Nonlinear Equations - UFRJ

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176 INDEX inner product Weyl’s, 64, 68 Kahler form, 48, 57 Kantorovich, 82 Legendre’s transform, 72 Legendre-Fenchel transform, 73 Lemma Noether normalization, 21, 29 lemma consequence of Hahn-Banach, 73 Dickson, 16 manifold abstract, 35 complex, 41 embedded, 34 embedded with boundary, 35 one dimensional, 36 orientation, 35 metric associated to a fewspace, 59 Fubini-Study, 59 Minkowski linear combinations, 9 momentum map, 75 Newton iteration, 121 plain, 82 Noetherian ring, 23 polarization bound, 85 projective space, 51 volume, 52 pseudo-inverse, 123 reproducing kernel, 57 short path, 155 singular value decomposition, 107 Smale’s 17th problem, 137 Smale’s 17th prolem, 11 Smale’s invariant gamma, 134 Smale’s invariants alpha, 97 beta, 97 gamma, 84 pseudo-Newton, 125 smooth analysis, 153 starting system, 149 Sylvester matrix, 13 resultant, 13 Sylvester’s resultant, 12 theorem, 48, 57, 60 alpha, 97, 130 robust, 105 sharp, 103 average conditioning, 149 Beltrán and Pardo, 136 Bernstein, 9 proof, 81 Bézout, 2, 23 average, 63 proof of multihomogeneous, 70 sketch of proof, 4 co-area formula, 49, 51 complex roots are lsc, 41
INDEX 177 complexity of homotopy, 140 proof, 147 condition number general, 116 homogeneous, 114 linear, 109 unmixed, 112 Eckart-Young, 109 gamma, 87, 128 robust, 94 sharp, 93 general root count, 69 Hahn-Banach, 73 Hilbert’s basis, 15, 16 Hilbert’s Nullstellensatz, 27 Kushnirenko, 8 proof, 79 Main theorem of elimination theory, 30 mu, 119 multihomogeneous Bezout, 7 primary decomposition, 25 root density, 68 Shub and Smale, 135 Smale, 87, 97, 128, 130 toric infinity, 80 variety algebraic, 29 degree, 29 dimension, 29 discriminant, 138 solution, 31, 138 wedge product, 43 Zariski topology, 1, 15
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Nonlinear Equations
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Copyright © 2011 by Gregorio Malaj
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Foreword I added together the ratio
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ix another book with a systematic p
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Contents Foreword vii 1 Counting so
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CONTENTS xiii 10 Homotopy 135 10.1
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2 [CH. 1: COUNTING SOLUTIONS if and
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4 [CH. 1: COUNTING SOLUTIONS connec
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6 [CH. 1: COUNTING SOLUTIONS 1.2 Sh
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8 [CH. 1: COUNTING SOLUTIONS has ex
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10 [CH. 1: COUNTING SOLUTIONS equat
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Chapter 2 The Nullstellensatz The s
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14 [CH. 2: THE NULLSTELLENSATZ prin
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16 [CH. 2: THE NULLSTELLENSATZ or a
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18 [CH. 2: THE NULLSTELLENSATZ 3. T
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20 [CH. 2: THE NULLSTELLENSATZ and
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22 [CH. 2: THE NULLSTELLENSATZ 1. T
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24 [CH. 2: THE NULLSTELLENSATZ Proo
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26 [CH. 2: THE NULLSTELLENSATZ To e
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28 [CH. 2: THE NULLSTELLENSATZ for
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30 [CH. 2: THE NULLSTELLENSATZ 2.7
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32 [CH. 2: THE NULLSTELLENSATZ Coro
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34 [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
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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
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86 [CH. 7: NEWTON ITERATION Proposi
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88 [CH. 7: NEWTON ITERATION 1 y =
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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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124 [CH. 9: THE PSEUDO-NEWTON OPERA
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Page 166 and 167: 150 [CH. 10: HOMOTOPY Corollary 10.
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Page 173 and 174: Appendix A Open Problems, by Carlos
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Page 177 and 178: [SEC. A.5: EXTENSION OF THE ALGORIT
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Page 182 and 183: 166 BIBLIOGRAPHY [12] , Smale’s 1
Page 184 and 185: 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: Index algorithm discrete, x Homotop
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