Rational Algebraic Curves: A Computer Algebra Approach

272 Pages·2008·3.1 MB·English

by  J. Rafael Sendra

Algorithms and Computation
in Mathematics Volume 22
•
Editors
Arjeh M.Cohen Henri Cohen
DavidEisenbud MichaelF. Singer Bernd Sturmfels
J. Rafael Sendra
Franz Winkler
Sonia Pérez-Díaz
Rational Algebraic Curves
A Computer Algebra Approach
With 24 Figuresand 2 Tables
Authors
J.RafaelSendra
Departamento de Matemáticas
Universidad deAlcalá
28871AlcaládeHenares, Madrid
Spain
E-mail:[email protected]
FranzWinkler
RISC-Linz
J.KeplerUniversitätLinz
4040Linz
Austria
E-mail:[email protected]
SoniaPérez-Díaz
Departamento de Matemáticas
Universidad de Alcalá
28871AlcaládeHenares,Madrid
Spain
E-mail:[email protected]
LibraryofCongressControlNumber: 2007932190
MathematicsSubjectClassification(2000): 14H50, 14M20, 14Q05, 68W30
ISSN 1431-1550
ISBN 978-3-540-73724-7 SpringerBerlinHeidelbergNewYork
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Preface
Algebraic curves and surfaces are an old topic of geometric and algebraic
investigation.They have found applications for instance in ancient and mod-
ern architectural designs, in number theoretic problems, in models of bio-
logical shapes, in error-correcting codes, and in cryptographic algorithms.
Recently they have gained additional practical importance as central objects
in computer-aided geometric design. Modern airplanes, cars, and household
appliances would be unthinkable without the computational manipulation of
algebraic curves and surfaces. Algebraic curves and surfaces combine fasci-
nating mathematical beauty with challenging computational complexity and
wide spread practical applicability.
In this book we treat only algebraic curves, although many of the results
and methods can be and in fact have been generalized to surfaces. Being the
solution loci of algebraic, i.e., polynomial, equations in two variables, plane
algebraiccurvesarewellsuitedforbeinginvestigatedwithsymboliccomputer
algebramethods.Thisis exactlythe approachwetakeinourbook.We apply
algorithms from computer algebra to the analysis, and manipulation of alge-
braic curves. To a large extent this amounts to being able to represent these
algebraic curves in different ways, such as implicitly by defining polynomi-
als, parametrically by rational functions, or locally parametrically by power
series expansions around a point. All these representations have their indi-
vidual advantages; an implicit representation lets us decide easily whether a
given point actually lies on a given curve, a parametric representation allows
us to generate points of a given curve over the desired coordinate fields, and
with the help of a power series expansion we can for instance overcome the
numerical problems of tracing a curve through a singularity.
The central problem in this book is the determination of rational para-
metrizability of a curve, and, in case it exists, 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
VI Preface
parametrizations of algebraic curves are discussed. We also point to some
applicationsofthese techniques incomputeraidedgeometricdesign.Manyof
the symbolic algorithmic methods described in our book are implemented in
the program system CASA, which is based on the computer algebra system
Maple.
Our book is mainly intended for graduate students specializing in con-
structivealgebraiccurvegeometry.Wehopethatresearcherswantingtogeta
quickoverviewofwhatcanbedonewithalgebraiccurvesintermsofsymbolic
algebraic computation will also find this book helpful.
This book is the result of several years of research of the authors in the
topic,andinconsequencesomepartsofitarebasedonpreviousresearchpub-
lished in journal papers, surveys, and conference proceedings (see [ReS97a],
[Sen02], [Sen04], [SeW91], [SeW97], [SeW99], [SeW01a], [SeW01b]).
We gratefully acknowledge support of our work on this book by FWF
(Austria)SFBF013/F1304,O¨AD(Austria)Acc.Int.Proj.Nr.20/2002,(Spain)
Acc. Int. HU2001-0002,(Spain) BMF 2002-04402-C02-01,and (Spain) MTM
2005-08690-C02-01.
