Shellability and freeness of continuous splines
Let \(\Delta\) be a pure \(d\)-dimensional simplicial complex and \(C^r(\Delta)\) be the space of piecewise polynomial functions continuously differentiable of order \(r\) on \(\Delta\). A main question is to determine the dimension of the vector space of splines of degree at most \(k\), \(C_k^r(\Delta)\). \textit{L. Billera} and \textit{L. Rose} [Math. Z. 209, No. 4, 485--497 (1992; Zbl 0891.13004)] gave an answer to this question for \(r=0\). One may consider the same problem for a pure \(d\)-dimensional polytopal complex \(\mathcal{P}\subseteq\mathbb{R}^d\). The author provides an example of a shellable polyhedral complex \(\mathcal{P}\) in \(\mathbb{R}^2\) suct that the module \(C^0(\hat{\mathcal{P}})\) is not a free module over the polynomial ring in three variables. He shows that, for a pure, shellable \(d\)-dimensional polytopal complex \(\mathcal{P}\subseteq\mathbb{R}^d\), the freeness of \(C^0(\hat{\mathcal{P}})\) as an \(S\)-module depends on the embedding of in \(\mathbb{R}^d\).
scientific article; zbMATH DE number 6138029
scientific article