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Positive Polynomials and Sums of Squares
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Mathematical Surveys and Monographs
2008; 187 pp; hardcover
Volume: 146
ISBN-10: 0-8218-4402-4
ISBN-13: 978-0-8218-4402-1
List Price: US$67 Member Price: US$53.60
Order Code: SURV/146

The study of positive polynomials brings together algebra, geometry and analysis. The subject is of fundamental importance in real algebraic geometry when studying the properties of objects defined by polynomial inequalities. Hilbert's 17th problem and its solution in the first half of the 20th century were landmarks in the early days of the subject. More recently, new connections to the moment problem and to polynomial optimization have been discovered. The moment problem relates linear maps on the multidimensional polynomial ring to positive Borel measures.

This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmüdgen's solution to the moment problem in the compact case in 1991. The book is accessible to a well-motivated student at the beginning graduate level. The objects being dealt with are concrete and down-to-earth, namely polynomials in $$n$$ variables with real coefficients, and many examples are included. Proofs are presented as clearly and as simply as possible. Various new, simpler proofs appear in the book for the first time. Abstraction is employed only when it serves a useful purpose, but, at the same time, enough abstraction is included to allow the reader easy access to the literature. The book should be essential reading for any beginning student in the area.

Graduate students and research mathematicans interested in positive polynomials in algebra, geometry, and analysis; semialgebraic geometry.

Reviews

"Designed for students at the beginning graduate level, this concentrates on concrete objects, such as polynomials in $$n$$ variables with real coefficients, and Marshall includes plenty of examples and new, simple proofs. He also provides a very useful bibliography for further study."

-- SciTech Book News

"This book truly serves as both a textbook for beginners and a monograph for specialists. It guides the readers efficiently through a large part of the rapidly evolving subarea of real algebraic geometry which is concerned with sums of squares representations of positive polynomials. ... Almost all the results are stated in a way immediately accessible to people from outside the area."

-- Mathematical Reviews