目錄
Preface
1 A Gallery of Algebraic Curves
1.1 Curves of Degree One and Two
1.2 Curves of Degree Three and Higher
1.3 Six Basic Cubics
1.4 Some Curves in Polar Coordinates
1.5 Parametric Curves
1.6 The Resultant
1.7 Back to an Example
1.8 Lissajous Figures
1.9 Morphing Between Curves
1.10 Designer Curves
2 Points at Infinity
2.1 Adjoining Points at Infinity
2.2 Examples
2.3 A Basic Picture
2.4 Basic Definitions
2.5 Further Examples
3 From Real to Complex
3.1 Definitions
3.2 The Idea of Multiplicity; Examples
3.3 A Reality Check
3.4 A Factorization Theorem for Polynomials in C [x, y]
3.5 Local Parametrizations of a Plane Algebraic Curve
3.6 Definition of Intersection Multiplicity for Two Branches
3.7 An Example
3.8 Multiplicity at an Intersection Point of Two Plane Algebraic Curves
3.9 Intersection Multiplicity Without Parametrizations
3.10 B6zout's theorem
3.11 B6zout's theorem Generalizes the Fundamental Theorem of Algebra
3.12 An Application of B6zout's theorem: Pascal's theorem
4 Topology of Algebraic Curves in ]?2(C)
4.1 Introduction
4.2 Connectedness
4.3 Algebraic Curves are Connected
4.4 Orientable Two-Manifolds
4.5 Nonsingular Curves are Two-Manifolds
4.6 Algebraic Curves are Orientable
4.7 The Genus Formula
5 Singularities
5.1 Introduction
5.2 Definitions and Examples
5.3 Singularities at Infinity
5.4 Nonsingular Projective Curves
5.5 Singularities and Polynomial Degree
5.6 Singularities and Genus
5.7 A More General Genus Formula
5.8 Non-Ordinary Singularities
5.9 Further Examples
5.10 Singularities versus Doing Math on Curves
5.11 The Function Field of an Irreducible Curve
5.12 Birational Equivalence
5.13 Examples of Birational Equivalence
5.14 Space-Curve Models
5.15 Resolving a Higher-Order Ordinary Singularity
5.16 Examples of Resolving an Ordinary Singularity
5.17 Resolving Several Ordinary Singularities
5.18 Quadratic Transformations
6 The Big Three: C, K, S
6.1 Function Fields
6.2 Compact Riemann Surfaces
6.3 Projective Plane Curves
6.4 fl, f2, f: Curves and Function Fields
6.5 gl, g2, g: Compact Riemann Surfaces and Curves
6.6 h 1, h2, h: Function Fields and Compact Riemann Surfaces
6.7 Genus
6.8 Genus 0
6.9 Genus One
6.10 An Analogy
6.11 Equipotentials and Streamlines
6.12 Differentials Generate Vector Fields
6.13 A Major Difference
6.14 Divisors
6.15 The Riemann-Roch theorem
Bibliography
Index
About the Author
[