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Certificates of positivity for real polynomials : theory, practice, and applications / / Victoria Powers
| Certificates of positivity for real polynomials : theory, practice, and applications / / Victoria Powers |
| Autore | Powers Victoria |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2021] |
| Descrizione fisica | 1 online resource (161 pages) : illustrations |
| Disciplina | 512.942 |
| Collana | Developments in mathematics |
| Soggetto topico |
Polinomis
Polynomials |
| Soggetto genere / forma | Llibres electrònics |
| ISBN |
9783030855475
9783030855468 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Preface -- Contents -- 0 Introduction -- 1 Preliminaries -- 1.1 Basic Notation -- 1.2 The Real Numbers -- 1.3 Polynomials -- 1.4 Polynomials Versus Forms -- 1.5 Matrices and Quadratic Forms -- 1.6 Convex Sets and Cones -- Reference -- 2 Sums of Squares and Positive Polynomials -- 2.1 SOS and PSD Polynomials -- 2.2 Hilbert's 17th Problem -- 2.2.1 Uniform Denominators -- 2.2.2 Degree Bounds -- 2.3 The Gap Between PSD and SOS Polynomials -- References -- 3 Global Certificates of Positivity -- 3.1 The Gram Matrix Method -- 3.2 From Gram Matrices to Certificates -- 3.3 Application: Global Optimization of Polynomials -- References -- 4 Positive Semidefinite Ternary Quartics -- 4.1 Proofs of Hilbert's Theorem on Ternary Quartics -- 4.2 Finding and Counting Sum of Squares Representations -- References -- 5 Positivity on Semialgebraic Sets -- 5.1 Semialgebraic Sets -- 5.2 Preorders and Quadratic Modules -- 5.3 Positivity Properties -- 5.4 Saturated Quadratic Modules and Preorders -- 5.5 The Positivstellensatz -- 5.6 Sums of Squares and Positivity on Algebraic Sets -- 5.7 Application: The Moment Problem -- References -- 6 The Archimedean Property -- 6.1 Archimedean T-Modules -- 6.2 Marshall's Representation Theorem -- 6.3 Application: Theorems of Handelman and Pólya -- References -- 7 Theorems of Schmüdgen and Putinar -- 7.1 Schmüdgen's Positivstellensatz -- 7.1.1 Wörmann's Proof -- 7.2 Putinar's Positivstellenstz -- 7.3 Minimal Representations -- 7.4 Constructive Proofs -- 7.4.1 Degree Bounds -- 7.5 Application: Polynomial Optimization on Compact Semialgebraic Sets -- References -- 8 The Dimension One Case -- 8.1 Positivity on a Closed Interval -- 8.1.1 Bernstein Representations -- 8.2 Basic Closed Semialgebraic Subsets of mathbbR -- 8.3 Positivity on Curves in the Plane -- References -- 9 Positivity on Polytopes -- 9.1 Pólya's Theorem.
9.1.1 Pólya's Theorem with Zeros -- 9.2 Certificates in Preprimes and Quadratic Modules -- 9.3 Applications -- 9.3.1 Approximating the Stability Number of a Graph -- 9.3.2 Minimization of Polynomials on Polytopes -- References -- 10 The Noncompact Case -- 10.1 Saturation in the Noncompact Case -- 10.2 Stability and Open Cones -- 10.3 Cylinders with Compact Cross-Section -- References -- 11 Sums of Squares of Rational Polynomials -- 11.1 Weighted Sums of Squares Representations -- 11.2 The Univariate Case -- 11.3 Sums of Rational Squares -- 11.4 Algorithmic Approaches -- References -- 12 Positive Polynomials with Special Structure -- 12.1 Symmetric Polynomials -- 12.2 Diagonal-Tail Forms -- 12.3 Agiforms -- 12.4 Circuit Polynomials -- 12.5 Application: Global Optimization Using Geometric Programming -- References -- Appendix A Real Algebra and Algebraic Geometry -- A.1 Algebraic Sets -- A.2 Real Fields -- A.3 Tarski-Seidenberg Theorem -- A.4 The Real Spectrum -- References -- Appendix Index of Notation -- Index. |
| Record Nr. | UNISA-996466561803316 |
Powers Victoria
|
||
| Cham, Switzerland : , : Springer, , [2021] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
