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

✍ Scribed by Alexander Prestel, Charles N. Delzell (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2001
• Tongue: English
• Leaves: 268
• Series: Springer Monographs in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

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

Front Matter....Pages I-VIII
Introduction....Pages 1-5
Real Fields....Pages 7-29
Semialgebraic Sets....Pages 31-51
Quadratic Forms over Real Fields....Pages 53-80
Real Rings....Pages 81-111
Archimedean Rings....Pages 113-137
Positive Polynomials on Semialgebraic Sets....Pages 139-159
Sums of 2 m th Powers....Pages 161-178
Bounds....Pages 179-201
Back Matter....Pages 203-269

✦ Subjects

Algebra