Polynomials
(Sprache: Englisch)
This comprehensive book covers both long-standing results in the theory of polynomials and recent developments. After initial chapters on the location and separation of roots and on irreducibility criteria, the book covers more specialized polynomials.
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This comprehensive book covers both long-standing results in the theory of polynomials and recent developments. After initial chapters on the location and separation of roots and on irreducibility criteria, the book covers more specialized polynomials.
Klappentext zu „Polynomials “
Covers its topic in greater depth than the typical standard books on polynomial algebra
The theory of polynomials constitutes an essential part of university of algebra and calculus. Nevertheless, there are very few books entirely devoted to this theory.
This book provides an exposition of the main results in the theory of polynomials, both classical and modern. Many of the modern results have only been published in journals so far.
Considerable attention is given to Hilbert's 17th problem on the representation of non-negative polynomials by the sums of squares of rational functions and its generalizations. Galois theory is discussed primarily from the point of view of the theory of polynomials, not from that of the general theory of fields and their extensions.
This book provides an exposition of the main results in the theory of polynomials, both classical and modern. Many of the modern results have only been published in journals so far.
Considerable attention is given to Hilbert's 17th problem on the representation of non-negative polynomials by the sums of squares of rational functions and its generalizations. Galois theory is discussed primarily from the point of view of the theory of polynomials, not from that of the general theory of fields and their extensions.
Inhaltsverzeichnis zu „Polynomials “
ForewordNotational conventionsChapter 1. Roots of polynomials1. Inequalities for roots2. The roots of a polynomial and of its derivative3. The resultant and the discriminant4. Separation of roots5. Lagrange's series and estimates of the roots of a polynomial6. Problems to Chapter 17. Solutions of selected problemsChapter 2. Irreducible polynomials1. Main properties of irreducible polynomials2. Irreducibility criteria3. Irreducibility of trinomials and fournomials4. Hilbert's irreducibility theorem5. Algorithms for factorization into irreducible factors6. Problems to Chapter 27. Solutions of selected problemsChapter 3. Polynomials of a particular form1. Symmetric polynomials2. Integer-valued polynomials3. Cyclotomic polynomials4. Chebyshev polynomials5. Bernoulli's polynomials6. Problems to Chapter 37. Solutions of selected problemsChapter 4. Certain properties of polynomials1. Polynomials with prescribed values2. The height of a polynomial and other norms3. Equations for polynomials4. Transformations of polynomials5. Algebraic numbers6. Problems to Chapter 4Chapter 5. Galois theory1. Lagrange's theorem and the Galois resolvent2. Basic Galois theory3. How to solve equations by radicals4. Calculations of the Galois groupsChapter 6. Ideals in polynomial rings1. Hilbert's basis theorem and Hilbert's theorem on zeros2. Gröbner basesChapter 7. Hilbert's seventeenth problem1. The sums of squares: introduction2. Artin's theory3. Pfister's theoryChapter 8. Appendix1. The Lenstra-Lenstra-Lovasz algorithmBibliography
Bibliographische Angaben
- Autor: Victor V. Prasolov
- 2004, 2004, 301 Seiten, Maße: 16,5 x 24,4 cm, Gebunden, Englisch
- Übersetzung:Leites, Dimitry
- Übersetzer: D. Leites
- Verlag: Springer
- ISBN-10: 3540407146
- ISBN-13: 9783540407140
- Erscheinungsdatum: 09.07.2004
Sprache:
Englisch
Rezension zu „Polynomials “
From the reviews:"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 expers, 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"... 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 judgement 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. ..."Edward Barbeau, SIAM Review, Sept. 2005, Vol. 47, No. 3"... 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 thoughout the latter part of the Twentieth Century. [...] There is a plentiful of
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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 "Polynome bilden einen grundlegenden Baustein der Algebra gleichwie der Analysis. Nichtsdestotrotz werden sie in der herkömmlichen Literatur of als bloßes Mittel zum Zweck betrachtet, und es gibt nach wie vor wenige Bücher, die sich ausschließlich der Theorie der Polynome widmen. Das vorliegende Buch bildet einen wohltuenden Kontrast dazu. Es versteht sich als Sammlung der wichtigsten Resultate der Theorie der Polynome, klassischer ebenso wie moderner. [......] Als Einführung in die faszinierende Welt der Polynome ist es zweifellos jedem Interessierten wärmstens zu empfehlen."O.Pfeiffer (Kapfenberg), IMN - Internationale Mathematische Nachrichten, 59, Issue 198, 2005"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 specialised polynomials ... . Finally there is a detailed discussion of Hilbert's 17th problem ... ." Bulletin Bibliographique, Vol. 51 (1-2), 2005"This is an exposition of polynomial theory and results, both classical and modern. ... 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. ... it is all fascinating and relevant to a series devoted to 'algorithms and computations'. There is a plentiful supply of problems, some of which might be challenging even for polynomial people ... ." S. D. Cohen, Mathematical Reviews, 2005f"This volume is an excellent introduction to the main topics on polynomials. The author presents both classical and modern subjects. ... Each chapter contains a list of selected problems and their solutions. The book includes a rich bibliography and an useful index. It will be useful for undergraduate and graduate students in mathematics." Doru Stefanescu, Zentralblatt MATH, Vol. 1063, 2005"The theory of polynomials is a very important and interesting part of mathematics. ... We note that at the end of chapters 1-4 some interesting problems and their solutions can be found. This is an excellent book written about polynomials. We can recommend this book to all who are interested in the theory of polynomials." (Miklós Dormán, Acta Scientiarum Mathematicarum, Vol. 72, 2006)"This is an interesting, useful, well-organized, and well-written compendium of theorems and techniques about polynomials. ... The present volume is a translation of the 2001 Russian second edition. ... This is primarily a reference work ... it does include a set of interesting problems (with some solutions) at the end if each chapter." (Allen Stenger, The Mathematical Association of America, August, 2011)
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