Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- Notations -- 1 The algebra of polynomials -- 1.1 Complex polynomials -- 1.1.1 Definitions -- 1.1.2 Number of zeros -- 1.1.3 -- 1.1.4 -- 1.1.5 -- 1.1.6 -- 1.1.7 Representation for harmonic polynomials -- 1.2 The number of zeros of real analytic polynomial -- 1.2.1 -- 1.2.2 -- 1.2.3 -- 1.2.4 A complex approach -- 1.2.5 -- 1.2.6 The Sylvester resultant of P - w -- 1.2.7 Evaluating the Sylvester resultant -- 1.2.8 Further properties of the resultant -- 1.3 Real analytic polynomials at infinity -- 1.3.1 Resolving the singularity: the blow-up method -- 1.3.2 -- 1.3.3 Repetition of asymptotic values -- 1.3.4 -- 1.3.5 Inverting the transformation -- 1.3.6 The comparison principle -- 1.3.7 Solving algebraic equations and algebraic functions -- 2 The degree principle and the fundamental theorem of algebra -- 2.1 The fundamental theorem of algebra -- 2.1.1 -- 2.1.2 -- 2.1.3 -- 2.1.4 -- 2.1.5 -- 2.1.6 -- 2.1.7 Problems -- 2.2 Continuous functions in the plane -- 2.2.1 -- 2.2.2 -- 2.2.3 -- 2.2.4 -- 2.2.5 -- 2.2.6 Problem -- 2.2.7 -- 2.2.8 -- 2.2.9 Problems -- 2.3 The degree principle -- 2.3.1 -- 2.3.2 -- 2.3.3 -- 2.3.4 -- 2.3.5 -- 2.3.6 -- 2.3.7 Properties of the winding number -- 2.3.8 -- 2.3.9 -- 2.3.10 -- 2.3.11 -- 2.3.12 -- 2.3.13 -- 2.3.14 The degree principle for a simply-connected domain -- 2.3.15 Remarks on plane topology -- 2.3.16 Problems -- 2.4 The degree principle and homotopy -- 2.4.1 -- 2.4.2 -- 2.4.3 -- 2.4.4 -- 2.4.5 -- 2.4.6 Problems -- 2.5 The topological argument principle -- 2.5.1 -- 2.5.2 -- 2.5.3 -- 2.5.4 -- 2.5.5 -- 2.5.6 Problems -- 2.6 The coincidence theorem -- 2.6.1 -- 2.6.2 -- 2.6.3 -- 2.6.4 -- 2.6.5 -- 2.6.6 -- 2.6.7 -- 2.6.8 Completion of proof of fundamental theorem of algebra -- 2.6.9 Harmonic polynomials -- 2.6.10 -- 2.6.11 Wilmshurst's example.

2.6.12 Harmonic functions at a non-isolated zero -- 2.6.13 -- 2.6.14 Problems -- 2.7 Locally 1-1 functions -- 2.7.1 -- 2.7.2 -- 2.7.3 -- 2.7.4 -- 2.7.5 -- 2.7.6 -- 2.7.7 -- 2.7.8 The Jacobian determinant -- 2.7.9 -- 2.7.10 -- 2.7.11 -- 2.7.12 -- 2.7.13 -- 2.7.14 -- 2.7.15 Harmonic polynomials on the critical set -- 2.7.16 -- 2.7.17 -- 2.7.18 -- 2.7.19 -- 2.7.20 -- 2.7.21 The case n = 1 -- 2.7.22 The case n = 2 -- 2.7.23 -- 2.7.24 Problems -- 2.8 The Borsuk-Ulam theorem -- 2.8.1 -- 2.8.2 -- 2.8.3 Problem -- 3 The Jacobian problem -- 3.1 The Jacobian conjecture -- 3.1.1 -- 3.1.2 -- 3.1.3 -- 3.1.4 -- 3.1.5 -- 3.1.6 Proof of the conjecture in the case n = 2 -- 3.1.7 The leading term -- 3.1.8 Topology of the mapping -- 3.2 Pinchuk's example -- 3.2.1 -- 3.2.2 The Pinchuk surface -- 3.2.3 General observations for the R problem -- 3.2.4 Estimations of degree -- 3.2.5 -- 3.3 Polynomials with a constant Jacobian -- 3.3.1 -- 3.3.2 -- 3.3.3 -- 3.3.4 The case Omega = 1 -- 3.3.5 The case Omega = 2 -- 3.3.6 The closing gap problem -- 3.3.7 The case Omega = 3 -- 3.3.8 Problem -- 3.4 A topological approach -- 3.4.1 -- 3.4.2 -- 3.4.3 -- 3.4.4 -- 3.4.5 -- 3.4.6 -- 3.5 The resultant and the Jacobian -- 3.5.1 -- 3.5.2 Problem -- 4 Analytic and harmonic functions in the unit disc -- 4.1 Series representations -- 4.1.1 -- 4.1.2 Convolutions -- 4.1.3 -- 4.1.4 Inequalities of Fejér and Riesz, and of Hilbert -- 4.1.5 Linear functionals -- 4.1.6 Linear operators -- 4.1.7 Harmonic multiplication operators -- 4.1.8 -- 4.1.9 -- 4.1.10 Problems -- 4.2 Positive and bounded operators -- 4.2.1 -- 4.2.2 -- 4.2.3 -- 4.2.4 -- 4.2.5 Convolution operators and the extension property -- 4.2.6 -- 4.2.7 -- 4.2.8 -- 4.2.9 Problem -- 4.3 Positive trigonometric polynomials -- 4.3.1 -- 4.3.2 -- 4.3.3 -- 4.3.4 -- 4.3.5 -- 4.3.6 Problems.

