Regularity of Mixed Spline Spaces

We derive bounds on the regularity of the algebra $C^{\alpha }(\mathcal{P})$ of mixed splines over a central polytopal complex $\mathcal{P}\subset {\mathbb{R}}^3$. As a consequence we bound the largest integer $d$ (the postulation number) for which the Hilbert polynomial $HP(C^{\alpha }(\mathcal{P}),d)$ disagrees with the Hilbert function $HF(C^{\alpha }(\mathcal{P}),d)=\dim C^{\alpha }(\mathcal{P}{)}_d$. The polynomial $HP(C^{\alpha }(\mathcal{P}),d)$ has been computed in [DiPasquale 2014], building on [McDonald-Schenck 09] and [Geramita-Schenck 98]. Hence the regularity bounds obtained indicate when a known polynomial gives the correct dimension of the spline space $C^{\alpha }(\mathcal{P}{)}_d$. In the simplicial case with all smoothness parameters equal, we recover a bound originally due to [Hong 91] and [Ibrahim and Schumaker 91].