混合样条空间的正则性
Regularity of Mixed Spline Spaces
摘要：
本文给出了中心多面体复形$\mathcal{P}\subset\mathbb{R}^3$上混合样条代数$C^α(\mathcal{P})$的正则性的界。作为结果，我们绑定的最大整数$d$（假设数）的希尔伯特多项式$HP(C^α(\mathcal{P}),d)$不同意的希尔伯特函数$HF(C^α(\mathcal{P}),d)=\dim C^α(\mathcal{P})_d$。多项式$HP(C^α(\mathcal{P}),d)$已在[DiPasquale 2014]中计算，建立在[McDonald-Schenck 09]和[Geramita-Schenck 98]的基础上。因此，得到的正则性界限表明，当一个已知的多项式给出正确的尺寸样条空间$C^α(\mathcal{P})_d$。在所有光滑参数相等的单纯情况下，我们恢复了最初由于[Hong 91]和[Ibrahim and Schumaker 91]的界限。
英文摘要：
We derive bounds on the regularity of the algebra $C^α(\mathcal{P})$ 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 $HP(C^α(\mathcal{P}),d)$ disagrees with the Hilbert function $HF(C^α(\mathcal{P}),d)=\dim C^α(\mathcal{P})_d$. The polynomial $HP(C^α(\mathcal{P}),d)$ 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 $C^α(\mathcal{P})_d$. In the simplicial case with all smoothness parameters equal, we recover a bound originally due to [Hong 91] and [Ibrahim and Schumaker 91].
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