Some Researches On Piecewise Algebraic Curves, Piecewise Algebraic Varieties, And Piecewise Semialgebraic Sets

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...