TABLE OF CONTENTS
PREFACES vii
ABBREVIATIONS xiii
CHAPTER I
INTRODUCTION
1. Some basic theorems 1
2. The zeros of the derivative 6
3. Physical interpretations 7
4. Geometric interpretation 9
5. Function-theoretic interpretations. Infrapolynomials 13
CHAPTER II
THE CRITICAL POINTS OF A POLYNOMIAL
6. The convex hull of critical points 21
7. The critical points of a real polynomial 25
8. Some generalizations 29
9. Polynomial solutions of Lame's differential equation 36
CHAPTER III
INVARIANTIVE FORMULATION
10. The derivative under linear transformations 43
11. Covariant force fields 45
12. Circular regions 48
13. Zeros of the polar derivative 49
14. Generalization to abstract spaces 55
CHAPTER IV
COMPOSITE POLYNOMIALS
15. Apolar polynomials 60
16. Applications 65
17. Linear combinations of polynomials 74
18. Combinations of a polynomial and its derivatives 81
CHAPTER V
THE CRITICAL POINTS OF A RATIONAL FUNCTION WHICH HAS ITS ZEROS
AND POLES IN PRESCRIBED CIRCULAR REGIONS
19. A two-circle theorem for polynomials 89
20. Two-circle theorems for rational functions 93
21. The general case 96
22. Some important special cases 102
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