Positive Polynomials

1. Front Matter

   Pages I-VIII

   PDF

2. Introduction

   • Alexander Prestel, Charles N. Delzell

   Pages 1-5

3. Real Fields

   • Alexander Prestel, Charles N. Delzell

   Pages 7-29

4. Semialgebraic Sets

   • Alexander Prestel, Charles N. Delzell

   Pages 31-51

5. Quadratic Forms over Real Fields

   • Alexander Prestel, Charles N. Delzell

   Pages 53-80

6. Real Rings

   • Alexander Prestel, Charles N. Delzell

   Pages 81-111

7. Archimedean Rings

   • Alexander Prestel, Charles N. Delzell

   Pages 113-137

8. Positive Polynomials on Semialgebraic Sets

   • Alexander Prestel, Charles N. Delzell

   Pages 139-159

9. Sums of 2mth Powers

   • Alexander Prestel, Charles N. Delzell

   Pages 161-178

10. Bounds

    • Alexander Prestel, Charles N. Delzell

    Pages 179-201

11. Back Matter

    Pages 203-269

    PDF

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.

• Algebra
• Positive polynomials
• Real Algebra
• Semialgebraic sets
• functional analysis
• hilbert's 17th problem
• valued fields

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)

• Fachbereich Mathematik und Statistik, Universität Konstanz, Konstanz, Germany

  Alexander Prestel

• Department of Mathematics, Louisiana State University, Baton Rouge, USA

  Charles N. Delzell

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