AI 中文总结
该研究构造18三角形复形的单参数族，证明$C^1$三次样条空间维数的Schumaker下界猜想不成立，发现几何依赖的秩缺陷需考虑全局相容性，而非仅局部四星形修正。
AI 中文摘要
自20世纪70年代以来，确定任意非退化平面三角剖分上的$C^1$三次样条空间$S_3^1(\boldsymbol{\tau})$的维数一直是未解决的问题。Schumaker的下界包含了针对奇异内部四星形的局部修正项$\boldsymbol{\tau}$，且人们猜想该下界总能达到。我们通过构造固定18个三角形复形的单参数非退化实现族来反驳这一猜想，其中仅中心顶点在允许区间$I=(-3/4,24/55)$上按$v_6(t)=(t,0)$移动。该族呈现三种不同情形：当$t \neq 1/5,3/83$时，下界达到且$\text{dim}\thinspace S_3^1(\boldsymbol{\tau}(t))=33$；在$t=3/83$处，中心四星形奇异，$\boldsymbol{\tau}=1$，所得维数34恰好由经典局部修正项解释；而在$t=1/5$处，所有内部顶点非奇异且$\boldsymbol{\tau}=0$，但$\text{dim}\thinspace S_3^1(\boldsymbol{\tau}(1/5))=34 > P_{\boldsymbol{\tau}(1/5)}(1,3)=33$。平滑余因子计算表明，$t=3/83$处的相关性局限于中心顶点块，而$t=1/5$处的相关性耦合了全部7个内部顶点循环，尽管每个单独块均具有满行秩。互补的Bernstein--Bézier计算给出相同的维数分布。因此，奇异四星形修正项并未涵盖对$\text{dim}\thinspace S_3^1(\boldsymbol{\tau})$的所有依赖于几何的贡献，还必须考虑真正的全局相容性。
英文摘要
Determining the dimension of the $C^1$ cubic spline space $S_3^1(\mathcal{T})$ on an arbitrary nondegenerate planar triangulation has remained unresolved since the 1970s. Schumaker's lower bound includes a local correction $σ$ for singular interior four-stars, and it was conjectured that this bound is always attained. We disprove this conjecture by constructing a one-parameter family of nondegenerate realizations of a fixed 18-triangle complex, with only the central vertex moving as $v_6(t)=(t,0)$ on the admissible interval $I=(-3/4,24/55)$. The family exhibits three distinct cases. For $t\in I\setminus\{1/5,3/83\}$, the lower bound is attained and $\dim S_3^1(\mathcal{T}(t))=33$. At $t=3/83$, the central four-star is singular, $σ=1$, and the resulting dimension 34 is exactly accounted for by the classical local correction. At $t=1/5$, however, all interior vertices are nonsingular and $σ=0$, yet $\dim S_3^1(\mathcal{T}(1/5))=34>P_{\mathcal{T}(1/5)}(1,3)=33$. The smoothing-cofactor calculation shows that the dependence at $t=3/83$ is confined to the central vertex block, whereas the dependence at $t=1/5$ couples all seven interior vertex cycles even though every individual block has full row rank. A complementary Bernstein--Bézier calculation gives the same dimension profile. Thus the singular-four-star correction does not capture every geometry-dependent contribution to $\dim S_3^1(\mathcal{T})$; genuinely global compatibility must also be taken into account.
