Complex polynomials:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2002
|
| Edition: | 1. publ. |
| Series: | Cambridge studies in advanced mathematics
75 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 428 S. |
| ISBN: | 0521400686 9780521400688 |
Staff View
MARC
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface page xi
List of notation xix
1 The algebra of polynomials 1
1.1 Complex polynomials 1
1.2 The number of zeros of a real analytic polynomial 4
1.3 Real analytic polynomials at infinity 13
2 The degree principle and the fundamental theorem
of algebra 22
2.1 The fundamental theorem of algebra 22
2.2 Continuous functions in the plane 26
2.3 The degree principle 31
2.4 The degree principle and homotopy 40
2.5 The topological argument principle 43
2.6 The coincidence theorem 47
2.7 Locally 1 1 functions 56
2.8 The Borsuk Ulam theorem 79
3 The Jacobian problem 81
3.1 The Jacobian conjecture 81
3.2 Pinchuk s example 90
3.3 Polynomials with a constant Jacobian 105
3.4 A topological approach 118
3.5 The resultant and the Jacobian 124
4 Analytic and harmonic functions in the unit disc 125
4.1 Series representations 125
4.2 Positive and bounded operators 138
4.3 Positive trigonometric polynomials 144
4.4 Some inequalities for analytic and trigonometric
polynomials 151
vii
viii Contents
4.5 Cesaro means 156
4.6 De la Vallee Poussin means 160
4.7 Integral representations 167
4.8 Generalised convolution operators 168
5 Circular regions and Grace s theorem 172
5.1 Convolutions and duality 172
5.2 Circular regions 178
5.3 The polar derivative 184
5.4 Locating critical points 186
5.5 Critical points of rational functions 190
5.6 The Borwein Erdelyi inequality 193
5.7 Univalence properties of polynomials 196
5.8 Linear operators 203
6 The Ilieff Sendov conjecture 206
6.1 Introduction 206
6.2 Proof of the conjecture for those zeros on the unit circle 207
6.3 The direct application of Grace s theorem 208
6.4 A global upper bound 213
6.5 Inequalities relating the nearest critical point to the
nearest second zero 216
6.6 The extremal distance 221
6.7 Further remarks on the conjecture 223
7 Self inversive polynomials 228
7.1 Introduction 228
7.2 Polynomials with interspersed zeros on the unit circle 232
7.3 Relations with the maximum modulus 238
7.4 Univalent polynomials 241
7.5 A second necessary and sufficient condition for angular
separation of zeros 249
7.6 Suffridge s extremal polynomials 251
8 Duality and an extension of Grace s theorem to rational
functions 263
8.1 Linear operators and rational functions 263
8.2 Interpretations of the convolution conditions 270
8.3 The duality theorem for 7 (1,£) 275
8.4 The duality theorem for T(m, 0) 282
8.5 The duality principle 286
8.6 Duality and the class T(a, /3) 289
8.7 Properties of the Kaplan classes 293
8.8 The class S(a, p) 296
Contents ix
8.9 The classes To(a, 0) 300
8.10 The class 7(2, 2) 302
9 Real polynomials 304
9.1 Real polynomials 304
9.2 Descartes rule of signs 317
9.3 Strongly real rational functions 319
9.4 Critical points of real rational functions 323
9.5 Rational functions with real critical points 325
9.6 Real entire and meromorphic functions 326
10 Level curves 350
10.1 Level regions for polynomials 350
10.2 Level regions of rational functions 353
10.3 Partial fraction decomposition 355
10.4 Smale s conjecture 358
11 Miscellaneous topics 370
11.1 The abc theorem 370
11.2 Cohn s reduction method 372
11.3 Blaschke products 373
11.4 Blaschke products and harmonic mappings 377
11.5 Blaschke products and convex curves 382
11.6 Blaschke products and convex polygons 392
11.7 The mapping problem for Jordan polygons 402
11.8 Sudbery s theorem on zeros of successive derivatives 407
11.9 Extensions of Sudbery s theorem 413
References 416
Index 421
|
| any_adam_object | 1 |
| author | Sheil-Small, Terence 1937- |
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| ctrlnum | (OCoLC)48223532 (DE-599)BVBBV014650194 |
| dewey-full | 512.9/42 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/42 |
| dewey-search | 512.9/42 |
| dewey-sort | 3512.9 242 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV014650194 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:04:40Z |
| institution | BVB |
| isbn | 0521400686 9780521400688 |
| language | English |
| lccn | 2001052690 |
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| owner_facet | DE-20 DE-355 DE-BY-UBR DE-11 DE-83 |
| physical | XIX, 428 S. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Sheil-Small, Terence 1937- Verfasser (DE-588)142352241 aut Complex polynomials T. Sheil-Small 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2002 XIX, 428 S. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 75 Fonctions d'une variable complexe Polinômios larpcal Polynômes Álgebra larpcal Functions of complex variables Polynomials Komplexe Funktion (DE-588)4217733-9 gnd rswk-swf Polynom (DE-588)4046711-9 gnd rswk-swf Komplexe Funktion (DE-588)4217733-9 s DE-604 Polynom (DE-588)4046711-9 s Cambridge studies in advanced mathematics 75 (DE-604)BV000003678 75 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009945117&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sheil-Small, Terence 1937- Complex polynomials Cambridge studies in advanced mathematics Fonctions d'une variable complexe Polinômios larpcal Polynômes Álgebra larpcal Functions of complex variables Polynomials Komplexe Funktion (DE-588)4217733-9 gnd Polynom (DE-588)4046711-9 gnd |
| subject_GND | (DE-588)4217733-9 (DE-588)4046711-9 |
| title | Complex polynomials |
| title_auth | Complex polynomials |
| title_exact_search | Complex polynomials |
| title_full | Complex polynomials T. Sheil-Small |
| title_fullStr | Complex polynomials T. Sheil-Small |
| title_full_unstemmed | Complex polynomials T. Sheil-Small |
| title_short | Complex polynomials |
| title_sort | complex polynomials |
| topic | Fonctions d'une variable complexe Polinômios larpcal Polynômes Álgebra larpcal Functions of complex variables Polynomials Komplexe Funktion (DE-588)4217733-9 gnd Polynom (DE-588)4046711-9 gnd |
| topic_facet | Fonctions d'une variable complexe Polinômios Polynômes Álgebra Functions of complex variables Polynomials Komplexe Funktion Polynom |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009945117&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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