This book collects and explains the many theorems concerning the existence of certificates of positivity for polynomials that are positive globally or on semialgebraic sets. A certificate of positivity for a real polynomial is an algebraic identity that gives an immediate proof of a positivity condition for the polynomial. Certificates of positivity have their roots in fundamental work of David Hilbert from the late 19th century on positive polynomials and sums of squares. Because of the numerous applications of certificates of positivity in mathematics, applied mathematics, engineering, and other fields, it is desirable to have methods for finding, describing, and characterizing them. For many of the topics covered in this book, appropriate algorithms, computational methods, and applications are discussed.
This volume contains a comprehensive, accessible, up-to-date treatment of certificates of positivity, written by an expert in the field. It provides an overview of both the theory and computational aspects of the subject, and includes many of the recent and exciting developments in the area. Background information is given so that beginning graduate students and researchers who are not specialists can learn about this fascinating subject. Furthermore, researchers who work on certificates of positivity or use them in applications will find this a useful reference for their work.
Author(s): Victoria Powers
Series: Developments in Mathematics, 69
Publisher: Springer
Year: 2021
Language: English
Pages: 167
City: Cham
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
