【作者】 朱春钢；

【导师】 王仁宏；

【作者基本信息】 大连理工大学 ， 计算数学， 2005， 博士

【摘要】 利用多元样条进行散乱数据插值是计算几何中一个非常重要的课题。但由于多元样条空间的结构不但依赖于剖分拓朴性质,而且紧密地依赖于剖分的几何性质,这就使得对样条空间的插值结点的适定性的研究变得十分复杂。目前样条空间的插值(特别是Lagrange插值)适定性问题始终研究的热点问题。王仁宏为解决这一问题提出了分片代数曲线的概念。对于平面上(复或实平面)单连通区域Ω的剖分Δ,曲线 Z(f):={(x,y)|f(x,y)=0,f∈S_n~μ(Δ)} 称为Ω中关于剖分Δ的n次C~μ分片代数曲线。显然,分片代数曲线是经典代数曲线的自然推广。王仁宏指出:样条空间的Lagrange插值结点组适定的充要条件是这些结点不在同一条非零分片代数曲线上。因此,本质上解决插值结点的适定性问题关键在于研究分片代数曲线,在高维空间里就是研究分片代数簇。除此之外,分片代数曲线(簇)也与CAD、CAGD、CAE等领域中均有较为重要的应用。另一方面,人们发现它也是其他学科研究的一种有效工具。分片代数曲线(簇)作为二元(多元)样条的零点集合,它是代数几何与计算几何中一种新的重要概念,显然也是经典代数曲线(簇)的推广与补充。因此,研究分片代数曲线(簇)具有重要的理论与实用价值。本文的主要工作如下: 首先我们对多元样条空间的三种定义方式进行了回顾,并着重介绍了光滑余因子协调法。给出了分片代数曲线(簇)的定义,并对研究的理论与应用背景进行了阐述。 众所周知,Bezout定理,N(?)ther定理与Cayley-Bacharach定理是经典代数几何的基本定理。将它们推广到分片代数曲线上也有重要的理论与应用意义。王仁宏等对于分片代数曲线的Bezout定理多了大量的研究工作。第二章我们主要是对分片代数曲线的N(?)ther型定理与Cayley-Bacharach定理进行研究。首先对代数曲线的一些概念与主要定理进行了绍,并将一些概念推广到分片代数曲线上。然后对[27]中关于星形剖分下分片代数曲线的N(?)ther型定理改进,并利用贯穿剖分与样条的性质,得到了贯穿剖分下分片代数曲线的N(?)ther型定理。利用此结果与分片代数曲线的Bezout定理,将经典代数几何中的Cayley-Bacharach定理推广到分片代数曲线上,给出了0阶光滑分片代数曲线的Cayley-Bacharach定理分片代数曲线、分片代数簇与分片半代数集的某些问题研究与Hilbert函数,并得到一些有趣的结论. 对分片代数曲线研究的最初根源是二元样条的插值问题,但是将分片代数曲线的理论应用于二元样条插值的研究还非常少.第三章中,我们首先给出了沿分片代数曲线插值的概念.利用第二章中得到的分片代数曲线的N仪her型定理与,我们得到了一种崭新的构造二元样条Lagrange插值适定结点组的方法.它类似于构造一般多项式Lagrange插值适定结点组的迭代方法. 与代数曲线类似,在进行分片代数曲线的绘制时也会遇到很多问题.目前,一般都借助计算机来绘制分片代数曲线.实际上,计算机绘制出来的图形某些时候是不一定准确的.例如,当计算机屏幕显示不出来图形时,你并不能确定曲线就是空集,而且曲线在奇点附近的显示也是非常不精确的.因此对实分片代数曲线进行理论上的研究是非常必要和重要的.第四章主要对实分片代数曲线进行了研究.首先给出了实分片代数曲线的一些性质,然后定义了实二元样条的特征,利用实代数几何与代数学的基本知识,对某些二元样条及其定义的实分片代数曲线进行了研究,并给出了一种实分片代数曲线孤立点的判断方法.为了研究实代数曲线在三角域上的拓扑结构提出了代数曲线局部G一P的概念,利用实多项式的Sturm一Habicht序列,分析了实代数曲线在三角形:域上的正则点与关键点,并给出一种生成实分片代数曲线线性拓扑图的算法. 分片代数簇作为一些多元样条的公共零点集合,同样也是代数几何中一种新的重要概念,是经典代数簇的推广,丰富和发展.它不仅与许多实际问题如:多元样条插值,代数簇的光滑拼接,CAD,CAM和CAGD紧密相联,而且还为研究经典代数几何提供了理论依据.第五章中,我们利用代数几何的有关结果,对分片代数簇进行了研究,得到了分片代数簇的一些性质与维数公式,并且对分片代数簇的坐标环、正则函数与同构定理进行了研究. 半代数集为一些实多项式等式与不等式的公共零点集合.半代数集与半代数函数为实代数几何中的重要内容,在很多方面具有应用(如多项式实根计数,实体造型等).第六章中我们首次引入了分片半代数集的概念.对它的投影稳定性,维数等问题近行了初步的讨论,并给出了分片半代数集的Tarski一Seidenberg基本定理与维数公式.关键词:分片代数曲线;分片代数簇;样条插值;二元样条;多元样条一n一更多还原

