Nonlinear Equations - UFRJ

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Chapter 6 Exponential sums and sparse polynomial systems The objective of this chapter is to prove Kushnirenko’s and Bernstein’s theorems. We will need a few preliminaries of convex geometry. 6.1 Legendre’s transform Through this section, let E be a Hilbert space. Definition 6.1. Recall that a subset U of E is convex if and only if, for all v 0 , v 1 ∈ U and for all t ∈ [0, 1], (1 − t)v 0 + tv 1 ∈ U. Gregorio Malajovich, <strong>Nonlinear</strong> equations. 28 o Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 2011. Copyright c○ Gregorio Malajovich, 2011. 72
[SEC. 6.1: LEGENDRE’S TRANSFORM 73 Lemma 6.2. A set U is convex if and only if U is an intersection of closed half-spaces. In order to prove this Lemma we need a classical fact about Hilbert spaces: Lemma 6.3. Let U be a convex subset in a Hilbert space, and let p ∉ U. Then there is a hyperplane separating U and p, namely where α ∈ E ∗ . x ∈ U ⇒ α(x) < α(p) This is a consequence of the Hahn-Banach theorem, see [23] Lemma I.3 p.6. Proof of Lemma 6.2. Assume that U is convex. Then, let S be the collection of all half-spaces H α,α0 = {α(x)−α 0 ≥ 0}, α ∈ E ∗ , α 0 ∈ R, such that U ⊆ H α,α0 . Clearly U ⊆ ⋂ H α,α0 . α,α 0∈S Equality follows from Lemma 6.3. The reciprocal is easy and left to the reader. Definition 6.4. A function f : U ⊆ E → R is convex if and only if its epigraph Epi f = {(x, y) : f(x) ≤ y} is convex. Note that from this definition, the domain of a convex function is always convex. In this book we shall convention that a convex function has non-empty domain. Definition 6.5. The Legendre-Fenchel transform of a function f : U ⊆ E → R is the function f ∗ : U ∗ ⊆ E ∗ → R given by f ∗ (α) = sup α(x) − f(x). x∈U
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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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Page 98 and 99: Chapter 7 Newton Iteration and Alph
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Page 112 and 113: 96 [CH. 7: NEWTON ITERATION Exercis
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Page 122 and 123: 106 [CH. 7: NEWTON ITERATION Proof.
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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
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142 [CH. 10: HOMOTOPY P n+1 x i [N(
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144 [CH. 10: HOMOTOPY Let d Riem de
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146 [CH. 10: HOMOTOPY Lemma 10.13.
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148 [CH. 10: HOMOTOPY above, the se
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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
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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
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[SEC. A.7: INTEGER ZEROS OF A POLYN
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166 BIBLIOGRAPHY [12] , Smale’s 1
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168 BIBLIOGRAPHY [42] Michael R. Ga
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170 BIBLIOGRAPHY [69] , Complexity
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Glossary of notations As a general
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Index algorithm discrete, x Homotop
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INDEX 177 complexity of homotopy, 1
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