Nonlinear Equations - UFRJ

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CONTENTS
4.4 The co-area formula . . . . . . . . . . . . . . . . . . . 48
4.5 Projective space . . . . . . . . . . . . . . . . . . . . . 51
5 Reproducing kernel spaces 55
5.1 Fewspaces . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Metric structure on root space . . . . . . . . . . . . . 58
5.3 Root density . . . . . . . . . . . . . . . . . . . . . . . 60
5.4 Affine and multi-homogeneous setting . . . . . . . . . 63
5.5 Compactifications . . . . . . . . . . . . . . . . . . . . . 65
6 Exponential sums and sparse polynomial systems 72
6.1 Legendre’s transform . . . . . . . . . . . . . . . . . . . 72
6.2 The momentum map . . . . . . . . . . . . . . . . . . . 75
6.3 Geometric considerations . . . . . . . . . . . . . . . . 77
6.4 Calculus of polytopes and kernels . . . . . . . . . . . . 79
7 Newton iteration 82
7.1 The gamma invariant . . . . . . . . . . . . . . . . . . . 83
7.2 The γ-Theorems . . . . . . . . . . . . . . . . . . . . . 87
7.3 Estimates from data at a point . . . . . . . . . . . . . 96
8 Condition number theory 107
8.1 Linear equations . . . . . . . . . . . . . . . . . . . . . 107
8.2 The linear term . . . . . . . . . . . . . . . . . . . . . . 110
8.3 The condition number for unmixed systems . . . . . . 111
8.4 Condition numbers for homogeneous systems . . . . . 113
8.5 Condition numbers in general . . . . . . . . . . . . . . 114
8.6 Inequalities about the condition number . . . . . . . . 118
9 The pseudo-Newton operator 121
9.1 The pseudo-inverse . . . . . . . . . . . . . . . . . . . . 123
9.2 Alpha theory . . . . . . . . . . . . . . . . . . . . . . . 125
9.3 Approximate zeros . . . . . . . . . . . . . . . . . . . . 127
9.4 The alpha theorem . . . . . . . . . . . . . . . . . . . . 130
9.5 Alpha-theory and conditioning . . . . . . . . . . . . . 133