Certificates of Positivity for Real Polynomials

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 19 th 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.

Includes extensive background information for increased accessibility Contains discussion of computational and algorithmic aspects of the subject Features an extensive bibliography

Auteur

Résumé

"The book under review is a very nice introduction to the central topic of real algebra ... . The book is very well written with many examples demonstrating the statements of the main results. ... In the reviewer's opinion this is a very nice and concise presentation of the most important pillars of real algebra up to the present time." (Aljaz Zalar, Mathematical Reviews, June, 2023)

"This book introduces several theories with applications ... . Also, the book includes introductions to various recent developments ... . The author described the results as friendly as possible with sufficient intuitions and examples. ... This book provides a good introduction to certificates of positivity who want to learn this area as a beginner." (Jaewoo Jung, zbMATH 1483.14001, 2022)

Contenu

1. Preliminaries.- 2. Sums of Squares and Positive Polynomials. - 3. Global Certificates of Positivity.- 4. Positive Semidefinite Ternary Quartics.- 5. Positivity on Semialgebraic Sets.- 6. The Archimedean Property.- 7. Theorems of Schmudgen and Putinar.- 8. The Dimension One Case.- 9. Positivity on Polytopes.- 10. The Noncompact Case.- 11. Sums of Squares of Rational Polynomials.- 12. Positive Polynomials with Special Structure.- Real Algebra and Algebraic Geometry.- Index of Notation.- Index.