发表机构
School of Mathematical Sciences, Dalian University of Technology; Dalian University of Technology（大连理工大学数学科学学院; 大连理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
构造一个41顶点56三角形的三角剖分，证明其上三次C^1样条无法插值所有顶点数据，通过显式线性关系否证Alfeld猜想，无需矩阵秩计算。
AI 中文摘要
我们构造了一个多边形圆盘的非退化协调直线三角剖分，在该剖分上，某些顶点数据不存在总次数至多为三的连续可微分段多项式插值函数。该三角剖分有41个顶点和56个三角形，其内部顶点诱导的图是一个具有八条长度为二的臂的树。在此三角剖分上的每个三次 $C^1$ 样条都满足其顶点值之间一个具有整数系数的显式线性关系。我们通过 Bernstein–Bézier 光滑性方程的加权和推导出该关系，并使用有理坐标和权重进行验证。这否证了 Alfeld 关于顶点插值的猜想。不存在性证明不需要矩阵秩计算。另一次精确算术计算给出顶点求值映射的秩为40，样条空间维数为107，达到了该三角剖分的经典维数下界。
英文摘要
We construct a nondegenerate conforming straight-line triangulation of a polygonal disk on which some vertex data admit no continuously differentiable piecewise polynomial interpolant of total degree at most three. The triangulation has 41 vertices and 56 triangles, and the graph induced by its interior vertices is a tree with eight arms of length two. Every cubic $C^1$ spline on this triangulation satisfies an explicit linear relation with integer coefficients among its vertex values. We derive the relation by a weighted sum of Bernstein--Bézier smoothness equations and verify it using rational coordinates and weights. This disproves Alfeld's conjecture on vertex interpolation. The nonexistence proof does not require a matrix rank computation. A separate computation in exact arithmetic gives rank 40 for the vertex evaluation map and dimension 107 for the spline space, attaining the classical dimension lower bound for this triangulation.
Comments9 pages, 1 figure. Exact verification code is included as an ancillary Python file
