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Positive Polynomials

From Hilbert’s 17th Problem to Real Algebra

  • Book
  • © 2001

Overview

Authors:
  1. Alexander Prestel

    1. Fachbereich Mathematik und Statistik, Universität Konstanz, Konstanz, Germany

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  2. Charles N. Delzell

    1. Department of Mathematics, Louisiana State University, Baton Rouge, USA

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-VIII
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  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
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Keywords

About this book

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.

Reviews

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)

Authors and Affiliations

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

    Alexander Prestel

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

    Charles N. Delzell

Bibliographic Information

  • Book Title: Positive Polynomials

  • Book Subtitle: From Hilbert’s 17th Problem to Real Algebra

  • Authors: Alexander Prestel, Charles N. Delzell

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-662-04648-7

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2001

  • Hardcover ISBN: 978-3-540-41215-1Published: 24 April 2001

  • Softcover ISBN: 978-3-642-07445-5Published: 22 September 2011

  • eBook ISBN: 978-3-662-04648-7Published: 17 April 2013

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: VIII, 268

  • Topics: Algebra, Algebraic Geometry, Functional Analysis

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