B-splines, Polytopes, and Their Characteristic D-modules

Abstract

Given a polytope σ ⊂ ℝ ^m , its characteristic distribution δ_σ generates a D-module which we call the characteristic D-module of σ and denote by M _σ. 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 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.

Key Words:

• B-splines
• D-modules

2010 Mathematics Subject Classification:

• 14F10
• 32C38
• 52B11

ACKNOWLEDGMENTS

I am grateful to my advisor Rikard B\ootgvad, for all the usual reasons; I would also like to thank Rolf Källström and Jan-Erik Björk for helpful discussions.

Notes

Communicated by V. Walther.