A lower bound for the dimension of tetrahedral splines in large degree

TL;DR

This paper establishes a new lower bound for the dimension of large-degree, differentiably continuous tetrahedral splines, which can be exact under generic vertex configurations, using algebraic methods.

Contribution

It introduces a novel algebraic formula for the lower bound of spline space dimensions that is accurate in large degree for generic vertex arrangements.

Findings

The formula provides an exact dimension in large degree for generic vertex positions.

Existing bounds diverge significantly from actual dimensions in large degree.

The approach uses commutative and homological algebra techniques.

Abstract

We derive a formula which is a lower bound on the dimension of trivariate splines on a tetrahedral partition which are continuously differentiable of order $r$ in large enough degree. While this formula may fail to be a lower bound on the dimension of the spline space in low degree, we illustrate in several examples considered by Alfeld and Schumaker that our formula may give the exact dimension of the spline space in large enough degree if vertex positions are generic. In contrast, for splines continuously differentiable of order $r>1$, every lower bound in the literature diverges (often significantly) in large degree from the dimension of the spline space in these examples. We derive the bound using commutative and homological algebra.