Polynomials

  • Victor V. Prasolov

Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 11)

Table of contents

  1. Front Matter
    Pages I-XIII
  2. Victor V. Prasolov
    Pages 1-45
  3. Victor V. Prasolov
    Pages 47-76
  4. Victor V. Prasolov
    Pages 77-131
  5. Victor V. Prasolov
    Pages 133-180
  6. Victor V. Prasolov
    Pages 181-218
  7. Victor V. Prasolov
    Pages 219-241
  8. Victor V. Prasolov
    Pages 243-278
  9. Back Matter
    Pages 289-301

About this book

Introduction

This comprehensive book covers both long-standing results in the theory of polynomials and recent developments which have until now only been available in the research literature. After initial chapters on the location and separation of roots and on irreducibility criteria, the book covers more specialized polynomials, including those which are symmetric, integer-value or cyclotomic, and those of Chebyshev and Bernoulli. There follow chapters on Galois theory and ideals in polynomial rings. Finally there is a detailed discussion of Hilbert’s 17th problem on the representation of non-negative polynomials as sums of squares of rational functions and generalizations.

From the reviews:

"... Despite the appearance of this book in a series titled Algorithms and Computation of Mathematics, computation occupies only a small part of the monograph. It is best described as a useful reference for one's personal collection and a text for a full-year course given to graduate or even senior undergraduate students. [.....] the book under review is worth purchasing for the library and possibly even for one's own collection. The author's interest in the history and development of this area is evident, and we have pleasant glimpses of progress over the last three centuries. He exercises nice judgment in selection of arguments, with respect to both representativeness of approaches and elegance, so that the reader gains a synopsis of and guide to the literature, in which more detail can be found. ..."

E. Barbeau, SIAM Review 47, No. 3, 2005

"... the volume is packed with results and proofs that are well organised thematically into chapters and sections. What is unusual is to have a text that embraces and intermingles both analytic and algebraic aspects of the theory. Although the subject is about such basic objects, many tough results of considerable generality are incorporated and it is striking that refinements, both in theorems and proofs continued throughout the latter part of the Twentieth Century. [...] There is a plentiful of problems, some of which might be challenging even for polynomial people; solutions to selected problems are also included."

S.D.Cohen, MathSciNet, MR 2082772, 2005

"Problems concerning polynomials have impulsed resp. accompanied the development of algebra from its very beginning until today and over the centuries a lot of mathematical gems have been brought to light. This book presents a few of them, some being classical, but partly probably unknown even to experts, some being quite recently discovered. […] Many historical comments and a clear style make the book very readable, so it can be recommended warmly to non-experts already at an undergraduate level and, because of its contents, to experts as well."

G.Kowol, Monatshefte für Mathematik 146, Issue 4, 2005

Keywords

Galois theory Gröbner bases Hilberts seventeenth problem algebra irreducible polynomials

Authors and affiliations

  • Victor V. Prasolov
    • 1
  1. 1.Department MathematicsIndependent University of MoscowMoscowRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-03980-5
  • Copyright Information Springer-Verlag Berlin Heidelberg 2004
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-03979-9
  • Online ISBN 978-3-642-03980-5
  • Series Print ISSN 1431-1550
  • About this book
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