| Multivariate splines are important tools in approximation theory, CAGD and wavelets. Moreover, multivariate splines are closely related with some topics in pure mathematics, such as, abstract algebraic, algebraic geometry and combinatorics. In this thesis, some problems about multivariate splines are discussed.(1) The dimension of multivariate splines spaces is very important. In general, multivariate splines space with smoothness order r and degree d on A is denoted by Srd(A). The dimension of Srd(Δ) is denoted by dimSrd(Δ). Morgan and Scott gave an example of a triangulation Δms on which the dimension of S12(Δms depends on such geometric conditions. This problem has caused much interest of many experts and scholars in multivariate splines. In [66], dimS12(Δms) is discussed. In [44], Diener showed that the dimension of Sr2r(Ams) depended on the same type of geometric condition for all values of r > 1 and the dimension of Srd(Δms), d > 2r was stable. But there is not any result about the dimension of srd(ΔmS),D < 2r - 1. In this thesis, the dimension of Srd(Δms),d < 2r - 1 is discussed. The following results are presented: When d < 5/3r,dimSrd(Δms) = fd+2 2). When d > 5/3r, dzmSrd(Δms) becomes singularity. The singularity of dimSJ(Ams) increases along with d increasing firstly, then it decreases along with d increasing. Near 6r, the singularity reaches the maximum. When d > 2r, the singularity vanishes.(2) A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. It is obvious that the piecewise algebraic curve is a generalization of the classical algebraic curve. The piecewise algebraic curve is not only very important for the interpolation by the bivariate splines (cf. [82]), but also a useful tool for studying traditional algebraic curves (cf.[51]). It is well known that Bezout's theorem is an important and classical theorem in the algebraic geometry[81]. Its weak form says that two algebraic curves will have infinitely many intersection points provided that the number of their intersection points more than the product of their degrees. Denote by BN = BN(m, r; n, t; A) the so-called Bezout's number. It means any two piecewise algebraic curvesmust have infinitely many intersection points provided that they have more thanBN intersection points. In [69], an upper boundary of BN = BN(m, 0; n, 0; Δ) is presented. In this thesis, a conjecture about triangulation which is presented in [69] is confirmed. The relation between Piecewise Algebraic Curve and Four Color Conjecture is presented. By Morgan-Scott triangulation, we show the Be-zout number of piecewise algebraic curve is instability. By using the combinatorial method which is different with the method in [69] an upper bound of the BN(m, r; n, t, Δ) is presented.(3) Discrete truncated powers are defined as the number of nonnegative integer solution of linear equations. It is closely related with multivariate splines and multivariate truncated powers. It is well known that the number of nonnegative integer solutions of linear equations is very important in some mathematical subjects. The research about discrete truncated powers will influence the subjects. In [37], the concept of discrete truncated powers was presented by Dahmen and Micchelli. In [38] , Dahmen and Micchelli showed the piecewise structure of discrete truncated powers. The leading term of discrete truncated powers were also presented. In [24], by combinatorics, an explicit formulation of the number of nonnegative integer solutions of linear equation was presented. The formulation can be considered as the explicit formulation of discrete truncated powers of one variable. But the method in [24] is difficult to be generalized to several variables. In this thesis, by multivariate truncated powers and multivariate Box splines, an explicit formulation of discrete truncated powers of several variables is presented. In [54], a conjecture about magic square was confirmed. The key Lemma for confirming the conjecture is a spe...
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