4.4 Some inequalities for analytic and trigonometric polynomials -- 4.4.1 -- 4.4.2 -- 4.4.3 -- 4.4.4 -- 4.4.5 -- 4.4.6 -- 4.4.7 -- 4.4.8 Problems -- 4.5 Cesàro means -- 4.5.1 -- 4.5.2 -- 4.5.3 -- 4.5.4 Problems -- 4.6 De la Vallée Poussin means -- 4.6.1 -- 4.6.2 -- 4.6.3 -- 4.6.4 -- 4.6.5 -- 4.7 Integral representations -- 4.7.1 -- 4.7.2 -- 4.7.3 -- 4.8 Generalised convolution operators -- 4.8.1 -- 4.8.2 -- 4.8.3 -- 4.8.4 -- 4.8.5 -- 5 Circular regions and Grace's theorem -- 5.1 Convolutions and duality -- 5.1.1 -- 5.1.2 -- 5.1.3 -- 5.1.4 Proof of the Grace-Szegö theorem -- 5.1.5 -- 5.2 Circular regions -- 5.2.1 -- 5.2.2 -- 5.2.3 -- 5.2.4 Apolar polynomials -- 5.2.5 -- 5.2.6 Symmetric linear forms -- 5.2.7 -- 5.2.8 Weak apolarity -- 5.2.9 -- 5.2.10 -- 5.2.11 -- 5.3 The polar derivative -- 5.3.1 -- 5.3.2 -- 5.4 Locating critical points -- 5.4.1 -- 5.4.2 -- 5.4.3 -- 5.4.4 -- 5.4.5 -- 5.4.6 -- 5.5 Critical points of rational functions -- 5.5.1 -- 5.5.2 -- 5.6 The Borwein-Erdélyi inequality -- 5.6.1 -- 5.6.2 -- 5.6.3 -- 5.7 Univalence properties of polynomials -- 5.7.1 -- 5.7.2 -- 5.7.3 -- 5.7.4 -- 5.7.5 -- 5.7.6 -- 5.7.7 -- 5.7.8 -- 5.7.9 -- 5.7.10 Problems -- 5.8 Linear operators -- 5.8.1 Polynomials in the unit disc -- 5.8.2 -- 5.8.3 -- 6 The Ilieff-Sendov conjecture -- 6.1 Introduction -- 6.1.1 -- 6.2 Proof of the conjecture for those zeros on the unit circle -- 6.2.1 -- 6.2.2 -- 6.2.3 Problem -- 6.3 The direct application of Grace's theorem -- 6.3.1 -- 6.3.2 -- 6.3.3 -- 6.3.4 -- 6.3.5 -- 6.3.6 -- 6.4 A global upper bound -- 6.4.1 -- 6.4.2 -- 6.4.3 -- 6.5 Inequalities relating the nearest critical point to the nearest second zero -- 6.5.1 -- 6.5.2 -- 6.5.3 -- 6.5.4 -- 6.5.5 Proof of theorems 6.5.2 and 6.5.3 -- 6.5.6 -- 6.5.7 -- 6.6 The extremal distance -- 6.6.1 -- 6.7 Further remarks on the conjecture -- 6.7.1 -- 6.7.2 -- 6.7.3 -- 6.7.4 -- 6.7.5.

