四面体顶点星上样条的一个下界
A lower bound for splines on tetrahedral vertex stars
摘要：
所有四面体在一个公共顶点相交的四面体复体称为\textit{vertex star}。顶点星是平面三角剖分的自然推广，理解顶点星上的样条是分析三元样条的关键一步。计算顶点完全被四面体包围的顶点星上的样条的维数是特别困难的--我们称这些为\textit{closed}顶点星。一个公式，由于阿尔菲尔德，内姆图，和Schumaker给出了尺寸的$C^r$样条封闭的顶点星的程度至少$3r+2$。证明了该公式是次数至少为$(3r+2)/2$的$C^r$样条维数的下界.我们的证明使用非极性和所谓的\textit{Waldschmidt constant}的点集对偶的内面的顶点星。我们还使用一个参数的Whiteley表明，只有样条的程度最多$(3r+1)/2$上的一个通用的闭顶点星是全球性的多项式。
英文摘要：
A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a \textit{vertex star}. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly difficult to compute the dimension of splines on vertex stars in which the vertex is completely surrounded by tetrahedra -- we call these \textit{closed} vertex stars. A formula due to Alfeld, Neamtu, and Schumaker gives the dimension of $C^r$ splines on closed vertex stars of degree at least $3r+2$. We show that this formula is a lower bound on the dimension of $C^r$ splines of degree at least $(3r+2)/2$. Our proof uses apolarity and the so-called \textit{Waldschmidt constant} of the set of points dual to the interior faces of the vertex star. We also use an argument of Whiteley to show that the only splines of degree at most $(3r+1)/2$ on a generic closed vertex star are global polynomials.
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