Universal Dimension Formula for Planar $C^r$ Splines at Degree $3r+1$

Abstract

We prove that Schumaker's lower bound gives the dimension of $S_{3r+1}^r(Δ)$ for every nondegenerate triangulation $Δ$ of a polygonal disk in $\mathbb{R}^2$ and every integer $r\ge3$. After accounting for the local vertex contributions, we reduce the remaining dimension defect to compatible vertex functionals. Apolarity associates a unique polynomial with each face, and compatibility forces differences of adjacent face polynomials to contain powers of both endpoint forms. We eliminate the leading residual coefficient using Hermite relations when $r$ is odd and an additional coefficient comparison when $r$ is even. A uniform induction then raises the endpoint powers until all face differences vanish; the zero boundary face potentials imply that every vertex functional vanishes and the global defect is zero. Together with the known cases $r=1,2$ and Hong's higher-degree theorem, this yields the dimension formula for all $r\ge1$ and $d\ge3r+1$.