论文标题
在四面体顶点星上花键的下限
A lower bound for splines on tetrahedral vertex stars
论文作者
论文摘要
四面体复合物在公共顶点相遇的四面体都称为\ textit {vertex star}。顶点恒星是平面三角剖分的自然概括，了解顶点恒星上的花键是分析三元花键的关键步骤。在顶点恒星上计算花纹的尺寸特别困难，其中顶点完全被四面体包围 - 我们称这些\ textIt {note}顶点恒星。由于Alfeld，Neamtu和Schumaker引起的公式提供了$ C^r $ spline的尺寸，其封闭的顶点星星的尺寸至少为$ 3R+2 $。我们表明，该公式是$ c^r $ spline的尺寸的下限，至少$（3R+2）/2 $。我们的证明使用可极性和在顶点恒星的内部面上的点双偶的\ textit {waldschmidt常数}。我们还使用Whiteley的论点来表明，通用封闭顶点星上最多$ $（3R+1）/2 $的唯一的样物是全局多项式。
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.
