Geometry of Polynomials

During the years since the first edition of this well-known monograph appeared, the subject (the geometry of the zeros of a complex polynomial) has continued to display the same outstanding vitality as it did in the first 150 years of its history, beginning with the contributions of Cauchy and Gauss. Thus, the number of entries in the bibliography of this edition had to be increased from about 300 to about 600 and the book enlarged by one third. It now includes a more extensive treatment of Hurwitz polynomials and other topics. The new material on infrapolynomials, abstract polynomials, and matrix methods is of particular interest. Cover......Page 1 Title page......Page 2 Contents......Page 6 Preface to the Second Edition......Page 8 Preface to the First Edition......Page 10 Abbreviations......Page 14 1. Some basic theorems......Page 16 2. The zeros of the derivative......Page 21 3. Physical interpretations......Page 22 4. Geometric interpretation......Page 24 5. Function-theoretic interpretations. Infrapolynomials......Page 28 6. The convex hull of critical points......Page 36 7. The critical points of a real polynomial......Page 40 8. Some generalizations......Page 44 9. Polynomial solutions of Lamé's differential equation......Page 51 10. The derivative under linear transformations......Page 58 11. Covariant force fields......Page 60 12 Circular regions......Page 63 13. Zeros of the polar derivative......Page 64 14. Generalization to abstract spaces......Page 70 15. Apolar polynomials......Page 75 16. Applications......Page 80 17. Linear combinations of polynomials......Page 89 18. Combinations of a polynomial and its derivatives......Page 96 19 A two-circle theorem for polynomials......Page 104 20. Two-circle theorems for rational functions......Page 108 21. The general case......Page 111 22. Some important special cases......Page 117 23. Polynomials with two given zeros......Page 122 24. Mean-Value Theorems......Page 125 25. Polynomials with p known zeros......Page 128 26. Alternative treatment......Page 133 27. The moduli of the zeros......Page 137 28. The ? zeros of smallest modulus......Page 143 29. Refinement of the bounds......Page 145 30. Applications......Page 148 31. Matrix methods......Page 154 32. Construction of bounds......Page 162 33. Further bounds......Page 166 34. Lacunary polynomials......Page 171 35. Other bounds for lacunary polynomials......Page 178 36. Dynamic stability......Page 181 37. Cauchy indices......Page 183 38. Sturm sequences......Page 186 39. Determinant sequences......Page 189 40. The number of zeros with negative real parts......Page 194 41. The number of zeros in a sector......Page 204 42. An algorithm......Page 209 43. Determinant sequences......Page 213 44. Polynomials with zeros on or symmetric in the unit circle......Page 216 45. Singular determinant sequences......Page 218 Bibliography......Page 222 Index......Page 256 Includes an extensive treatment of Hurwitz polynomials and other topics. This title features material on infrapolynomials, abstract polynomials, and matrix methods.