A lower bound for splines on tetrahedral vertex stars

TL;DR

This paper establishes a new lower bound for the dimension of splines on closed vertex stars in tetrahedral complexes, extending previous formulas and employing algebraic methods like apolarity and Waldschmidt constants.

Contribution

It introduces a lower bound for spline dimensions on closed vertex stars that improves upon existing formulas, using algebraic techniques and geometric arguments.

Findings

The dimension of $C^r$ splines is at least as large as the Alfeld-Neamtu-Schumaker formula for degrees ≥ (3r+2)/2.

For degrees ≤ (3r+1)/2, the only splines are global polynomials.

The proof employs apolarity and Waldschmidt constants to establish the lower bound.

Abstract

A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a \textit{vertex star}. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly difficult to compute the dimension of splines on vertex stars in which the vertex is completely surrounded by tetrahedra -- we call these \textit{closed} vertex stars. A formula due to Alfeld, Neamtu, and Schumaker gives the dimension of $C^r$ splines on closed vertex stars of degree at least $3r+2$. We show that this formula is a lower bound on the dimension of $C^r$ splines of degree at least $(3r+2)/2$. Our proof uses apolarity and the so-called \textit{Waldschmidt constant} of the set of points dual to the interior faces of the vertex star. We also use an argument of Whiteley to show that the only splines of degree at most $(3r+1)/2$ on a generic closed vertex star are global polynomials.