Rational Algebraic Curves: A Computer Algebra Approach / Edition 1

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The central problem considered in this introduction for graduate students is the determination of rational parametrizability of an algebraic curve and, in the positive case, the computation of a good rational parametrization. This amounts to determining the genus of a curve: its complete singularity structure, computing regular points of the curve in small coordinate fields, and constructing linear systems of curves with prescribed intersection multiplicities. The book discusses various optimality criteria for rational parametrizations of algebraic curves.

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Editorial Reviews

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From the reviews:

"The central problem in the book is the question whether a given algebraic curve is parametrizable and in case to compute a good rational parametrization. This involves an analysis of the singularities to compute the genus of the curve. … There are several exercises at the end of each chapter. … The book is a good basis for graduate students specializing in constructive algebraic curve geometry. It can be used for lectures and seminars in computer algebra based on Maple and CASA." (Gerhard Pfister, Zentralblatt MATH, Vol. 1129 (7), 2008)

"Algebraic geometry is a central field within mathematics which is often viewed as difficult for outsiders to enter. Recent years have seen many books which present various topics of algebraic geometry from a computational viewpoint, thereby making the subject more accessible. The book under review, RAC for short, is part of this positive trend. Its main focus is parametrizing rational plane algebraic curves. It is a graduate level text aimed at fairly wide readership, including for example readers interested primarily in computer-aided geometric design." (David P. Roberts, MathDL, May, 2008)

"The book Rational algebraic curves: a computer algebra approach is a very complete text on rational curves. … The book is really an excellent reference on genus zero curves. It is very useful, mainly due to the computational material presented. It looks to be intended for graduate students, but anyone who wants to know how symbolic algebraic computation can clarify the study of algebraic curves will find it very helpful. A long list of references is provided at the end of the book." (Valmecir A. Bayer, Mathematical Reviews, Issue 2009 a)

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Product Details

  • ISBN-13: 9783540737247
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 10/28/2007
  • Series: Algorithms and Computation in Mathematics Series, #22
  • Edition description: 2008
  • Edition number: 1
  • Pages: 270
  • Product dimensions: 6.20 (w) x 9.40 (h) x 0.70 (d)

Table of Contents

Introduction and Motivation     1
Intersection of Curves     4
Generating Points on a Curve     5
Solving Diophantine Equations     5
Computing the General Solution of First-Order Ordinary Differential Equations     8
Applications in CAGD     9
Plane Algebraic Curves     15
Basic Notions     15
Affine Plane Curves     16
Projective Plane Curves     19
Polynomial and Rational Functions     24
Coordinate Rings and Polynomial Functions     24
Polynomial Mappings     26
Rational Functions and Local Rings     28
Degree of a Rational Mapping     32
Intersection of Curves     34
Linear Systems of Curves     41
Local Parametrizations and Puiseux Series     50
Power Series, Places, and Branches     51
Puiseux's Theorem and the Newton Polygon Method     55
Rational Newton Polygon Method     61
Exercises     62
The Genus of a Curve     67
Divisor Spaces and Genus     67
Computation of the Genus     69
Symbolic Computation of the Genus     78
Exercises     85
Rational Parametrization     87
Rational Curves and Parametrizations     88
Proper Parametrizations     95
Tracing Index     100
Computation of the Index of a Parametrization     101
Tracing Index Under Reparametrizations     104
Inversion of Proper Parametrizations     105
Implicitization     108
Parametrization by Lines     114
Parametrization of Conics     114
Parametrization of Curves with a Point of High Multiplicity     116
The Class of Curves Parametrizable by Lines     118
Parametrization by Adjoint Curves     119
Symbolic Treatment of Parametrization     136
Exercises     145
Algebraically Optimal Parametrization     149
Fields of Parametrization     150
Rational Points on Conics     154
The Parabolic Case     155
The Hyperbolic and the Elliptic Case     156
Solving the Legendre Equation     157
Optimal Parametrization of Rational Curves     169
Exercises     185
Rational Reparametrization     187
Making a Parametrization Proper     188
Luroth's Theorem and Proper Reparametrizations      188
Proper Reparametrization Algorithm     190
Making a Parametrization Polynomial     194
Making a Parametrization Normal     200
Exercises     207
Real Curves     209
Parametrization     209
Reparametrization     217
Analytic Polynomial and Analytic Rational Functions     217
Real Reparametrization     220
Normal Parametrization     226
Exercises     235
The System CASA     239
Algebraic Preliminaries     247
Basic Ring and Field Theory     247
Polynomials and Power Series     250
Polynomial Ideals and Elimination Theory     253
Grobner Bases     253
Resultants     254
Algebraic Sets     256
References     257
Index     265
Table of Algorithms     269
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