B-Splines, Polytopes and their Characteristic D-Modules

TL;DR

This paper explores the algebraic structure of distributions associated with polytopes and cell complexes, establishing their properties and connections to B-splines through $D$-modules, with methods for computing related ideals.

Contribution

It introduces the characteristic $D$-module for polytopes and cell complexes, linking it to B-splines and providing techniques for calculating $D$-annihilators.

Findings

Characteristic $D$-modules of polytopes and complexes are well-defined and have key properties.

Under certain conditions, the $D$-module direct image matches the B-spline generated module.

Provides methods for computing $D$-annihilator ideals of polytopes.

Abstract

Given a polytope $\sigma\subset \mathbb{R}^m$, its characteristic distribution $\delta_\sigma$ generates a $D$-module which we call the characteristic $D$-module of $\sigma$ and denote by $M_\sigma$. More generally, the characteristic distributions of a cell complex $K$ with polyhedral cells generate a $D$-module $M_K$, which we call the characteristic $D$-module of the cell complex. We prove various basic properties of $M_K$, and show that under certain mild topological conditions on $K$, the $D$-module theoretic direct image of $M_K$ coincides with the module generated by the $B$-splines associated to the cells of $K$ (considered as distributions). We also give techniques for computing $D$-annihilator ideals of polytopes.