Focus Group on Approximation Theory during Applications of Commutative Algebra Workshop, Fields Institute 2025
Hyperplane arrangements and deletion-restriction methods for splines
Recommended Reading
Other articles we may reference
Slides and notes from introductory talks
Two main questions
- What is the Hilbert function for the module of splines on the chambers of generic hyperplane arrangement in real three-dimensional space? Preliminary notes and approach are outlined in Notes on presenting the non-trivial homology module for splines on fans in three dimensions . One of these notes is to prove that the module of continuous splines on the chambers of a three-dimensional central hyperplane arrangement is free if and only if the hyperplane arrangement is '3-generated.' This gives a starting point if the goal is to inductively construct free spline modules.
- Can we define a 'restriction map' for splines on the chambers of a hyperplane arrangement which yields 'addition-deletion theorems' for splines? Preliminary notes and approach are outlined in A restriction map and applications to freeness, with emphasis on hyperplane arrangements .
Macaulay2 tutorials
Each of the following links will take you to a tutorial that you can run on the web version of Macaulay2.
Macaulay2 supplementary script
The following script contains useful functions. They can be run by downloading the file loading it locally in Macaulay2 or by uploading to the editor in the Macaulay2 web interface.