Alcala´ de Henares and Linz, J. Rafael Sendra
June 2007 Franz Winkler
Sonia P´erez-D´ıaz
Contents
1 Introduction and Motivation............................... 1
1.1 Intersection of Curves.................................... 4
1.2 Generating Points on a Curve............................. 5
1.3 Solving Diophantine Equations............................ 6
1.4 Computing the General Solution
of First-Order Ordinary Differential Equations .............. 8
1.5 Applications in CAGD ................................... 9
2 Plane Algebraic Curves.................................... 15
2.1 Basic Notions ........................................... 15
2.1.1 Affine Plane Curves ............................... 16
2.1.2 Projective Plane Curves............................ 19
2.2 Polynomialand Rational Functions ........................ 24
2.2.1 Coordinate Rings and Polynomial Functions .......... 24
2.2.2 Polynomial Mappings .............................. 26
2.2.3 Rational Functions and Local Rings ................. 28
2.2.4 Degree of a Rational Mapping ...................... 32
2.3 Intersection of Curves.................................... 34
2.4 Linear Systems of Curves................................. 41
2.5 Local Parametrizationsand Puiseux Series.................. 50
2.5.1 Power Series, Places, and Branches .................. 51
2.5.2 Puiseux’s Theorem and the Newton PolygonMethod .. 55
2.5.3 Rational Newton Polygon Method ................... 61
Exercises ................................................... 62
3 The Genus of a Curve ..................................... 67
3.1 Divisor Spaces and Genus ................................ 67
3.2 Computation of the Genus................................ 69
3.3 Symbolic Computation of the Genus ....................... 78
Exercises ................................................... 85
VIII Contents
4 Rational Parametrization .................................. 87
4.1 Rational Curves and Parametrizations ..................... 88
4.2 Proper Parametrizations ................................. 95
4.3 Tracing Index...........................................100
4.3.1 Computation of the Index of a Parametrization .......101
4.3.2 Tracing Index Under Reparametrizations .............104
4.4 Inversion of Proper Parametrizations.......................105
4.5 Implicitization ..........................................108
4.6 Parametrizationby Lines.................................114
4.6.1 Parametrizationof Conics ..........................114
4.6.2 Parametrization of Curves with a Point of High
Multiplicity.......................................116
4.6.3 The Class of Curves Parametrizable by Lines .........118
4.7 Parametrizationby Adjoint Curves ........................119
4.8 Symbolic Treatment of Parametrization ....................136
Exercises ...................................................145
5 Algebraically Optimal Parametrization ....................149
5.1 Fields of Parametrization.................................150
5.2 Rational Points on Conics ................................154
5.2.1 The Parabolic Case................................155
5.2.2 The Hyperbolic and the Elliptic Case ................156
5.2.3 Solving the Legendre Equation......................157
5.3 Optimal Parametrizationof Rational Curves ................169
Exercises ...................................................185
6 Rational Reparametrization ...............................187
6.1 Making a ParametrizationProper .........................188
6.1.1 Lu¨roth’s Theorem and Proper Reparametrizations.....188
6.1.2 Proper Reparametrization Algorithm ................190
6.2 Making a ParametrizationPolynomial......................194
6.3 Making a ParametrizationNormal.........................200
Exercises ...................................................207
7 Real Curves ...............................................209
7.1 Parametrization.........................................209
7.2 Reparametrization.......................................217
7.2.1 Analytic Polynomialand Analytic Rational Functions..217
7.2.2 Real Reparametrization ............................220
7.3 Normal Parametrization..................................226
Exercises ...................................................235
A The System CASA ........................................239
Contents IX
B Algebraic Preliminaries....................................247
B.1 Basic Ring and Field Theory..............................247
B.2 Polynomials and Power Series.............................250
B.3 PolynomialIdeals and Elimination Theory..................253
B.3.1 Gro¨bner Bases ....................................253
B.3.2 Resultants........................................254
B.4 Algebraic Sets ..........................................256
References.....................................................257
Index..........................................................265
Table of Algorithms ...........................................269
1
Introduction and Motivation
Summary. Inthisfirstchapter,weinformallyintroducethenotionofrationalalge-
braiccurves,andwemotivatetheirusebymeansofsomeexamplesofapplications.
