Format:
1 Online-Ressource (XI, 156 Seiten)
ISBN:
9783030855475
Series Statement:
Developments in mathematics volume 69
Content:
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.
Note:
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.
Additional Edition:
ISBN 9783030855468
Additional Edition:
ISBN 9783030855482
Additional Edition:
ISBN 9783030855499
Additional Edition:
Erscheint auch als Druck-Ausgabe Powers, Victoria Certificates of positivity for real polynomials Cham, Switzerlans : Springer Nature, 2021 ISBN 9783030855468
Language:
English
Subjects:
Mathematics
Keywords:
Positives Polynom
;
Quadratsumme
;
Hilbertsches Problem 17
;
Semialgebraische Menge
;
Quadratischer Modul
DOI:
10.1007/978-3-030-85547-5
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