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

ISBN-13: 9783642074455 / Angielski / Miękka / 2011 / 268 str.

Alexander Prestel;Charles Delzell

Opis

Positivity is one of the most basic mathematical concepts, involved in many areas of mathematics (analysis, real algebraic geometry, functional analysis, etc.). The main objective of the book is to give useful characterizations of polynomials. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed.

Recenzje

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)

Spis treści

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.