Regularity of Mixed Spline Spaces

TL;DR

This paper establishes bounds on the regularity of mixed spline spaces over polytopal complexes, helping determine when the Hilbert polynomial accurately predicts the dimension of spline spaces.

Contribution

It provides new bounds on the regularity of mixed spline spaces, extending previous results and clarifying when the Hilbert polynomial matches the Hilbert function.

Findings

Bounds on the regularity of mixed spline spaces derived.

Identification of the postulation number for spline spaces.

Recovery of classical bounds in the simplicial case with uniform smoothness.

Abstract

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].