Smooth planar rr-splines of degree 2r2r

In \cite{as}, Alfeld and Schumaker give a formula for the dimension of the space of piecewise polynomial functions (splines) of degree $d$ and smoothness $r$ on a generic triangulation of a planar simplicial complex $\Delta$ (for $d \ge 3r+1$) and any triangulation (for $d\geq 3r+2$). In \cite{ss}, it was conjectured that the Alfeld-Schumaker formula actually holds for all $d \ge 2r+1$. In this note, we show that this is the best result possible; in particular, there exists a simplicial complex $\Delta$ such that for any $r$, the dimension of the spline space in degree $d=2r$ is not given by the formula of \cite{as}. The proof relies on the explicit computation of the nonvanishing of the first local cohomology module described in \cite{ss2}.Comment: 6 pages, 1 figur

A note on the dimension of the bivariate spline space over the Morgan-Scott triangulation

Abstract. In [D. Diener, SIAM J. Numer. Anal., 27 (1990), pp. 543–551], a conjecture on the dimension of the bivariate spline space Sr 2r (△) over the Morgan–Scott triangulation was posed. In this paper, it is proved that the conjecture should be modified for all even r>2