770