| ||
Certificates of positivity for real polynomials : theory, practice, and applications / / Victoria Powers
| Certificates of positivity for real polynomials : theory, practice, and applications / / Victoria Powers |
| Autore | Powers Victoria |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2021] |
| Descrizione fisica | 1 online resource (161 pages) : illustrations |
| Disciplina | 512.942 |
| Collana | Developments in mathematics |
| Soggetto topico |
Polinomis
Polynomials |
| Soggetto genere / forma | Llibres electrònics |
| ISBN |
9783030855475
9783030855468 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Preface -- Contents -- 0 Introduction -- 1 Preliminaries -- 1.1 Basic Notation -- 1.2 The Real Numbers -- 1.3 Polynomials -- 1.4 Polynomials Versus Forms -- 1.5 Matrices and Quadratic Forms -- 1.6 Convex Sets and Cones -- Reference -- 2 Sums of Squares and Positive Polynomials -- 2.1 SOS and PSD Polynomials -- 2.2 Hilbert's 17th Problem -- 2.2.1 Uniform Denominators -- 2.2.2 Degree Bounds -- 2.3 The Gap Between PSD and SOS Polynomials -- References -- 3 Global Certificates of Positivity -- 3.1 The Gram Matrix Method -- 3.2 From Gram Matrices to Certificates -- 3.3 Application: Global Optimization of Polynomials -- References -- 4 Positive Semidefinite Ternary Quartics -- 4.1 Proofs of Hilbert's Theorem on Ternary Quartics -- 4.2 Finding and Counting Sum of Squares Representations -- References -- 5 Positivity on Semialgebraic Sets -- 5.1 Semialgebraic Sets -- 5.2 Preorders and Quadratic Modules -- 5.3 Positivity Properties -- 5.4 Saturated Quadratic Modules and Preorders -- 5.5 The Positivstellensatz -- 5.6 Sums of Squares and Positivity on Algebraic Sets -- 5.7 Application: The Moment Problem -- References -- 6 The Archimedean Property -- 6.1 Archimedean T-Modules -- 6.2 Marshall's Representation Theorem -- 6.3 Application: Theorems of Handelman and Pólya -- References -- 7 Theorems of Schmüdgen and Putinar -- 7.1 Schmüdgen's Positivstellensatz -- 7.1.1 Wörmann's Proof -- 7.2 Putinar's Positivstellenstz -- 7.3 Minimal Representations -- 7.4 Constructive Proofs -- 7.4.1 Degree Bounds -- 7.5 Application: Polynomial Optimization on Compact Semialgebraic Sets -- References -- 8 The Dimension One Case -- 8.1 Positivity on a Closed Interval -- 8.1.1 Bernstein Representations -- 8.2 Basic Closed Semialgebraic Subsets of mathbbR -- 8.3 Positivity on Curves in the Plane -- References -- 9 Positivity on Polytopes -- 9.1 Pólya's Theorem.
9.1.1 Pólya's Theorem with Zeros -- 9.2 Certificates in Preprimes and Quadratic Modules -- 9.3 Applications -- 9.3.1 Approximating the Stability Number of a Graph -- 9.3.2 Minimization of Polynomials on Polytopes -- References -- 10 The Noncompact Case -- 10.1 Saturation in the Noncompact Case -- 10.2 Stability and Open Cones -- 10.3 Cylinders with Compact Cross-Section -- References -- 11 Sums of Squares of Rational Polynomials -- 11.1 Weighted Sums of Squares Representations -- 11.2 The Univariate Case -- 11.3 Sums of Rational Squares -- 11.4 Algorithmic Approaches -- References -- 12 Positive Polynomials with Special Structure -- 12.1 Symmetric Polynomials -- 12.2 Diagonal-Tail Forms -- 12.3 Agiforms -- 12.4 Circuit Polynomials -- 12.5 Application: Global Optimization Using Geometric Programming -- References -- Appendix A Real Algebra and Algebraic Geometry -- A.1 Algebraic Sets -- A.2 Real Fields -- A.3 Tarski-Seidenberg Theorem -- A.4 The Real Spectrum -- References -- Appendix Index of Notation -- Index. |
| Record Nr. | UNINA-9910510570603321 |
|
Powers Victoria
|
||
| Cham, Switzerland : , : Springer, , [2021] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||