A Homological Decomposition for the Dimension and Dimensional Stability of Polynomial Spline Spaces over T-Meshes

Abstract

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.