| Description |
xxiv, 607 pages : illustrations ; 24 cm + 1 CD-ROM (4 3/4 in.) |
|
4 3/4 in |
| Contents |
Contents note continued: 11.7.1.Applying Estimates to a Specific Polynomial -- 11.8.TFA: Summary and Comments -- 11.9.The Fancy Fundamental Theorem and the Complex Viete Map -- 11.9.1.Deep Reflections on the Viete Formulas and the Fancy FTA -- 11.10.Braids and Roots -- 11.11.The Galois Theory and the Abel-Ruffini Theorem -- 12.The Polynomial Kalidoscope with Holomorphic Mirrors -- 12.1.The Derivative -- 12.2.Roots of a polynomial and its derivative -- 12.3.More about the Taylor Formula for Polynomials -- 12.4.Geometry of complex polynomial maps in the vicinity of their singular points -- 12.5.Fixed and Periodic Points of Polynomial Maps -- 12.6.Polynomial Maps, Vector Fields and Fixed Points -- 12.7.The Complex Exponential Map -- 12.8.Commuting Polynomial Maps -- 12.9.Line Integrals and Cauchy Formulas: an Analytical Proof of Theorem A |
|
Contents note continued: 2.2.2.The Discriminant Curve and its Tangential Investigations -- 2.3.Playing Dice with Viete -- 2.4.Completing the Square, Substitution Flows, and other Quadratic Magic -- 2.5.The Inverse Viete Map -- 2.5.1.Investigating the Geometry of the Coefficient-to-Root Map -- 2.5.2.Families in the Coefficient Space under the Inverse Viete Map -- 2.6.Adding a New Dimension -- 3.The Complex Numbers and Other Fields -- 3.1.From Integers to Rational Numbers -- 3.2.How to Manufacture the Real Numbers from the Rational Ones -- 3.3.From Real to Complex Number -- 3.3.1.The Need to Extend Real Numbers and the Incompleteness of the Root Space -- 3.3.2.Adding Points of the Coordinate Plane -- 3.3.3.Challenge of Multiplication -- 3.4.Back to the Drawing Board -- 3.4.1.Replacing Points by Vectors -- 3.4.2.Cartesian and Polar Coordinates -- 3.4.3.Adding Vectors Geometrically -- 3.4.4.Approaching Multiplication of Plane Vectors in Stages -- |
|
Contents note continued: 3.5.Issues with the Polar Form. Multivalued Representations -- 3.6.Properties of the Complex Numbers -- 3.6.1.The Equality of Two Complex Number -- 3.6.2.Operations with the Cartesian Representation of Complex Numbers -- 3.6.3.A Short Historical Note -- 3.6.4.Division, Conjugation and Absolute Value -- 3.6.5.The n-th Roots of Unity -- 3.7.A Short Field Trip -- 3.8.Conclusions -- 4.The Geometry of Complex Linear Polynomial Mappings -- 4.1.Complex Polynomial Maps Considered -- 4.2.Plane Geometry and Complex Linear Polynomials -- 4.3.Addition and Multiplication as Transformations -- 4.4.Composing the Addition and Multiplication Maps -- 4.5.Invariants of the Complex Linear Polynomial Maps -- 4.6.On Distances and Isometries -- 4.7.Decomposing Complex Multiplication -- 4.8.Composing Complex Linear Polynomial Maps -- 4.9.A Detour: About Orientation -- 4.10.How to Reconstruct an Isometry of the Plane from its Action on Points? -- 4.11.Reflections -- |
|
Contents note continued: 4.12.The Structure of Complex Linear Polynomial Transformations -- 4.13.The Group, a Fundamental Algebraic Structure -- 4.13.1.Old Familiar Examples of Groups -- 4.13.2.Subgroups of the Group L -- 4.14.Fixed Points of Linear Polynomial Maps -- 5.Complex Polynomial Maps, Closed Plane Curves, and a Few Topological Exhibits -- 5.1.Polynomial Equations and Maps -- 5.1.1.From Polynomial Maps to Plane Curves -- 5.1.2.Loops within Loops -- 5.2.The World of Plane Curves -- 5.2.1.The Rope Model -- 5.2.2.Self-intersections -- 5.2.3.Simplifying Intersections -- 5.2.4.Identifying Intersection Points on a Plane Curve -- 5.2.5.Moving the Rope and Deforming the Curve -- 5.2.6.The Viewpoint of a Nail -- 5.2.7.The Cutting Number -- 5.2.8.The Algebraic Cutting Number -- 5.3.The Train Model --- Parameterizing Plane Curves -- 5.3.1.Loops of Traveling Light -- 5.3.2.Motion and Angular Accumulation -- 5.3.3.Index of a Curve with Respect to a Point---the Winding Number -- |
|
