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Sheil-Small T. — Complex polynomials

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Название: Complex polynomials

Автор: Sheil-Small T.

Аннотация:

Complex Polynomials explores the geometric theory of polynomials and rational functions in the plane. Early chapters build the foundations of complex variable theory, melding together ideas from algebra, topology, and analysis. Throughout the book, the author introduces a variety of ideas and constructs theories around them, incorporating much of the classical theory of polynomials as he proceeds. These ideas are used to study a number of unsolved problems. Several solutions to problems are given, including a comprehensive account of the geometric convolution theory.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 428

Добавлена в каталог: 07.12.2009

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Предметный указатель

Homology group of punctured plane      43
Homotopic      39
Homotopic curves      40
Homotopy classes      41
Homotopy group      42
Homotopy invariant      41
Hurwitz's theorem      65
Hyperbolic non-euclidean geometry      375
Ilieff — Sendov conjecture      206
Ilieff — Sendov conjecture, application of Grace's theorem      208—211
Ilieff — Sendov conjecture, case of real polynomial      216
Ilieff — Sendov conjecture, critical circle      225
Ilieff — Sendov conjecture, extremal distance      221
Ilieff — Sendov conjecture, extremal polynomial      223
Ilieff — Sendov conjecture, independent of particular zero      225
Ilieff — Sendov conjecture, nearest second zero      216
Ilieff — Sendov conjecture, proof for zeros on unit circle      207
Ilieff — Sendov conjecture, proof up to degree 5      211—212
Ilieff — Sendov conjecture, proof when zero at origin      216
Ilieff — Sendov conjecture, remaining zeros on unit circle      220
Ilieff — Sendov conjecture, upper bound on distance      215
Ilieff — Sendov problem      see "Ilieff — Sendov conjecture"
Imaginary zeros      311—317
Implicit function theorem      63
Inequality for analytic polynomials      151—152
Inequality for harmonic polynomials      154
Inequality, implies surjectivity      101
Instantaneous double reversal      100—101
Integral mean theorem, zeros on unit circle      239
Integral representation of bounded convolution operator      168
Integral representation of Kaplan classes      293
Integral representation of non-vanishing polynomials      175
Interpretation of convolution conditions      270
Interspersed zeros and poles on unit circle      234
Interspersed zeros and poles, real rational functions      319
Interspersion lemma      232
Interspersion of zeros and local maxima      238
Interspersion theorem real meromorphic function      327
Interspersion theorem real rational functions      320
Interspersion theorem, unit circle      235
Inverse function      56
Inverse function inequality      356
Inverse function theorem      61
Inverse function, branch      58
Inverse function, continuation      57—58
Invertibility of harmonic multiplication operator      137
Inverting tract equation      18—19
Inverting transformation      18—19
Isolated zeros      8
Iterate      49
Jack, I.S.      266
Jacobian      61 124
Jacobian conjecture      81
Jacobian conjecture, algebraic resolution      124
Jacobian conjecture, condition in polar coordinates      88
Jacobian conjecture, degree of counter-examples      102—104
Jacobian conjecture, examples when true      87
Jacobian conjecture, proof for degree 2      85—86
Jacobian conjecture, proof under additional hypothesis      85
Jacobian conjecture, weak form      104—105
Jacobian determinant      61
Jacobian matrix      61
Jacobian operator, algebraic properties      91
Jacobian problem      89
Jacobian problem, geometric transformation      105
Jensen circle      304 305
Jordan curve      39
Jordan curve theorem      39
Jordan domain      39
Jordan polygon and dilatation criterion      404
Jordan polygon, mapping problem      402
Julia — Caratheodory lemma      330
Kaplan class      244 277 382 385
Kaplan class, coefficient bounds      294
Kaplan class, extreme points      293
Kaplan class, factorisation theorem      246
Kaplan class, integral representation      293
Kaplan class, K(1,1)      271
Kaplan class, linear functionals      293
Kaplan, W.      241
Keller Jacobian conjecture      87
Kernel      132
Knot      120
Krein — Milman theorem      287
Kristiansen, G.K.      240
Lagrange's interpolation formula      2
Laguerre — Polya class      347
Laurent expansion      126
Lax's theorem      153—154 186
Leading term      88
Length of curve      85
Level curve      66 306—307 350
Level curve of polynomials      351
Level curve, geometry      310
Level region      306 352
Level region and Smale's conjecture      364
Level region of rational function      353
Level region, convexity      360
Levin representation      332
Levin representation lemma      333
Lewandowski, Z.      242
Lewy's theorem      64
Limits of Suffridge's extremal polynomials      258
Linear form      181
Linear functional      131 177
Linear functional on Kaplan class      293
Linear functional on rational function      264—265
Linear operator      132
Linear operator lemma      272—273
Linear operator on polynomials      203
Linear operator on rational functions      264
Linear operator, Grace theorem      205
Linearly accessible domain      302
Liouville's theorem      26
Littlewood, J.E.      239
Lobachevsky      375
Local multiplicity      56
Local uniform convergence      128
Locally 1-1      24 82
Locally 1-1, function      56
Locally 1-1, harmonic functions      64
Locally 1-1, polynomial, topology      100
Locally bounded      128
Locating critical points      186
Location of zeros given critical points      201—202
Logarithm      28
Logarithmic derivative lemma      187
Logarithmic derivative, algebra      407
Logarithmic derivative, critical points      306
Logarithmic derivative, existence of non-real critical points      307
Logarithmic differentiation      23
Loop      28
Loop lemma      121
Lyzzaik, A.      392
Magnus theorem      109
Majorisation      171
Mapping problem for Jordan polygons      402
Marden, M.      206
Mason's theorem      370
Max-min inequalities      200—201
Maximum modulus      238
Maximum principle      57
Mean      145
Meromorphic function      27 45
Monodromy theorem      58
Multiple zeros      187
Multiplicity      22
Multiplicity of function at point      44
Multiplicity of harmonic function with polynomial co-analytic part      47
Multiplicity, analytic expression      24

