Certificates of Positivity for Real Polynomials: Theory, Practice, and Applications

The book "Certificates of Positivity" collects and explains theorems about the existence of certificates of positivity for polynomials, which are algebraic identities that provide immediate proof of positivity conditions. These certificates have applications in mathematics, applied mathematics, engineering, and other fields. The book discusses algorithms, computational methods, and applications, making it a valuable resource for researchers and beginners in the field.

Format: Paperback / softback
Length: 156 pages
Publication date: 27 November 2022
Publisher: Springer Nature Switzerland AG

This book is a valuable resource for anyone interested in the study of polynomials and their positivity properties. It collects and explains a wide range of theorems related to 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 provides an immediate proof of a positivity condition for the polynomial. These certificates have their origins in the fundamental work of David Hilbert in the late 19th century on positive polynomials and sums of squares. Due to their numerous applications in mathematics, applied mathematics, engineering, and other fields, it is desirable to have methods for finding, describing, and characterizing these certificates. This book discusses appropriate algorithms, computational methods, and applications for many of the topics covered. It is written in a clear and concise manner, making it accessible to beginning graduate students and researchers who are not specialists in the field. Additionally, researchers who work on certificates of positivity or use them in their applications will find this book a useful reference for their work.

The book begins by introducing the basic concepts and definitions related to certificates of positivity. It then discusses the historical background and the fundamental results that led to the development of this area of study. Next, the book presents a comprehensive survey of the existing literature on certificates of positivity, including both theoretical and computational aspects. It covers various topics such as positivity conditions, certificates of positivity for real polynomials, certificates of positivity for rational functions, and certificates of positivity for semialgebraic sets. Each topic is discussed in detail, with examples and exercises to help the reader understand the concepts better. The book also includes a discussion of the computational methods used to find certificates of positivity, including algorithms for linear systems, semidefinite programming, and convex optimization. Computational examples are provided to illustrate the effectiveness of these methods.

In addition to the theoretical aspects, the book also emphasizes the practical applications of certificates of positivity. It discusses how certificates of positivity are used in various fields such as mathematics, applied mathematics, engineering, and computer science. Examples are given to illustrate the importance and usefulness of these certificates in solving real-world problems. The book also highlights the open research problems in the area and encourages researchers to contribute to the development of this field. Overall, this book is a must-read for anyone interested in the study of polynomials and their positivity properties. It provides a comprehensive and up-to-date treatment of the subject, and includes both theoretical and computational aspects. It is accessible to beginning graduate students and researchers, and will also be useful for researchers who work on certificates of positivity or use them in their applications.

Weight: 267g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030855499
Edition number: 1st ed. 2021