AI 中文总结
该研究针对平面T网格上特定双次数的多项式样条空间，通过同调分解推导维数公式，证明加权删除定理与CNDC的性质，给出维数稳定性条件及校正项上界并与已有结果比较。
AI 中文摘要
我们研究平面T网格上具有指定光滑度阶的双次数$(m,m')$多项式样条空间的维数与维数稳定性。同调维数公式将样条维数表示为欧拉示性数项与校正项之和。对于固定的有序双次数，欧拉示性数项由网格结构和指定的光滑度阶决定。校正项可表示为附属于极大内部段（MIS）的系数空间的商。我们证明了加权删除定理：当可用顶点关系生成某个MIS的系数空间时，可从商中移除该MIS的求和项而不改变校正项，此操作既不改变T网格也不改变其链复形。重复删除后得到加权完全不可对角化分量（CNDC）。我们证明加权CNDC与符合条件的MIS的删除顺序无关，且在结构类的对应对中相同。因此校正项可仅用加权CNDC中的MIS表示，同时保留原始T网格的所有关系。维数稳定性等价于剩余关系空间维数的不变性，特别地，空的加权CNDC足以保证稳定性。我们还推导了剩余校正项的上界，并与基于所有MIS的Mourrain上界进行了比较。
英文摘要
We study the dimension and dimensional stability of polynomial spline spaces of bi-degree $(m,m')$ with prescribed smoothness orders over planar T-meshes. The homological dimension formula writes the spline dimension as the sum of an Euler characteristic term and a correction term. For a fixed ordered bi-degree, the Euler characteristic term is determined by the mesh structure and the prescribed smoothness orders. The correction term can be written as a quotient of coefficient spaces attached to maximal interior segments (MISs). We prove a weighted deletion theorem: when the available vertex relations generate the coefficient space of an MIS, its summand can be removed from the quotient without changing the correction term. This operation changes neither the T-mesh nor its chain complexes. Repeating the deletion leaves a weighted completely non-diagonalizable component (CNDC). We prove that the weighted CNDC is independent of the order in which eligible MISs are removed and is the same for corresponding pairs in the structural class. The correction term can therefore be represented using only the MISs in the weighted CNDC, while all relations from the original T-mesh are retained. Dimensional stability is then equivalent to constancy of the dimension of the remaining relation space. In particular, an empty weighted CNDC is sufficient for stability. We also derive an upper bound for the remaining correction term and compare it with Mourrain's upper bound based on all MISs.
