AI 中文总结
本文针对不含可消失T l-边的T网格，通过解耦技术推导了最高光滑度阶双次数多项式样条空间的尖锐维数界，该新公式与Mourrain同调维数公式一致且下界更优。
AI 中文摘要
T网格$\boldsymbol{\tau}$上双次数$(d_1,d_2)$的多项式样条空间的维数，在最高光滑度阶为$(d_1-1,d_2-1)$时，同时取决于网格拓扑与几何构型。在T网格$\boldsymbol{\tau}$的T连通分支不含可消失T l-边的假设下，本文推导了该多项式样条空间维数的显式上下界。通过在T网格$\boldsymbol{\tau}$的完全不可对角化分量（CNDC）中引入解耦技术，本文分离了紧密耦合的多顶点约束，并将全局保形条件转化为沿每条内部大边的局部线性方程。基于该解耦技术，本文提出了多项式样条空间的新维数公式，并由此得到了维数的尖锐上下界；该界的尖锐性体现在，具有相同拓扑的T网格的不同几何实现均可达到该多项式样条空间维数的上下界。本文进一步证明，该新公式与Mourrain的同调维数公式一致，且通过本文方法得到了更尖锐的下界。
英文摘要
The dimension of a polynomial spline space of bi-degree $(d_1,d_2)$ over a T-mesh $\mathscr{T}$ with the highest order of smoothness $(d_1-1,d_2-1)$ depends on both mesh topology and geometric configurations. Under the assumption that the T-connected components of the T-mesh $\mathscr{T}$ contain no vanishable T $l$-edges, we develop explicit upper and lower bounds of the dimension of the polynomial spline space. By introducing a decoupling technique within the completely non-diagonalizable component (CNDC) of the T-mesh $\mathscr{T}$, we separate tightly coupled multi-vertex constraints and transform global conformality conditions into localized linear equations along each interior large edge. Based on the decoupling technique, a new dimension formula of the polynomial spline space is then presented, and from which sharp upper and lower bounds of the dimension are obtained. The bounds are sharp in the sense that different geometric realizations of T-meshes with the same topology can attain the lower and upper bounds for the dimension of the polynomial spline space. We further prove that the new formula is consistent with Mourrain's homological dimension formula, and a sharper lower bound is obtained by our method.
