Positive Polynomials

by Prestel, Alexander; Delzell, Charles N.

• ISBN13: 9783540412151
• ISBN10: 3540412158
• Format: Hardcover
• Copyright: 2001-06-01
• Publisher: Springer Nature

Summary

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.

Table of Contents

Introduction	1	(6)
Real Fields	7	(24)
Ordered Fields	7	(5)
Extensions of Orderings	12	(4)
The Real Closure	16	(8)
Exercises	24	(4)
Bibliographical and Historical Comments	28	(3)
Semialgebraic Sets	31	(22)
Semialgebraic Sets	31	(5)
Ultraproducts	36	(5)
Elimination of Quantifiers	41	(4)
The ``Finiteness Theorem''	45	(2)
Exercises	47	(1)
Bibliographical and Historical Comments	48	(5)
Quadratic Forms over Real Fields	53	(28)
Witt Decomposition	53	(6)
The Witt Ring of a Field	59	(3)
Signatures	62	(6)
Quadratic Forms Over Real Function Fields	68	(6)
Generalization of Hilbert's 17th Problem	74	(3)
Exercises	77	(2)
Bibliographical and Historical Comments	79	(2)
Real Rings	81	(32)
The Real Spectrum of a Commutative Ring	81	(5)
The Positivstellensatz	86	(5)
``Continuous'' Representation of Polynomials	91	(3)
ηα-Fields	94	(7)
The Real Spectrum of R(X1,...,Xn)	101	(6)
Exercises	107	(2)
Bibliographical and Historical Comments	109	(4)
Archimedean Rings	113	(26)
Quadratic Modules and Semiorderings	113	(6)
Rings with Archimedean Preorderings	119	(5)
Rings with Archimedean Quadratic Modules	124	(6)
Rings with Archimedean Preprimes	130	(4)
Exercises	134	(2)
Bibliographical and Historical Comments	136	(3)
Positive Polynomials on Semialgebraic Sets	139	(22)
Semiorderings and Weak Isotropy	139	(3)
Archimedean Quadratic Modules on R[X1,...,Xn]	142	(3)
Distinguished Representations of Positive Polynomials	145	(7)
Applications to the Moment Problem	152	(5)
Exercises	157	(1)
Bibliographical and Historical Comments	158	(3)
Sums of 2mth Powers	161	(18)
Preorderings and Semiorderings of Level 2m	161	(5)
Semiorderings of Level 2m on Fields	166	(3)
Archimedean Modules of Level 2m	169	(7)
Exercises	176	(1)
Bibliographical and Historical Comments	177	(2)
Bounds	179	(24)
Length of Sums of Squares	179	(4)
Existence of Degree Bounds	183	(6)
Positive Polynomials over Non-Archimedean Fields	189	(7)
Distinguished Representations in the Non-Archimedean Case	196	(5)
Exercise	201	(1)
Bibliographical and Historical Comments	201	(2)
Appendix: Valued Fields	203	(44)
A.1 Valuations	203	(4)
A.2 Algebraic Extensions	207	(6)
A.3 Henselian Fields	213	(10)
A.4 Complete Fields	223	(7)
A.5 Dependence and Composition of Valuations	230	(6)
A.6 Transcendental Extensions	236	(6)
A.7 Exercises	242	(3)
A.8 Bibliographical Comments	245	(2)
References	247	(8)
Glossary of Notations	255	(4)
Index	259