Positive Polynomials and Sums of Squares (Mathematical Surveys and Monographs)

The study of positive polynomials brings together algebra, geometry and analysis. The subject is of fundamental importance in real algebraic geometry, when studying the properties of objects defined by polynomial inequalities. Hilbert's 17th problem and its solution in the first half of the 20th century were landmarks in the early days of the subject. More recently, new connections to the moment problem and to polynomial optimization have been discovered. The moment problem relates linear maps on the multidimensional polynomial ring to positive Borel measures. This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmudgen's solution to the moment problem in the compact case in 1991. The book is accessible to a well-motivated student at the beginning graduate level. The objects being dealt with are concrete and down-to-earth, namely polynomials in $n$ variables with real coefficients, and many examples are included. Proofs are presented as clearly and as simply as possible. Various new, simpler proofs appear in the book for the first time. Abstraction is employed only when it serves a useful purpose, but, at the same time, enough abstraction is included to allow the reader easy access to the literature. The book should be essential reading for any beginning student in the area. Cover......Page 1 Title Page......Page 2 Copyright Page......Page 3 Contents......Page 4 Preface......Page 8 Introduction......Page 10 0.2 Positive Semidefinite Matrices......Page 14 1.1 Preliminaries on Polynomials......Page 16 1.2 Positive Polynomials......Page 17 1.3 Extending Positive Polynomials......Page 21 1.4 Hilbert's 17th Problem......Page 24 1.5 Baer-Krull Theorem......Page 27 1.6 Formal Power Series Rings......Page 30 2.1 Quadratic Modules and Preorderings......Page 34 2.2 Positivstellensatz......Page 38 2.3 The Proof......Page 40 2.4 The Real Spectrum......Page 42 2.5 Abstract Positivstellensatz......Page 44 2.6 Saturation......Page 46 2.7 Low-Dimensional Examples......Page 48 3.1 Introduction......Page 54 3.2 Proof of Haviland's Theorem......Page 57 3.3 Uniqueness Question......Page 59 3.4 The Conditions (SMP) and (MP)......Page 60 3.5 Schmudgen's Theorem......Page 61 3.6 Countable Dimensional Vector Spaces......Page 63 4.1 Stability......Page 68 4.2 Examples where (SMP) and (MP) fail......Page 74 4.3 Examples where (SMP) and (MP) hold......Page 77 4.4 Direct Integral Decomposition......Page 78 5.1 Preprimes......Page 84 5.2 T-modules......Page 85 5.3 Semiorderings and Valuations......Page 88 5.4 Representation Theorem......Page 91 5.5 Theorems of Polya and Reznick......Page 93 5.6 Other Applications......Page 96 5.7 Topology on VA = Hom(A, R)......Page 97 6.1 Wormann's Trick......Page 100 6.2 Non-Compact Case......Page 102 6.3 Remarks and Examples......Page 105 7.1 Introduction......Page 110 7.2 Stable Compactness......Page 113 7.3 Jacobi-Prestel Counterexample......Page 116 7.4 The case ...< 1......Page 118 8.1 Isotropy and Weak Isotropy......Page 122 8.2 Residue Forms......Page 123 8.3 Local-Global Principle for Weak Isotropy......Page 126 8.4 Pfister Forms......Page 129 8.5 Application to Putinar's Question......Page 130 9.1 Basic Lemma......Page 136 9.2 Local-Global Principle......Page 138 9.3 The Case n = 1......Page 141 9.4 The Case n = 2......Page 143 9.5 Hessian Conditions......Page 146 9.6 Second Local-Global Principle......Page 147 10.1 The Cone of PSD Matrices......Page 150 10.2 Semidefinite Programming......Page 151 10.3 Max-Cut Problem......Page 155 10.4 Global Optimization......Page 158 10.5 Constrained Optimization......Page 161 10.6 Exploiting the Gradient Ideal......Page 164 10.7 Existence of Feasible Solutions......Page 169 11.1 Basic Version......Page 174 11.2 Tarski's Transfer Principle......Page 175 11.3 Lang's Homomorphism Theorem......Page 176 11.4 Geometric Version......Page 178 11.5 General Version......Page 180 12.1 Transcendence Degree and Krull Dimension......Page 182 12.2 Non-Singular Zeros......Page 184 12.3 Algebraic Sets......Page 186 12.4 Dimension......Page 188 12.5 Radical Ideals and Real Ideals......Page 190 12.7 Sign-Changing Criterion......Page 191 Bibliography......Page 196 "The study of positive polynomials brings together algebra, geometry and analysis. The subject is of fundamental importance in real algebraic geometry when studying the properties of objects defined by polynomial inequalities. Hilbert's 17th problem and its solution in the first half of the 20th century were landmarks in the early days of the subject. More recently, new connections to the moment problem and to polynomial optimization have been discovered. The moment problem relates linear maps on the multidimensional polynomial ring to positive Borel measures." "This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmudgen's solution to the moment problem in the compact case in 1991. The book is accessible to a well-motivated student at the beginning graduate level. The objects being dealt with are concrete and down-to-earth, namely polynomials in n variables with real coefficients, and many examples are included. Proofs are presented as clearly and as simply as possible. Various new, simpler proofs appear in the book for the first time. Abstraction is employed only when it serves a useful purpose, but, at the same time, enough abstraction is included to allow the reader easy access to the literature. The book should be essential reading for any beginning student in the area."--Jacket The study of positive polynomials brings together algebra, geometry and analysis. The subject is of fundamental importance in real algebraic geometry when studying the properties of objects defined by polynomial inequalities. Hilbert's 17th problem and its solution in the first half of the 20th century were landmarks in the early days of the subject. More recently, new connections to the moment problem and to polynomial optimization have been discovered. The moment problem relates linear maps on the multidimensional polynomial ring to positive Borel measures. This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmüdgen's solution to the moment problem in the compact case in 1991. The book is accessible to a well-motivated student at the beginning graduate level. The objects being dealt with are concrete and down-to-earth, namely polynomials in $n$ variables with real coefficients, and many examples are included. Proofs are presented as clearly and as simply as possible. Various new, simpler proofs appear in the book for the first time. Abstraction is employed only when it serves a useful purpose, but, at the same time, enough abstraction is included to allow the reader easy access to the literature. The book should be essential reading for any beginning student in the area.