Nonlinear Equations - UFRJ

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Copyright © 2011 by Gregorio Malajovich Impresso no Brasil / Printed in Brazil Capa: Noni Geiger / Sérgio R. Vaz 28 o Colóquio Brasileiro de Matemática • Cadeias de Markov e Teoria do Potencial - Johel Beltrán • Cálculo e Estimação de Invariantes Geométricos: Uma Introdução às Geometrias Euclidiana e Afim - M. Andrade e T. Lewiner • De Newton a Boltzmann: o Teorema de Lanford - Sérgio B. Volchan • Extremal and Probabilistic Combinatorics - Robert Morris e Roberto Imbuzeiro Oliveira • Fluxos Estrela - Alexander Arbieto, Bruno Santiago e Tatiana Sodero • Geometria Aritmética em Retas e Cônicas - Rodrigo Gondim • Hydrodynamical Methods in Last Passage Percolation Models - E. A. Cator e L. P. R. Pimentel • Introduction to Optimal Transport: Theory and Applications - Nicola Gigli • Introdução à Aproximação Numérica de Equações Diferenciais Parciais Via o Método de Elementos Finitos - Juan Galvis e Henrique Versieux • Matrizes Especiais em Matemática Numérica - Licio Hernanes Bezerra • Mecânica Quântica para Matemáticos em Formação - Bárbara Amaral, Alexandre Tavares Baraviera e Marcelo O. Terra Cunha • Multiple Integrals and Modular Differential Equations - Hossein Movasati • Nonlinear Equations - Gregorio Malajovich • Partially Hyperbolic Dynamics - Federico Rodriguez Hertz, Jana Rodriguez Hertz e Raúl Ures • Processos Aleatórios com Comprimento Variável - A. Toom, A. Ramos, A. Rocha e A. Simas • Um Primeiro Contato com Bases de Gröbner - Marcelo Escudeiro Hernandes ISBN: 978-85-244-329-3 Distribuição: IMPA Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ E-mail: ddic@impa.br http://www.impa.br
To Beatriz
Page 3: Nonlinear Equations
Page 9 and 10: Foreword I added together the ratio
Page 11 and 12: ix another book with a systematic p
Page 13 and 14: Contents Foreword vii 1 Counting so
Page 15: CONTENTS xiii 10 Homotopy 135 10.1
Page 18 and 19: 2 [CH. 1: COUNTING SOLUTIONS if and
Page 20 and 21: 4 [CH. 1: COUNTING SOLUTIONS connec
Page 22 and 23: 6 [CH. 1: COUNTING SOLUTIONS 1.2 Sh
Page 24 and 25: 8 [CH. 1: COUNTING SOLUTIONS has ex
Page 26 and 27: 10 [CH. 1: COUNTING SOLUTIONS equat
Page 28 and 29: Chapter 2 The Nullstellensatz The s
Page 30 and 31: 14 [CH. 2: THE NULLSTELLENSATZ prin
Page 32 and 33: 16 [CH. 2: THE NULLSTELLENSATZ or a
Page 34 and 35: 18 [CH. 2: THE NULLSTELLENSATZ 3. T
Page 36 and 37: 20 [CH. 2: THE NULLSTELLENSATZ and
Page 38 and 39: 22 [CH. 2: THE NULLSTELLENSATZ 1. T
Page 40 and 41: 24 [CH. 2: THE NULLSTELLENSATZ Proo
Page 42 and 43: 26 [CH. 2: THE NULLSTELLENSATZ To e
Page 44 and 45: 28 [CH. 2: THE NULLSTELLENSATZ for
Page 46 and 47: 30 [CH. 2: THE NULLSTELLENSATZ 2.7
Page 48 and 49: 32 [CH. 2: THE NULLSTELLENSATZ Coro
Page 50 and 51: 34 [CH. 3: TOPOLOGY AND ZERO COUNTI
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40 [CH. 3: TOPOLOGY AND ZERO COUNTI
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Chapter 4 Differential forms Throug
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44 [CH. 4: DIFFERENTIAL FORMS Prope
Page 62 and 63:
46 [CH. 4: DIFFERENTIAL FORMS Lemma
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48 [CH. 4: DIFFERENTIAL FORMS 2. cl
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50 [CH. 4: DIFFERENTIAL FORMS be me
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52 [CH. 4: DIFFERENTIAL FORMS Expan
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54 [CH. 4: DIFFERENTIAL FORMS We co
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56 [CH. 5: REPRODUCING KERNEL SPACE
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Chapter 6 Exponential sums and spar
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74 [CH. 6: EXPONENTIAL SUMS AND SPA
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Chapter 7 Newton Iteration and Alph
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84 [CH. 7: NEWTON ITERATION As long
Page 102 and 103:
86 [CH. 7: NEWTON ITERATION Proposi
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88 [CH. 7: NEWTON ITERATION 1 y =
Page 106 and 107:
90 [CH. 7: NEWTON ITERATION t 1 t 2
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92 [CH. 7: NEWTON ITERATION The Tay
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94 [CH. 7: NEWTON ITERATION 2 63 2
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96 [CH. 7: NEWTON ITERATION Exercis
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98 [CH. 7: NEWTON ITERATION The con
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100 [CH. 7: NEWTON ITERATION Let us
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102 [CH. 7: NEWTON ITERATION Passin
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104 [CH. 7: NEWTON ITERATION 13−3
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106 [CH. 7: NEWTON ITERATION Proof.
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108 [CH. 8: CONDITION NUMBER THEORY
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122 [CH. 9: THE PSEUDO-NEWTON OPERA
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136 [CH. 10: HOMOTOPY form for x i+
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138 [CH. 10: HOMOTOPY Again, f t is
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140 [CH. 10: HOMOTOPY We will need
Page 158 and 159:
142 [CH. 10: HOMOTOPY P n+1 x i [N(
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144 [CH. 10: HOMOTOPY Let d Riem de
Page 162 and 163:
146 [CH. 10: HOMOTOPY Lemma 10.13.
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148 [CH. 10: HOMOTOPY above, the se
Page 166 and 167:
150 [CH. 10: HOMOTOPY Corollary 10.
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152 [CH. 10: HOMOTOPY by the geomet
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154 [CH. 10: HOMOTOPY 2. The condit
Page 173 and 174:
Appendix A Open Problems, by Carlos
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[SEC. A.3: EQUIDISTRIBUTION OF ROOT
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[SEC. A.5: EXTENSION OF THE ALGORIT
Page 179:
[SEC. A.7: INTEGER ZEROS OF A POLYN
Page 182 and 183:
166 BIBLIOGRAPHY [12] , Smale’s 1
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168 BIBLIOGRAPHY [42] Michael R. Ga
Page 186 and 187:
170 BIBLIOGRAPHY [69] , Complexity
Page 189 and 190:
Glossary of notations As a general
Page 191 and 192:
Index algorithm discrete, x Homotop
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INDEX 177 complexity of homotopy, 1
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