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200 10$aCertificates of Positivity for Real Polynomials $eTheory, Practice, and Applications /$fby Victoria Powers
205   $a1st ed. 2021.
210  1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2021.
215   $a1 online resource (161 pages) $cillustrations
225 1 $aDevelopments in Mathematics,$x2197-795X ;$v69
311 08$aPrint version: Powers, Victoria Certificates of Positivity for Real Polynomials Cham : Springer International Publishing AG,c2022 9783030855468 
320   $aIncludes bibliographical references and index.
327   $a1. 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.
330   $aThis book collects and explains the many theorems concerning the existence of certificates of positivity for polynomials that are positive globally or on semialgebraic sets. A certificate of positivity for a real polynomial is an algebraic identity that gives an immediate proof of a positivity condition for the polynomial. Certificates of positivity have their roots in fundamental work of David Hilbert from the late 19th century on positive polynomials and sums of squares. Because of the numerous applications of certificates of positivity in mathematics, applied mathematics, engineering, and other fields, it is desirable to have methods for finding, describing, and characterizing them. For many of the topics covered in this book, appropriate algorithms, computational methods, and applications are discussed. This volume contains a comprehensive, accessible, up-to-date treatment of certificates of positivity, written by an expert in the field. It provides an overview of both the theory and computational aspects of the subject, and includes many of the recent and exciting developments in the area. Background information is given so that beginning graduate students and researchers who are not specialists can learn about this fascinating subject. Furthermore, researchers who work on certificates of positivity or use them in applications will find this a useful reference for their work.
410  0$aDevelopments in Mathematics,$x2197-795X ;$v69
606   $aGeometry, Algebraic
606   $aNumber theory
606   $aComputer science
606   $aAlgebraic Geometry
606   $aNumber Theory
606   $aComputer Science
615  0$aGeometry, Algebraic.
615  0$aNumber theory.
615  0$aComputer science.
615 14$aAlgebraic Geometry.
615 24$aNumber Theory.
615 24$aComputer Science.
676   $a512.942
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996   $aCertificates of Positivity for Real Polynomials$92553541
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