【Abstract】 Suppose A is a partition of Ω, where Ω is a simply connected domain in C2 or R2. The curveis called a Cu piecewise algebraic curve. It is obvious that the piecewise algebraic curve is a. generalization of the classical algebraic curve. Wang[1,12,109,110] gave the result that a set of points is the Lagrange interpolation set for Sun(△) if and only if there is no spline g ∈ Sun(△) \ {0} such that it lies on the piecewise algebraic curve g. A piecewise algebraic variety is the zero set of some multivariate splines. It is a kind of generalization of the classical algebraic variety. Then the study of piecewise algebraic curves(varieties) is important for the interpolation by bivariate(multivariate) spline space. Moreover, the piecewise algebraic curve(variety) is not only a myriad of applications in CAD, CAGD, CAE et al., but also a useful tool for studying.traditional algebraic curves and other subjects. In this thesis, some problems about piecewise algebraic curves, piecewise algebraic varieties and piecewise semialgebraic sets are discussed.In chapter 1, we first introduce three methods about studying multivariate splines: the Wang’s method(the smoothing cofactor-conformality method), the Bernstein-Bezier method, and the multivariate B-splines method. Moreover, by Wang’s method, the definition of piecewise algebraic curves(varieties) is given. Last, we present the recent researches on piecewise algebraic curves and piecewise algebraic varieties.It is well known that Bezout’s theorem, Nother’s theorem, and Cayley-Bacharach theorem are important and classical results in algebraic geometry. To generate them to piecewise algebraic curves is important for studying the piecewise algebraic curves and the bivariate spline interpolation problems. Wang et al. have studied the Bezout’s theorem of piecewise algebraic curves in many ways. In chapter 2, we discuss the Nother-type theorem and Cayley-Bacharach theorem of piecewise algebraic curves on some partitions. First, we generate some concepts of algebraic curves to piecewise algebraic curves. Next, we describes the improvement of theNother-type theorem of piecewise algebraic curves on the star region in [27]. Moreover, by the properties of cross-cut partitions, the Nother-type theorem of piecewise algebraic curves on the cross-cut partition is discussed. Using Bezout’s theorem and Nother-type theorem of piecewise algebraic curves, the Cayley-Bacharach theorem and Hilbert function of C0 piecewise algebraic curves are presented last.A natural problem of the interpolation by Sun(△) is to construct interpolation sets for Sun(△). Unfortunately, interpolation by spline spaces are strongly connected with the problem on the dimensions of these spaces. Therefore, this kind of interpolation problems will be very complicated. For studying the multivariate spline interpolation, wang presented the concept of piecewise algebraic curves and piecewise algebraic varieties about 30 years ago. Unfortunately, few researches of piecewise algebraic curves and piecewise algebraic varieties has been applied to solve the multivariate spline interpolation problems. In chapter 3, we study the Lagrange interpolation set for interpolating along a piecewise algebraic curve by using the results of chapter 2. Moreover, the recursive construction theorem and geometric structure for constructing Lagrange interpolation sets for S0n(△) are provided.One can plot real piecewise algebraic curves with the help of a computer. Indeed, these computer plots are somewhat unreliable. For instance, one cannot be totally sure that a curve is empty, based on the fact that the plot on the screen looks empty. Also, plots are not very precise near singular points of piecewise algebraic curves. These facts make it necessary to have some theoretical results about real real piecewise algebraic curves at hand. In chapter 4, we study the real piecewise algebraic curves. Chapter 4 starts with some properties of piecewise algebraic curves. Next, we define the character of real bivariate spline and, using elementary methods in更多还原