Rational Algebraic Curves

by Sendra, J. Rafael; Winkler, Franz; Perez-diaz, Sonia

• ISBN13: 9783540737247
• ISBN10: 3540737243
• Format: Hardcover
• Copyright: 2007-12-04
• Publisher: Springer Nature

Summary

The central problem considered in this book 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, i.e. its complete singularity structure, computing regular points of the curve in small coordinate fields, and constructing linear systems of curves with prescribed intersection multiplicities. Various optimality criteria for rational parametrizations of algebraic curves are discussed. This book is mainly intended for graduate students and researchers in constructive algebraic curve geometry.

Table of Contents

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