From spline approximation to Roth's equation and Schur functors (Q361851)

Let \(\Delta\) be a connected finite simplicial complex whose realization is a topological disk in \(\mathbb R^2\). Consider the space \(C^r_d(\Delta)\) of splines of smoothness \(r\in \mathbb{N}_0\) and degree \(d\in \mathbb{N}\) over \(\Delta\). The authors prove the conjecture by Schenck and Stiller that the dimension of \(C^r_d(\Delta)\) is equal to the Alfeld-Schumaker formula for any \(r\) and \(d\geq 2r+1\) in the case of a simplicial complex with exactly one non-pseudoedge. The proof combines methods from representation theory, matrix theory, and the theories of commutative and homological algebras.