Rational Algebraic Curves: A Computer Algebra Approach / Edition 1

Rational Algebraic Curves: A Computer Algebra Approach / Edition 1

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by J. Rafael Sendra, Franz Winkler, Sonia Perez-Diaz
     
 

ISBN-10: 3540737243

ISBN-13: 9783540737247

Pub. Date: 10/28/2007

Publisher: Springer Berlin Heidelberg

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

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Overview

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.

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
Pages:
270
Product dimensions:
6.20(w) x 9.40(h) x 0.70(d)

Related Subjects

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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