Geometry of polynomials

Cover......Page 1
Title page\0......Page 2
Contents\0......Page 6
Preface to the Second Edition\0......Page 8
Preface to the First Edition\0......Page 10
Abbreviations\0......Page 14
1. Some basic theorems\0......Page 16
2. The zeros of the derivative\0......Page 21
3. Physical interpretations\0......Page 22
4. Geometric interpretation\0......Page 24
5. Function-theoretic interpretations. Infrapolynomials\0......Page 28
6. The convex hull of critical points\0......Page 36
7. The critical points of a real polynomial\0......Page 40
8. Some generalizations\0......Page 44
9. Polynomial solutions of Lamé\'s differential equation\0......Page 51
10. The derivative under linear transformations\0......Page 58
11. Covariant force fields\0......Page 60
12 Circular regions\0......Page 63
13. Zeros of the polar derivative\0......Page 64
14. Generalization to abstract spaces\0......Page 70
15. Apolar polynomials\0......Page 75
16. Applications\0......Page 80
17. Linear combinations of polynomials\0......Page 89
18. Combinations of a polynomial and its derivatives\0......Page 96
19 A two-circle theorem for polynomials\0......Page 104
20. Two-circle theorems for rational functions\0......Page 108
21. The general case\0......Page 111
22. Some important special cases\0......Page 117
23. Polynomials with two given zeros\0......Page 122
24. Mean-Value Theorems\0......Page 125
25. Polynomials with p known zeros\0......Page 128
26. Alternative treatment\0......Page 133
27. The moduli of the zeros\0......Page 137
28. The ? zeros of smallest modulus\0......Page 143
29. Refinement of the bounds\0......Page 145
30. Applications\0......Page 148
31. Matrix methods\0......Page 154
32. Construction of bounds\0......Page 162
33. Further bounds\0......Page 166
34. Lacunary polynomials\0......Page 171
35. Other bounds for lacunary polynomials\0......Page 178
36. Dynamic stability\0......Page 181
37. Cauchy indices\0......Page 183
38. Sturm sequences\0......Page 186
39. Determinant sequences\0......Page 189
40. The number of zeros with negative real parts\0......Page 194
41. The number of zeros in a sector\0......Page 204
42. An algorithm\0......Page 209
43. Determinant sequences\0......Page 213
44. Polynomials with zeros on or symmetric in the unit circle\0......Page 216
45. Singular determinant sequences\0......Page 218
Bibliography\0......Page 222
Index\0......Page 256