6.7.6 -- 6.7.7 -- 6.7.8 -- 7 Self-inversive polynomials -- 7.1 Introduction -- 7.1.1 -- 7.1.2 Relations on the unit circle -- 7.1.3 -- 7.1.4 -- 7.1.5 Problems -- 7.2 Polynomials with interspersed zeros on the unit circle -- 7.2.1 -- 7.2.2 -- 7.2.3 -- 7.2.4 -- 7.2.5 Problems -- 7.3 Relations with the maximum modulus -- 7.3.1 -- 7.3.2 -- 7.3.3 -- 7.3.4 Problems -- 7.4 Univalent polynomials -- 7.4.1 -- 7.4.2 Close-to-convex functions -- 7.4.3 -- 7.4.4 Kaplan classes -- 7.4.5 -- 7.4.6 -- 7.4.7 -- 7.5 A second necessary and sufficient condition for angular separation of zeros -- 7.5.1 -- 7.5.2 -- 7.6 Suffridge's extremal polynomials -- 7.6.1 -- 7.6.2 -- 7.6.3 -- 7.6.4 -- 7.6.5 -- 7.6.6 -- 7.6.7 -- 7.6.8 -- 7.6.9 -- 7.6.10 -- 7.6.11 -- 7.6.12 Limiting versions of Suffridge's convolution theorem -- 8 Duality and an extension of Grace's theorem to rational functions -- 8.1 Linear operators and rational functions -- 8.1.1 -- 8.1.2 -- 8.1.3 -- 8.1.4 -- 8.1.5 -- 8.1.6 -- 8.1.7 -- 8.1.8 -- 8.1.9 Proof of lemma 8.1.7 -- 8.1.10 -- 8.1.11 -- 8.1.12 Proof of theorem 8.1.2 -- 8.2 Interpretations of the convolution conditions -- 8.2.1 -- 8.2.2 The class T(1, 1) -- 8.2.3 -- 8.2.4 -- 8.2.5 -- 8.2.6 -- 8.2.7 The classes T(1, Beta) -- 8.2.8 -- 8.2.9 The cases Beta = 2 and Beta = 3 -- 8.3 The duality theorem for T(1, Beta) -- 8.3.1 -- 8.3.2 -- 8.3.3 -- 8.3.4 The convolution inequality -- 8.3.5 -- 8.3.6 -- 8.3.7 -- 8.3.8 A generalisation of the Eneström-Kakeya theorem -- 8.3.9 -- 8.4 The duality theorem for T(m, Beta) -- 8.4.1 -- 8.4.2 -- 8.4.3 -- 8.4.4 The convolution inequality -- 8.4.5 -- 8.4.6 -- 8.5 The duality principle -- 8.5.1 -- 8.5.2 -- 8.5.3 -- 8.6 Duality and the class T(Alpha, Beta) -- 8.6.1 -- 8.6.2 -- 8.6.3 The convolution inequality -- 8.6.4 -- 8.6.5 -- 8.7 Properties of the Kaplan classes -- 8.7.1 -- 8.7.2 -- 8.7.3 Coefficient bounds for K(Alpha, Beta).

8.7.4 -- 8.8 The class S(Alpha, Beta) -- 8.8.1 -- 8.8.2 -- 8.8.3 -- 8.9 The classes T(Alpha, Beta) -- 8.9.1 Robinson's conjectures -- 8.10 The class T(2, 2) -- 8.10.1 -- 9 Real polynomials -- 9.1 Real polynomials -- 9.1.1 -- 9.1.2 -- 9.1.3 -- 9.1.4 -- 9.1.5 -- 9.1.6 -- 9.1.7 -- 9.1.8 The conjecture for polynomials with exactly two non-real zeros -- 9.1.9 -- 9.1.10 Geometry of the level curves -- 9.1.11 The case for real polynomials with purely imaginary zeros -- 9.1.12 -- 9.1.13 -- 9.1.14 -- 9.1.15 A general result -- 9.1.16 The case for real polynomials with purely imaginary zeros except for a zero at 0 -- 9.1.17 -- 9.2 Descartes' rule of signs -- 9.2.1 -- 9.2.2 -- 9.2.3 -- 9.3 Strongly real rational functions -- 9.3.1 -- 9.3.2 -- 9.3.3 -- 9.3.4 -- 9.3.5 -- 9.3.6 -- 9.3.7 -- 9.3.8 -- 9.3.9 -- 9.4 Critical points of real rational functions -- 9.4.1 -- 9.4.2 -- 9.4.3 -- 9.4.4 -- 9.4.5 -- 9.5 Rational functions with real critical points -- 9.5.1 -- 9.6 Real entire and meromorphic functions -- 9.6.1 -- 9.6.2 Strongly real meromorphic functions -- 9.6.3 -- 9.6.4 -- 9.6.5 -- 9.6.6 -- 9.6.7 -- 9.6.8 -- 9.6.9 Theorem on critical points -- 9.6.10 We begin with some remarks -- 9.6.11 -- 9.6.12 -- 9.6.13 -- 9.6.14 -- 9.6.15 -- 9.6.16 -- 9.6.17 Proof of theorem 9.6.9 -- 9.6.18 -- 9.6.19 -- 9.6.20 -- 9.6.21 The Laguerre-Pólya class -- 9.6.22 -- 9.6.23 -- 9.6.24 -- 10 Level curves -- 10.1 Level regions for polynomials -- 10.1.1 -- 10.1.2 -- 10.1.3 -- 10.2 Level regions of rational functions -- 10.2.1 -- 10.2.2 -- 10.3 Partial fraction decomposition -- 10.3.1 -- 10.3.2 -- 10.3.3 Problem -- 10.3.4 -- 10.3.5 -- 10.4 Smale's conjecture -- 10.4.1 -- 10.4.2 -- 10.4.3 -- 10.4.4 The Cordova-Ruscheweyh method [14] -- 10.4.5 -- 10.4.6 -- 10.4.7 Some tentative conjectures -- 10.4.8 A Blaschke product approach -- 10.4.9 -- 10.4.10 -- 10.4.11 -- 11 Miscellaneous topics.

11.1 The abc theorem.