# A new bound for smooth spline spaces

### Hal Schenck

Auburn University, USA### Mike Stillman

Cornell University, Ithaca, USA### Beihui Yuan

Cornell University, Ithaca, USA

## Abstract

For a planar simplicial complex $\Delta \subseteq \mathbb{R}^2$, Schumaker proves in [22] that a lower bound on the dimension of the space $C^r_k(\Delta)$ of planar splines of smoothness $r$ and degree $k$ on $\Delta$ is given by a polynomial $P_\Delta(r,k)$, and Alfeld–Schumaker show in [2] that $P_\Delta(r,k)$ gives the correct dimension when $k \ge 4r+1$. Examples due to Morgan–Scott, Tohaneanu, and Yuan show that the equality $\Crk =P_\Delta(r,k)$ can fail for $k \in \{2r, 2r+1\}$. In this note we prove that the equality $\Crk= P_\Delta(r,k)$ cannot hold in general for $k \le (22r+7)/10$.

## Cite this article

Hal Schenck, Mike Stillman, Beihui Yuan, A new bound for smooth spline spaces. J. Comb. Algebra 4 (2020), no. 4, pp. 359–367

DOI 10.4171/JCA/43