Moment Varieties of Measures on Polytopes

TL;DR

This paper investigates the algebraic structure of moment varieties derived from measures on polytopes, revealing their complex nature and the algebraic relations among moments for different polytope types.

Contribution

It characterizes the prime ideals of moment varieties associated with measures on polytopes, extending understanding beyond special cases like splines and cumulants.

Findings

Identified prime ideals for moment varieties of polytopes.

Connected moment varieties to Hankel determinantal ideals and multisymmetric functions.

Highlighted computational challenges in algebraic geometry related to these varieties.

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.