Contents note continued: 5.3.4.The Cutting Number and the Index -- 5.3.5.Index Invariance under Deformations -- 5.4.The Fundamental Group, the Index Invariant, and a Crush Course in Topology -- 5.4.1.A Light Speed Flight through the Language of Topology -- 5.4.2.The Fundamental Group -- 6.Geometry of Complex Polynomial Maps of Degree Two and Three -- 6.1.Zeros and Preimages of Complex Polynomial Maps -- 6.2.Geometrical Tools for Locating Preimages of Points under Complex Polynomial Maps -- 6.3.The Quadratic Map -- 6.4.Root and Coefficient Representations -- 6.5.Lassoing Roots -- 6.5.1.Reflections on Exercise 6.3 -- 6.6.The Quadratic Polynomial with Roots of Multiplicity 2 -- 6.6.1.Reflections on Exercise 6.4 -- 6.7.The Local Geometry of Complex Quadratic Maps -- 6.7.1.Reflections on Exercise 6.5 -- 6.7.2.More Reflections -- 6.8.The Monic Cubic Polynomial Equation -- 7.Modular Spaces of Cubic and Quartic Polynomials -- 7.1.Cubic Polynomials and the Viete Map -- |
|
Contents note continued: 7.2.Cubic Polynomials and their Discriminants -- 7.2.1.Playing Dice with the Real Cubic Viete -- 7.3.The Discriminant Surface: Stratified and Ruled -- 7.4.The Reduction Flow and the Depressed Cubic Polynomials -- 7.5.The Whitney Projection: Adding Dimension to the Coefficient Space -- 7.6.A High Altitude Flight over the Quartic Planet: Surveying the Discriminant Geometry of Quartic Polynomials -- 8.Cubic and Quartic Equations: Solutions in Radicals and a Field Theory Trip -- 8.1.Radicals as Multi-valued Functions -- 8.2.Cubic Formula: the Renaissance Art of Solving Cubic Equations -- 8.3.Ferrari's Quartic Formula -- 8.4.Algebra Fields Forever -- 8.5.What Kind of Fields Do the Cubic and Quartic Polynomials Grow? -- 8.6.More History of Cubic Equations than You Want to Know -- 8.6.1.The Case of the Disappearing Proof -- 8.6.2.Viete Comes Back -- 9.Solutions of Cubic Equations in Radicals: their Geometry and Symmetry -- |
|
Contents note continued: 9.1.Radicals, Multi-valued Functions, and Riemann Surfaces -- 9.2.Visualizing the Cubic Formula: the Cardano Map and the S3-symmetry -- 10.The Cubic and Quartic Equations and the Plane Curves -- 10.1.How Algebraic Geometry Does Not Help to Solve Cubic Equations, but Gives Hope -- 10.2.How Algebraic Geometry Does Help to Solve Quartic Equations -- 11.Complex Polynomial Maps of High Degrees and the Fundamental Theorem of Algebra -- 11.1.The Two Faces of Polynomials -- 11.2.Lassoing the Preimages of Points under Complex Polynomial Maps -- 11.3.The Taylor Formula for Polynomials -- 11.3.1.Planets, Generalized Cycloids, and Complex Polynomials -- 11.3.2.Concluding the Proof of Theorem A -- 11.4.The Fundamental Theorem of Algebra and its Implications -- 11.5.Proof of the Fundamental Theorem of Algebra -- 11.6."The Lady with a Dog" is enjoying her new algebraic life -- 11.7.Estimating Polynomials and Measuring their Tails -- |
|
Machine generated contents note: 1.Maps, Models, and the Coordinate Plane -- 1.1.Rectangles and the Cartesian Plane: Consider a Rectangular Point -- 1.1.1.The Length-Height Model -- 1.1.2.The Perimeter-Area Model -- 1.2.On Parametric Representations -- 1.3.Modeling the Universe of Lines in the Plane -- 1.3.1.The Slope-Intercept Model -- 1.3.2.The xy-Intercept Model -- 1.3.3.The Coefficient Model -- 1.4.Parameterizing the Universe of Ellipses -- 1.5.On Mappings -- 1.5.1.How VusuMatica Helps to Connect Alternative Representations of the Modular Space of Rectangles -- 1.5.2.The Geometry of the Connecting Map -- 1.6.Reflections on the Approach -- 2.The Universe of Quadratic Polynomials, the Viete Map, and its Inverse -- 2.1.Quadratic Equations and Quadratic Polynomials -- 2.1.1.A Minor Digression: Polynomials, Equations, Roots, and Zeros -- 2.2.The Viete Map for Quadratics -- 2.2.1.The Viete Map and the Mystery of White Region -- |
| Notes |
CD-ROM contains the VisuMatica software and instructions |
| Bibliography |
Includes bibliographical references (pages 603-607) |
| Subject |
Polynomials -- Data processing.
|
|
Polynomials -- Mathematical models.
|
| Author |
Nodelman, Vladimir.
|
| LC no. |
2012359659 |
| ISBN |
9789814313599 |
|
9814313599 |
|