N-fold mapping and finite valence      390
N-fold mapping of circle      382—387
N-fold mapping, Fourier coefficients      391
n-valent      66
Nakai and Bab a theorem      111
NE convex      376
NE line      375
Nearest second zero      216
Negative type: strongly real rational function      321
Non-Euclidean line      375
Non-isolated zero      53
Non-real critical points of real rational function      323—324
Non-real zeros theorem      325
Non-separating lemma      119
Non-vanishing polynomials      173
Norm of operator      140
Norm of self-inversive polynomial      153
Normal family      128
Null homotopic      41
Number of isolated zeros of real analytic polynomial      8
Number of zeros      1
Number of zeros of real analytic polynomial      4
Open mapping      57
Order of critical point      353
Order of entire function      327
Order of meromorphic function      327
Orevkov, S.Yu.      118
Orientation      56
p-mean preserving operator      169
parabola      78
Parabolic region      78
Parametrisation      27
Parseval's formula      131
Partial fraction decomposition of rational function      355
Periodic      129
Pinchuk surface      100
Pinchuk's example      90—93
Pinchuk, S.      81 118
Plane topology      39
Poincare, H.      375
Poisson's formula      128
Polar coordinate      28
Polar coordinate form      82
Polar coordinate, Jacobian conjecture      88
Polar derivative      185
Pole      27 44
Polya and Schoenberg      161
Polya and Schoenberg's proof      161—165
Polya and Schoenberg's theorem      161
Polya — Schoenberg conjecture      260 275 276
Polya's theorem      347
Polynomial mean      145
Polynomial, constant on curve      84
Polynomial, non-vanishing      173
Polynomial, with all real zeros, Descartes' rule      318
Polynomial, with zeros on unit circle      231
Polynomially invertible      87 105
Popoviciu conjecture      407
Positive convolution operator      142
Positive harmonic function      142
Positive operator      138
Positive trigonometric polynomial      144—151
Positive trigonometric polynomial and convexity preserving operator      148
Positive trigonometric polynomial, representation      149
Positive trigonometric polynomial, representation theorem      150
Positive type, strongly real rational function      321
Positivity lemma      157
Prime degree      109—110
primitive      31
Problems      25—26 29 31 40 43 46—47 55—56 78—79 80 117 124 138 144 151—152 155—156 159—160 203 208 232 237—238 240 391—392
Projection onto plane      89
Projective plane      123
Properties of Sylvester resultant      12
Pseudo-surface      121 123
Pythagorean triples      372
Quadratic polynomials      373
Quadrilateral, harmonic mapping problem      405
Radical      23
Rado — Kneser — Choquet theorem      59 392
Ratio of linear functionals      180
Rational function in unit disc      193
Rational function with real critical points      325—326
Rational function, convolution containment theorem      265
Rational function, critical points      190
Rational function, distinct solutions      370
Rational function, Grace theorem      263
Rational function, linear functional theorem      264—265
Rational function, linear operator theorem      264
Rational function, proof of extended Grace theorem      269
Rational function, strongly real/interspersion theorem      320
Real analytic polynomial      4 27 81
Real analytic polynomial at infinity      13—19
Real critical point      325
Real polynomial      1 304
Real polynomial with imaginary zeros      311—317
Real polynomial, critical point      305
Real polynomial, Ilieff — Sendov conjecture      216
Real polynomial, representation as sum of polynomials with real zeros      322
Real rational function, critical point theorem      323—324
Real rational function, interspersed zeros and poles on real axis      319
Real rational function, representation in terms of strongly real rational functions      321
Real zeros, Descartes' rule      318
Rectilinear polygon      35
Relatively prime degrees      109—110
Removable singularity      44
Repeated asymptotic value curve      18
Repetition of asymptotic values      16
Repetition of reduction process      15
Repetition property      102 117
Repetition, third method      16—17
Representation for harmonic polynomial      3
Representation of complex polynomial      2 3
Representation of linear operator      132—133
Residue      43
Residue theorem      24 27
Resolving the singularity      13
Resultant      124
Resultant and Jacobian      124
Reverse inequalities      216
Reversed asymptotic value curve      18
Riemann mapping theorem      35 39
Riemann — Hurwitz formula      353
Robinson's conjectures      300—301
Robinson, S.      300 303
Rogosinski's coefficient theorem      170
Rogosinski's lemma      157
Rotation conjecture      392
Rouche's theorem      48
Rubinstein, Z.      207
Rule of signs, Descartes'      317
Ruscheweyh theorem for $S(\alpha,\beta)$       297
Ruscheweyh, St.      281
Schmeisser, G.      216
Schoenflies theorem      59
Schwarz function      168
Schwarz's lemma      60 234
Second derivative, non-real zeros      325
Second dual      172
Second zero      216
Self-intersections      123
Self-inversive polynomial      149 152 228
Self-inversive polynomial on unit circle      229
Self-inversive polynomial, norm theorem      153
Self-inversive polynomial, representation lemma      254
Self-inversive polynomial, zeros and critical points      230
Sendov, B.      206
Sense preserving      56 61
Sense reversing      56 61
Separated sets      30
Separation by a line      186

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