Positive Polynomials

From Hilbert's 17th Problem to Real Algebra

By: Alexander Prestel, Charles Delzell

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Hardcover | 24 April 2001

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282 Pages

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Positivity is one of the most basic mathematical concepts. In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of R n which itself is often given by polynomial inequalities. The main objective of the book is to give useful characterizations of such polynomials. It takes as starting point Hilbert's 17th Problem from 1900 and explains how E. Artin's solution of that problem eventually led to the development of real algebra towards the end of the 20th century. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed. Thus the monograph can also serve as the basis for a 2-semester course in real algebra.

Industry Reviews

From the reviews of the first edition:

"This is a nicely written introduction to 'reality' and 'positivity' in rings, and besides students and researchers it can also be interesting for anyone who would like to learn more on positivity and orderings." (Vilmos Totik, Acta Scientiarum Mathematicarum, Vol. 68, 2002)

"A book on 'real algebra' that serves as an introduction to the subject in addition to the main theme of the text. ... Well written with exercises for every chapter." (ASLIB Book Guide, Vol. 66 (11), 2001)

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Non-Fiction Mathematics Algebra Topology Geometry Algebraic Geometry Calculus & Mathematical Analysis Functional Analysis & Transforms

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Paperback

Published: 22nd September 2011

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Introduction	p. 1
Real Fields	p. 7
Ordered Fields	p. 7
Extensions of Orderings	p. 12
The Real Closure	p. 16
Exercises	p. 24
Bibliographical and Historical Comments	p. 28
Semialgebraic Sets	p. 31
Semialgebraic Sets	p. 31
Ultraproducts	p. 36
Elimination of Quanti ers	p. 41
The ôFiniteness Theoremö	p. 45
Exercises	p. 47
Bibliographical and Historical Comments	p. 48
Quadratic Forms over Real Fields	p. 53
Witt Decomposition	p. 53
The Witt Ring of a Field	p. 59
Signatures	p. 62
Quadratic Forms Over Real Function Fields	p. 68
Generalization of HilbertÆs 17th Problem	p. 74
Exercises	p. 77
Bibliographical and Historical Comments	p. 79
Real Rings	p. 81
The Real Spectrum of a Commutative Ring	p. 81
The Positivstellensatz	p. 86
öContinuousö Representation of Polynomials	p. 91
-Fields	p. 94
The Real Spectrum of R[X1;: : : ;Xn	p. 101
Exercises	p. 107
Bibliographical and Historical Comments	p. 109
Archimedean Rings	p. 113
Quadratic Modules and Semiorderings	p. 113
Rings with Archimedean Preorderings	p. 119
Rings with Archimedean Quadratic Modules	p. 124
Rings with Archimedean Preprimes	p. 130
Exercises	p. 134
Bibliographical and Historical Comments	p. 136
Positive Polynomials on Semialgebraic Sets	p. 139
Semiorderings and Weak Isotropy	p. 139
Archimedean Quadratic Modules on R[X1;: : : ;Xn	p. 142
Distinguished Representations of Positive Polynomials	p. 145
Applications to the Moment Problem	p. 152
Exercises	p. 157
Bibliographical and Historical Comments	p. 158
Sums of 2mth Powers	p. 161
Preorderings and Semiorderings of Level 2m	p. 161
Semiorderings of Level 2m on Fields	p. 166
Archimedean Modules of Level 2m	p. 169
Exercises	p. 176
Bibliographical and Historical Comments	p. 177
Bounds	p. 179
Length of Sums of Squares	p. 179
Existence of Degree Bounds	p. 183
Positive Polynomials over Non-Archimedean Fields	p. 189
Distinguished Representations in the Non-Archimedean Case	p. 196
Exercise	p. 201
Bibliographical and Historical Comments	p. 201
Appendix: Valued Fields	p. 203
Valuations	p. 203
Algebraic Extensions	p. 207
Henselian Fields	p. 213
Complete Fields	p. 223
Dependence and Composition of Valuations	p. 230
Transcendental Extensions	p. 236
Exercises	p. 242
Bibliographical Comments	p. 245
References	p. 247
Glossary of Notations	p. 255
Index	p. 259
Table of Contents provided by Publisher. All Rights Reserved.

Positive Polynomials

From Hilbert's 17th Problem to Real Algebra

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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