A Guide to Plane Algebraic Curves /

This book can be used in a one semester undergraduate course or senior capstone course, or as a useful companion in studying algebraic geometry at the graduate level. This Guide is a friendly introduction to plane algebraic curves. It emphasizes geometry and intuition, and the presentation is kept c...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Kendig, Keith
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge : Cambridge University Press, 2011.
Colección:Dolciani mathematical expositions.
Temas:
Curves, Plane.
Curves, Algebraic.
Courbes planes.
Courbes algébriques.
Curves, Algebraic
Curves, Plane
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • copyright page
  • title page
  • Preface
  • Itâ€?s for. . .
  • What this book is, and what it isnâ€?t.
  • What are the prerequisites for this book?
  • Why should I be interested, in algebraic curves?
  • A Bit of Perspective.
  • The Bookâ€?s Story Line . . .
  • Many thanks to . . .
  • Contents
  • CHAPTER 1 A Gallery of Algebraic Curves
  • 1.1 Curves of Degree One and Two
  • Degree One
  • Degree Two
  • 1.2 Curves of Degree Three and Higher
  • Degree Three
  • Higher Degrees
  • 1.3 Six Basic Cubics
  • 1.4 Some Curves in Polar Coordinates
  • Rectangular versus Polar CoordinatesAlgebraic versus Not Algebraic
  • The Oppositeness Idea
  • 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
  • Linkages
  • CHAPTER 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
  • CHAPTER 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 Bézoutâ€?s theorem
  • 3.11 Bézoutâ€?s theorem Generalizes the Fundamental Theorem of Algebra
  • 3.12 An Application of Bézoutâ€?s theorem: Pascalâ€?s theorem
  • CHAPTER 4 Topology of AlgebraicCurves in P^2(C)
  • 4.1 Introduction
  • 4.2 Connectedness4.3 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
  • CHAPTER 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
  • Curves of the Form y^m = x^ n
  • An Example with Repeated Tangent Lines5.10 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-OrderOrdinary Singularity
  • 5.16 Examples of Resolving an Ordinary Singularity
  • 5.17 Resolving Several Ordinary Singularities
  • 5.18 Quadratic Transformations
  • CHAPTER 6 The Big Three: C, K, S
  • 6.1 Function Fields
  • 6.2 Compact Riemann Surfaces
  • 6.3 Projective Plane Curves

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