【作者】 许志强；

【导师】 王仁宏；

【作者基本信息】 大连理工大学 ， 计算数学， 2003， 博士

【摘要】 多元样条函数在函数逼近、计算几何及小波等领域中均有较为重要的应用。另一方面，多元样条与基础数学的一些领域，如：抽象代数、代数几何、微分方程及组合数学等，亦有着密切关联。本文主要针对多元样条在应用中及与其相关的基础数学领域中提出的一些问题进行研究。考虑的问题主要为：样条函数空间维数的奇异性、分片代数曲线Bezout定理、整系数线性方程组非负整数解个数及与其相关的组合数学问题。主要工作如下： （1）利用多元样条对散乱数据插值是多元样条一个重要的应用领域。要使插值的多元样条函数存在且唯一，一个必要条件是插值点数与多元样条函数空间的维数一致。另一方面，人们对多元样条维数的研究亦有理论上的兴趣。因此，多元样条维数的研究是较重要的。通常的，我们将剖分△上k次μ阶光滑的样条函数空间记为S_k^μ（△），其维数记为dimS_k^μ（△）。对于Morgan-Scott剖分△_ms，人们发现dimS₂¹(△_ms)严重依赖于剖分的几何特征[58]，这个性质称为样条空间维数的奇异性。因此，Morgan-Scott剖分上样条空间维数的研究一直令人感兴趣。施锡泉在[66]中讨论了dimS₂¹(△_ms)的变化特征，并对高维Morgan-Scott剖分上的样条空间维数奇异性进行了讨论。Diener在[44]中对dimS_2r^r(△_ms)，r＞0进行了讨论，发现dimS_2r^r(△_ms)，r＞0也依赖于剖分的几何特征。除此之外，Diener证明了dimS_d^r(△_ms)，d＞2r不依赖于剖分的几何特征。文[46]中对dimS_2r^r(△_ms)，r＞0进行了进一步研究。但迄今为止我们并不知道当d＜2r时，dimS_d^r(△_ms)的特征。本文提出并讨论了dimS_d^r(△_ms)，d＜2r的奇异性，发现此时样条空间维数的奇异性变化较为复杂。特别地，d≤5/3 r，dimS_d^r(△_ms)不具有奇异性，d＞5/3 r，dimS_d^r(△_ms)奇异性开始出现，且奇异性随着d的增加而增加，当d到11/6 r附近奇异性达最大，随后下降，至2r+1处消失。 （2）分片代数曲线定义为二元样条函数的零点集合。利用样条函数对散乱数据插值时，插值适定的充要条件即为节点数与样条空间维数一致且所有节点不落在同一条分片代数曲线上。分片代数曲线的研究不仅对二元样条插值有重要的意义，而且对于传统的代数曲线理论研究也是较为重要的。众所周知，Bezout定理是传统代数几何的开卷定理。其弱形式是：两条交点有限的代数曲线交点上界不超过其次数的乘积，我们将两条代数曲线次数的乘积称为其Bezout数。鉴于Bezout定理在传统代数曲线理论中的重要地位，考虑Bezou七定理在分片代数曲线中的推广对于分片代数曲线的研究十分重要.施锡泉与王仁宏在文【网中对任意三角剖分上，两条。阶光滑的分片代数曲线交点有限的前提下，相交数所能达到的上界进行了估计，即考虑了。阶光滑的分片代数曲线的Bezout定理.我们首先证明了价9]中提出的关于三角剖分的猜想性结论.指出了分片线性代数曲线与四色猜想之间的内在联系.利用Morgan-scott剖分，指出了分片代数曲线Bezout数的不稳定性.最后，利用与文[60]完全不同的方法一一组合优化的方法，给出了任意三角剖分上任意光滑的分片代数曲线Besout数的上界估计，即考虑了任意阶光滑的分片代数曲线的Bezout定理. （3）离散截断幂定义为线性方程组非负整数解个数，其与多元Box样条和多元截断幂有着密切关系.线性方程组的整数解在多个数学分支中都有重要的应用，离散截断幂的研究亦会对这些学科产生影响.Dalunen与Micchelli在件刃提出了这一概念，并在1551中给出了离散截断幂的分片结构，且给出了离散截断幂解析表达形式的首项.M.Be汰，R.Diaz和S.Robins利用组合的方法在俘司中曾经给出整系数线性方程非负整数解个数的一个解析表达形式.这可看作一元的离散截断幂一个解析表达形式.但这种方法难以推广到多元.我们借助多元截断幂与多元Box样条，给出了多元离散截断幂的一个解析表达形式.贾在【54}中借助离散截断幂证明了stanley提出的一个关于幻方的猜想.其证明该猜想的关键性引理，亦可看作我们给出的关于离散截断幂表达形式结果的一个特殊情形. （4）多面体体积的计算在多个数学领域中均有重要的意义.借助离散截断幂，证明了空间凸多面体的体积等于多元截断幂在一点的函数值.通过这一结论，可用CAGD中快速计算多元样条函数的方法计算凸多面体体积.利用这一方法重新证明了2002年J.Pitman，R.Stanley在!601中给出的关于多面体体积的结论，且证明仅用了初等线性代数知识. （s）利用离散截断幂，重新证明了与有理多面体内整点数目相关的Ehthart拟多项式的一些经典结果.Ehrhart拟多项式的显式公式一直令人感兴趣，Ehrhart本人对整多面体的情形给出了Ehrhart多项式的头两项和最后一项.近来，Pommersherim[6一l，Kantor，Khovans地i[65]，eappell和Shaneson[32]与R.Diaz和5.Robins[42]对这一问题进行了研究.但是，他们主要考虑了有理多面体为整单纯形的情形，且采用的方法主要为代数几何的方法.我们借助多元截断幂与多元Box样条的函数值，给出了对一般有理多面体，Ehrha更多还原

【Abstract】 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更多还原