Complex Polynomials - Sheil-Small, T.
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Melding together ideas from algebra, topology and analysis, this book studies the geometric theory of polynomials and rational functions in the plane. Any theory in the plane should make full use of the complex numbers and thus the early chapters build the foundations of complex variable theory. In fact, throughout the book, the author introduces a variety of ideas and constructs theories around them, incorporating much of the classical theory of polynomials as he proceeds. These ideas are used to study a number of unsolved problems, bearing in mind that such problems indicate the current…mehr

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Produktbeschreibung
Melding together ideas from algebra, topology and analysis, this book studies the geometric theory of polynomials and rational functions in the plane. Any theory in the plane should make full use of the complex numbers and thus the early chapters build the foundations of complex variable theory. In fact, throughout the book, the author introduces a variety of ideas and constructs theories around them, incorporating much of the classical theory of polynomials as he proceeds. These ideas are used to study a number of unsolved problems, bearing in mind that such problems indicate the current limitations of our knowledge and present challenges for the future. However, theories also lead to solutions of some problems and several such solutions are given including a comprehensive account of the geometric convolution theory. This is an ideal reference for graduate students and researchers working in this area.

Table of contents:
1. The algebra of polynomials; 2. The degree principle and the fundamental theorem of algebra; 3. The Jacobian problem; 4. Analytic and harmonic functions in the unit disc; 5. Circular regions and Grace's theorem; 6. The Ilieff-Sendoff conjecture; 7. Self-inversive polynomials; 8. Duality and an extension of Grace's theorem to rational functions; 9. Real polynomials; 10. Level curves; 11. Miscellaneous topics.

This book studies the geometric theory of polynomials and rational functions in the plane. The theory is carefully constructed bearing in mind the needs of graduate students. Several unsolved problems are presented as well as the full solutions to some well known conjectures.

Geometric theory of polynomials and rational functions for graduate students and researchers.