These examples cover the intersection of curves in Section 1.1, the generation of
pointsoncurvesinSection1.2,thesolutionofDiophantineequationsinSection1.3,
thesolutionofcertaindifferentialequationsinSection1.4,andapplicationsincom-
puteraided geometric design in Section 1.5.
The theory of algebraic curves has a long and distinguished history, and
there is a huge number of excellent books on this topic. In our book we
concentrate on the computational aspects of algebraic curves, specially of
rational algebraic curves, and we will frequently refer to classical literature.
Moreover,ourcomputationalapproachisnotapproximativebutsymbolicand
based on computer algebra methods. That means we are dealing with exact
mathematical descriptions of geometric objects and both the input and the
output of algorithms are exact.
Our book is mainly intended for graduate students specializing in con-
structive algebraic curve geometry, as well as for researchers wanting to get
a quick overviewof what can be done with algebraic curves in terms of sym-
bolicalgebraiccomputation.Throughoutthisbookweonlyconsideralgebraic
curves. So, whenever we speak of a “curve” we mean an “algebraic curve.”
In this first chapter, we informally introduce the notion of rational al-
gebraic curves, and we motivate their use by means of some examples of
applications.
When speaking about algebraic curves one may distinguish between alge-
braic plane curves and algebraic space curves. Nevertheless, it is well known
(seeforinstance[Ful89],p.155)thatanyspacecurvecanbe birationallypro-
jected onto a plane curve.This means that there exists a rationallyinvertible
projection (in fact, almost all projections have this property), that maps the
spacecurveontoaplanecurve.Usingsuchaprojectionanditsinverse,which
can be computed by means of elimination theory techniques, one may reduce
the study of algebraic curves in arbitrary dimensional space to the study of
2 1 Introduction and Motivation
2
y1
−1−0.8 −0.4 0.2 0.4 0.6 0.8 1 1.2
x
−1
−2
Fig. 1.1. πz(C3)=C2 (left),C3 (right)
plane algebraic curves. In fact, throughout this book we will consider plane
algebraic curves, i.e., solution loci of nonconstant bivariate polynomials with
coefficients in a field, say C. In general, we will not work specifically over the
complex numbers C, but rather over an arbitrary algebraically closed field of
characteristic zero.
Let us see an example of a birational projection of a space curve onto a
plane curve. We consider in C3 the space curve C (see Fig.1.1), defined as
3
the intersection of the surfaces
g (x,y,z)=y+z−z3, g (x,y,z)=x+1−z2;
1 2
that is, C = {(x,y,z) ∈ C3|g (x,y,z) = g (x,y,z) = 0}. We consider the
3 1 2
projection along the z-axis
π :C3 −→C2; (x,y,z)(cid:3)→(x,y).
z
π (C ) is the plane curve C (see Fig.1.1) defined by the polynomial
z 3 2
f(x,y)=x3+x2−y2
(infact,inthiscase,f istheresultantofg andg w.r.t.z);i.e.C ={(x,y)∈
1 2 2
C2|f(x,y)=0}.The restriction of the projectionπ to the curveC is ratio-
z 3
nally invertible for all but finitely many points on C . Indeed, the inverse is
2
(cid:1) (cid:2)
y
π−1 :C −→C ; (x,y)(cid:3)→ x,y, .
z 2 3 x
Somealgebraicplanecurvescanberepresentedparametricallybymeansof
rationalfunctions. This means that a pair of rational functions χ (t),χ (t)∈
1 2
C(t)generatesall(exceptperhapsfinitelymany)pointsonthecurvewhenthe
parameterttakesvaluesinC.Thisrequirementisequivalenttothe condition
f(χ (t),χ (t)) = 0, assuming that not both rational functions are constant
1 2
and that f(x,y) = 0 is the equation of the curve. Plane curves with this
property are called rational curves, and their study is the central topic of
this book. Only irreducible curves can be rational. The simplest example of