This text presents easy to understand proofs of some of the most difficult results about polynomials. It encompasses a self-contained account of the properties of polynomials as analytic functions of a special kind.
The zeros of compositions of polynomials are also investigated along with their growth, and some of these considerations lead to the study of analogous questions for trigonometric polynomials and certain transcendental entire functions. The strength of methods are fully explained and demonstrated by means of applications.
Author(s): Qazi Ibadur Rahman, Gerhard Schmeisser
Series: London Mathematical Society Monographs
Publisher: Oxford University Press, USA
Year: 2002
Language: English
Pages: 758
Introduction
1.1 The fundamental theorem of algebra
1.2 Symmetric polynomials
1.3 The continuity theorem
1.4 Orthogonal polynomials: general properties
1.5 The classical orthogonal polynomials
1.6 Harmonic and subharmonic functions
1.7 Tools from matrix analysis
1.8 Notes
I CRITICAL POINTS IN TERMS OF ZEROS
2 Fundamental results on critical points 71
2.1 Convex hulls and the Gauss-Lucas theorem 71
2.2 Extensions of the Gauss-Lucas theorem 75
2.3 A verage distances from a line or a point 78
2.4 Real polynomials and Jensen's theorem 85
2.5 Extensions of Jensen's theorem 88
2.6 Notes 91
3 More sophisticated methods 96
3.1 Circular domains and polar derivative 96
3.2 Laguerre's theorem, its variants, and applications 98
3.3 Apolarity 102
3.4 Grace's theorem and equivalent forms 107
3.5 Notes 114
4 More specific results on critical points 117
4.1 Products and quotients of polynomials 117
4.2 Derivatives of reciprocals of polynomials 121
4.3 Complex analogues of Rolle's theorem 125
4.4 Bounds for some of the critical points 129
4.5 Converse results 132
4.6 Notes 137
5 Applications to compositions of polynomials 141
5.1 Linear combination of rational functions 142
5.2 Complex analogues of the intermediate-value theorem 143
5.3 Linear combination of derivatives: Walsh's approach 148
Linear combination of derivatives: recursive approach
Multiplicative composition: Schur-Szego approach
Multiplicative composition: Laguerre's approach
Multipliers preserving the reality of zeros
Notes
6 Polynomials with real zeros
6.1 The span of a polynomial
6.2 Largest zero and largest critical point
6.3 Interlacing and the Hermite-Biehler theorem
6.4 Consecutive zeros and critical points
6.5 Refinement of Rolle's theorem
6.6 Notes
7 Conjectures and solutions
7.1 A conjecture of Popoviciu
7.2 A conjecture of Smale
7.3 The conjecture of Sendov
7.4 Notes
II ZEROS IN TERMS OF COEFFICIENTS
8 Inclusion of all zeros
8.1 The Cauchy bound and its estimates
8.2 Various refinements
8.3 Multipliers and the Enestrom-Kakeya theorem
8.4 More general expansions
8.5 Orthogonal expansions with real coefficients
8.6 Alternative approach by matrix methods
8.7 Notes
9 Inclusion of some of the zeros
9.1 Inclusions in terms of a norm
9.2 Pellet's theorem and its consequences
9.3 Bounds in terms of some of the coefficients
9.4 Orthogonal expansions with real coefficients
9.5 The Landau-Montel problem
9.6 Notes
10 Number of zeros in an interval
10.1 The Budan-Fourier theorem and Descartes' rule
10.2 Exact count under a side condition
10.3 Extensions to pairs of conjugate zeros
10.4 More general expansions
10.5 Exact count by Sturm sequences
10.6 Exact count via quadratic forms
10.7 Notes
11 Number of zeros in a domain
11.1 General principles
11.2 Number of zeros in a sector
11.3 Number of zeros in a half-plane
11.4 The Routh-Hurwitz problem
11.5 Number of zeros in a disc
11.6 Distribution of zeros
11.7 Notes
III EXTREMAL PROPERTIES
12 Growth estimates
12.1 The Bernstein-Walsh lemma
12.2 The convolution method
12.3 The method of functionals
12.4 Various refinements
12.5 Local behaviour
12.6 Extensions to functions of exponential type
12.7 Notes
13 Mean values
13.1 Mean values on circles
13.2 A class of linear operators
13.3 Mean values on the unit interval
13.4 Notes
14 Derivative estimates on the unit disc
14.1 Bernstein's inequality and generalizations
14.2 Refinements
14.3 Conditions on the coefficients
14.4 Conditions on the zeros
14.5 Some special operators
14.6 Inequalities involving mean values
14.7 Notes
15 Derivative estimates on the unit interval
15.1 Inequalities of S. Bernstein and A. Markov
15.2 Extensions to higher-order derivatives
15.3 Two other extensions
15.4 Dependence of the bounds on the zeros
15.5 Some special classes
15.6 LP analogues of Markov's inequality
15.7 Notes
16 Coefficient estimates
16.1 Polynomials on the unit circle
16.2 Coefficients of real trigonometric polynomials
16.3 Polynomials on the unit interval
16.4 Notes
References
List of notation
Index
