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

v