Focus Group on Approximation Theory during Applications of Commutative Algebra Workshop, Fields Institute 2025
Hyperplane arrangements and deletion-restriction methods for splines

Recommended Reading

• The algebra of splines: duality, group actions, and homology . Survey article on splines and the recommended reading for our focus group.

Other articles we may reference

• Generalized Splines on Arbitrary Graphs . In this article the setup for generalized splines on graphs -- which we plan to use -- was first explained.
• Module Bases for Multivariate Splines . A generalization of Saito's criterion from hyperplane arrangements to the context of splines.
• The Euler multiplicity and addition-deletion theorems for multiarrangements. It would be great to find addition-deletion theorems for splines that mirror what has been done for multiarrangements.

Slides and notes from introductory talks

• Slides from Nelly's introductory talk: Commutative algebra in approximation theory.
• Notes for Michael's talk: Splines and hyperplane arrangements.

Two main questions

1. 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.
2. 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.

• Quadrants_And_Octants . This tutorial goes through a couple of basic examples (splines on quadrants and octants).
• Hyperplanes_And_Zonotopes . This tutorial goes through the process of computing splines on the chambers of a hyperplane arrangement.
• Ziegler_Pair . This tutorial goes through the correspondence between three-generated arrangements and continuous splines on chambers. The theoretical content for this is in the writeup Notes on presenting the non-trivial homology module for splines on fans in three dimensions .

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.

• Scripts_for_splines_on_hyperplanes.m2