**Mathematical Surveys and Monographs**

Volume: 146;
2008;
187 pp;

MSC: Primary 13; 14; 44;

**Electronic ISBN: 978-1-4704-1373-6
Product Code: SURV/146.E**

List Price: $71.00

AMS Member Price: $56.80

MAA Member Price: $63.90

#### Supplemental Materials

# Positive Polynomials and Sums of Squares

Share this page
*Murray Marshall*

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.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Positive Polynomials and Sums of Squares

- Contents iii4 free
- Preface vii8 free
- Introduction ix10 free
- Chapter 0. Preliminaries 114 free
- Chapter 1. Positive Polynomials and Sums of Squares 316
- Chapter 2. Krivine's Positivstellensatz 2134
- Chapter 3. The Moment Problem 4154
- Chapter 4. Non-Compact Case 5568
- Chapter 5. Archimedean T-modules 7184
- Chapter 6. Schmudgen's Positivstellensatz 87100
- Chapter 7. Putinar's Question 97110
- Chapter 8. Weak Isotropy of Quadratic Forms 109122
- Chapter 9. Scheiderer's Local-Global Principle 123136
- Chapter 10. Semidefinite Programming and Optimization 137150
- Appendix 1. Tarski-Seidenberg Theorem 161174
- Appendix 2. Algebraic Sets 169182
- Bibliography 183196