Positive Polynomials: From Hilbert's 17th Problem to Real Algebra

by Alexander Prestel, Charles Delzell

Overview

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 Rsubn 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.

Product Details

ISBN-13:	9783642074455
Publisher:	Springer Berlin Heidelberg
Publication date:	09/22/2011
Series:	Springer Monographs in Mathematics
Edition description:	Softcover reprint of the original 1st ed. 2001
Pages:	268
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

1. Real Fields.- 2. Semialgebraic Sets.- 3. Quadratic Forms over Real Fields.- 4. Real Rings.- 5. Archimedean Rings.- 6. Positive Polynomials on Semialgebraic Sets.- 7. Sums of 2mth Powers.- 8. Bounds.- Appendix: Valued Fields.- A.1 Valuations.- A.2 Algebraic Extensions.- A.3 Henselian Fields.- A.4 Complete Fields.- A.5 Dependence and Composition of Valuations.- A.6 Transcendental Extensions.- A.7 Exercises.- A.8 Bibliographical Comments.- References.- Glossary of Notations.

Editorial 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)

From the Publisher