Structure and dimension of multivariate spline space of lower degree on arbitrary triangulation

In this paper, we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules. On the base of analyzing the algebraic and geometric results about singularity of S21(ΔMS), we build the structure of triangulation and give some useful geometric conditions such that Sμ+1μ(Δ) space is singular, and we obtain an algebraic condition which is necessary and sufficient for the singularity of Sμ+1μ(Δ) spaces as well as their dimension formulae. Moreover, the structure matrix of spline spaces over any given partition is defined, which has been used to discuss the structure of S31(Δ) and S21(Δ) spaces over arbitrary triangulation and to prove the nonsingularity of S31(Δ) spaces. This partially settles a conjecture on the singularity of spline spaces in Wang et al., [Multivariate Spline and its Applications, Kluwer Press, Dordrecht, 2002; Academic Press, Beijing, 1994 (in Chinese)]. Meanwhile, the dimension formulae of S31(Δ), S21(Δ) spaces and the dimension formulae of Sμ+1μ(Δ)(μ⩾1) spaces are also given in this paper.