Positive Polynomials

From Hilbert’s 17th Problem to Real Algebra

Alexander Prestel, Charles Delzell (Autoren)

Buch | Hardcover

VIII, 268 Seiten

2001
Springer Berlin (Verlag)
978-3-540-41215-1 (ISBN)

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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.

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.

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)

Erscheint lt. Verlag	24.4.2001
Reihe/Serie	Springer Monographs in Mathematics
Zusatzinfo	VIII, 268 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	526 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Algebra • Functional Analysis • hilbert's 17th problem • Polynom • Positive polynomials • Real Algebra • semialgebraic sets • valued fields
ISBN-10	3-540-41215-8 / 3540412158
ISBN-13	978-3-540-41215-1 / 9783540412151
Zustand	Neuware