Preface -- Contents -- Conventions -- PART I Background -- Chapter 1 Polynomial Systems -- 1.1 Polynomials in One Variable -- 1.2 Multivariate Polynomial Systems -- 1.3 Trigonometric Equations as Polynomials -- 1.4 Solution Sets -- 1.5 Solution by Continuation -- 1.6 Overview -- 1.7 Exercises -- Chapter 2 Homotopy Continuation -- 2.1 Continuation for Polynomials in One Variable -- 2.2 Complex Versus Real Solutions -- 2.3 Path Tracking -- 2.4 Exercises -- Chapter 3 Projective Spaces -- 3.1 Motivation: Quadratic Equations -- 3.2 Definition of Projective Space -- 3.3 The Projective Line P1 -- 3.4 The Projective Plane P2 -- 3.5 Projective Algebraic Sets -- 3.6 Multiprojective Space -- 3.7 Tracking Solutions to Infinity -- 3.8 Exercises -- Chapter 4 Genericity and Probability One -- 4.1 Generic Points -- 4.2 Example: Generic Lines -- 4.3 Probabilistic Null Test -- 4.4 Algebraic Probability One -- 4.5 Numerical Certainty -- 4.6 Other Approaches to Genericity -- 4.7 Final Remarks -- 4.8 Exercises -- Chapter 5 Polynomials of One Variable -- 5.1 Some Algebraic Facts about Polynomials of One Complex Variable -- 5.2 Some Analytic Facts about Polynomials of One Complex Variable (Optional) -- 5.3 Some Numerical Aspects of Polynomials of One Variable -- 5.4 Exercises -- Chapter 6 Other Methods -- 6.1 Exclusion Methods -- 6.2 Elimination Methods -- 6.3 Gröbner Methods -- 6.4 More Methods -- 6.5 Floating Point vs. Exact Arithmetic -- 6.6 Discussion -- 6.7 Exercises -- PART II Isolated Solutions -- Chapter 7 Coefficient-Parameter Homotopy -- 7.1 Coefficient-Parameter Theory -- 7.2 Parameter Homotopy in Application -- 7.3 An Illustrative Example: Triangles -- 7.4 Nested Parameter Homotopies -- 7.5 Side Conditions -- 7.6 Homotopies that Respect Symmetry Groups -- 7.7 Case Study: Stewart-Gough Platforms -- 7.8 Historical Note: The Cheater's Homotopy -- 7.9 Exercises.

Chapter 8 Polynomial Structures -- 8.1 A Hierarchy of Structures -- 8.2 Notation -- 8.3 Homotopy Paths for Linearly Parameterized Families -- 8.4 Product Homotopies -- 8.5 Polytope Structures -- 8.6 A Summarizing Example -- 8.7 Exercises -- Chapter 9 Case Studies -- 9.1 Nash Equilibria -- 9.2 Chemical Equilibrium -- 9.3 Stewart-Gough Forward Kinematics -- 9.4 Six-Revolute Serial-Link Robots -- 9.5 Planar Seven-Bar Structures -- 9.6 Four-Bar Linkage Design -- 9.7 Exercises -- Chapter 10 Endpoint Estimation -- 10.1 Nonsingular Endpoints -- 10.2 Singular Endpoints -- 10.3 Singular Endgames -- 10.4 Losing the Endgame -- 10.5 Deflation of Isolated Singularities -- 10.6 Exercises -- Chapter 11 Checking Results and Other Implementation Tips -- 11.1 Checks -- 11.2 Corrective Actions -- 11.3 Exercises -- PART III Positive Dimensional Solutions -- Chapter 12 Basic Algebraic Geometry -- 12.1 Affine Algebraic Sets -- 12.2 The Irreducible Decomposition for Affine Algebraic Sets -- 12.3 Further Remarks on Projective Algebraic Sets -- 12.4 Quasiprojective Algebraic Sets -- 12.5 Constructible Algebraic Sets -- 12.6 Multiplicity -- 12.7 Exercises -- Chapter 13 Basic Numerical Algebraic Geometry -- 13.1 Introduction to Witness Sets -- 13.2 Linear Slicing -- 13.3 Witness Sets -- 13.4 Rank of a Polynomial System -- 13.5 Randomization and Nonsquare Systems -- 13.6 Witness Supersets -- 13.7 Probabilistic Algorithms About Algebraic Sets -- 13.8 Summary -- 13.9 Exercises -- Chapter 14 A Cascade Algorithm for Witness Supersets -- 14.1 The Cascade Algorithm -- 14.2 Examples -- 14.3 Exercises -- Chapter 15 The Numerical Irreducible Decomposition -- 15.1 Membership Tests and the Numerical Irreducible Decomposition -- 15.2 Sampling a Component -- 15.3 Numerical Elimination Theory -- 15.4 Homotopy Membership and Monodromy -- 15.5 The Trace Test -- 15.6 Singular Path Tracking.

15.7 Exercises -- Chapter 16 The Intersection Of Algebraic Sets -- 16.1 Intersection of Irreducible Algebraic Sets -- 16.2 Equation-by-Equation Solution of Polynomial Systems -- 16.3 Exercises -- Appendices -- Appendix A Algebraic Geometry -- A.1 Holomorphic Functions and Complex Analytic Spaces -- A.2 Some Further Results on Holomorphic Functions -- A.3 Germs of Complex Analytic Sets -- A.4 Useful Results About Algebraic and Complex Analytic Sets -- A.5 Rational Mappings -- A.6 The Rank and the Projective Rank of an Algebraic System -- A.7 Universal Functions and Systems -- A.8 Linear Projections -- A.9 Bertini's Theorem and Some Consequences -- A.10 Some Useful Embeddings -- A.11 The Dual Variety -- A.12 A Monodromy Result -- A.13 Line Bundles and Vector Bundles -- A.14 Generic Behavior of Solutions of Polynomial Systems -- Appendix B Software for Polynomial Continuation -- Appendix C HomLab User's Guide -- C.1 Preliminaries -- C.2 Overview of HOMLAB -- C.3 Defining the System to Solve -- C.4 Linear Product Homotopies -- C.5 Parameter Homotopies -- C.6 Defining a Homotopy Function -- C.7 The Workhorse: Endgamer -- C.8 Solutions at Infinity and Dehomogenization -- Bibliography -- Index.