A family of ideals of minimal regularity and the Hilbert series of \(C^r(\widehat\Delta)\) (Q1366268)

Let \(\Delta\) be a connected finite simplicial complex contained in \(\mathbb{R}^2\). The vector space \(C^r_k (\Delta)\) of splines on \(\Delta\) is the vector space of \(C^r\) functions on \(\Delta\), which are given on each 2-cell of \(\Delta\) by a polynomial of degree less than or equal to \(k\). In this paper, the authors consider situations in which the local cohomology vanishes, and they show that in these situations the dimension of \(C^r_k (\Delta)\) is completely determined by the combinatorial data and the local geometry (the number of lines of distinct slope incident to a given interior vertex).