Numericalization Research On Basic Theories Of Algebraic Curves

Algebraic curves are classic research objects of Mathematics. There are many valuable applications of algebraic curves in Mathematics and Applied Mathematics, and many fields of science and engineer, such as cryptography, image processing and computer vision. Recently the research on algebraic curves from the perspective of computation has become an active branch. The symbolic computation software Maple has a package that deals with algebraic curves. We can compute some fundamental quantities of algebraic curves by calling it. However, theoretical analysis and numerical experiments show that some fundamental quantities of algebraic curves are unstable under the effect of tiny perturbations and the random perturbations might change some properties of algebraic curves. Different from the traditional symbolic computation, we compute some fundamental quantities of algebraic curves numerically. These computation rely on solving polynomial systems by homotopy continuation method and computing the multiple roots of inexact univariate polynomial by the standard machine precision. Based on the compu-tation of fundamental quantities, we propose the numerical realization of the conditions of Max Nother’s residual intersection theorem.This dissertation has six chapters. The main work can be summarized as follows.In chapter1, we introduce the background for numericalization research on basic theo-ries of algebraic Curves, and then state the related research progress. Finally an outline of the dissertation is given.In chapter2, we present the preliminaries of algebraic curves and root-finding of equations involved in this dissertation.In chapter3, we compute the intersection and inflection of algebraic curves by homotopy continuation method. Compared with the symbolic operations, The proposed algorithms show advantage at computing time and dealing with algebraic curves of high degree. Theoretical anal-ysis and numerical experiments show that the algorithms keep excellent accuracy and robustness, even if the coefficients of algebraic curves are perturbed.In chapter4, an algorithm is proposed to compute singular points of algebraic curves, de-termine their multiplicities and characters. An example is introduced to show that random per-turbations of coefficients of algebraic curves will almost invariably destroy all singular points. We prove that singular points of algebraic curves are isolated. In view of a method of solving overdetermined polynomial systems and the property of singular points of being isolated, an efficient algorithm is presented to compute singular points and their multiplicities and charac-ters by homotopy continuation method. We also analyze the feasibility and robustness of the algorithm and prove that the algorithm has the polynomial time complexity on the degree of algebraic curve. Without using multiprecision arithmetic, numerical experiments and theoreti-cal analysis show that numerical procedures are accurate, efficient and robust. Even if there are tiny perturbations of coefficients of algebraic curves in certain range, our methods still obtain the approximate singular points and the characters of singular points of unperturbed algebraic curves.In chapter5, based on numerical realization of Duval’s rational Puisuex expansion, an al-gorithm is introduced to compute the orders of polynomials at places of algebraic curves. We also present an algorithm to realize conditions of Max Nother’s residual intersection theorem. The tiny perturbations of coefficients of algebraic curves may change some important informa-tion of places, such as the number of places and their ramification indexes. With the aid of a method of computation of multiple roots of inexact univariate polynomials, we give numerical realization of the algorithm of Duval’s rational Puisuex expansion, and then compute the places of an inexact algebraic curve by double floating arithmetic. The precision of the coefficients of places is enough to compute the orders of a polynomial at these approximate places. Based on the computation of previous fundamental quantities of algebraic curves, such as intersec-tion, singular point, multiplicity, character, place and order, we finally present an algorithm to realize conditions of Max Nother’s residual intersection theorem numerically. Computational complexity of the algorithm is polynomial time on the degrees of algebraic curves. Numerical experiments show that it is accurate and efficient to check whether three algebraic curves satisfy conditions of Max Nother’s residual intersection theorem.In chapter6, we conclude the thesis and prospect the future research.