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Polynomials : Algorithms and Computation in Mathematics - Dimitry Leites


Algorithms and Computation in Mathematics


Published: 9th July 2004
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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.

From the reviews of the first edition: "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 fur 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 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 herkommlichen Literatur of als blosses Mittel zum Zweck betrachtet, und es gibt nach wie vor wenige Bucher, 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 Einfuhrung in die faszinierende Welt der Polynome ist es zweifellos jedem Interessierten warmstens 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." (Miklos Dorman, Acta Scientiarum Mathematicarum, Vol. 72, 2006)

Foreword Notational conventions
Roots of polynomials
Inequalities for roots
The roots of a polynomial and of its derivative
The resultant and the discriminant
Separation of roots
Lagrange's series and estimates of the roots of a polynomial
Problems to
Solutions of selected problems
Irreducible polynomials
Main properties of irreducible polynomials
Irreducibility criteria
Irreducibility of trinomials and fournomials
Hilbert's irreducibility theorem
Algorithms for factorization into irreducible factors
Problems to
Solutions of selected problems
Polynomials of a particular form
Symmetric polynomials
Integer-valued polynomials
Cyclotomic polynomials
Chebyshev polynomials
Bernoulli's polynomials
Problems to
Solutions of selected problems
Certain properties of polynomials
Polynomials with prescribed values
The height of a polynomial and other norms
Equations for polynomials
Transformations of polynomials
Algebraic numbers
Problems to
Galois theory
Lagrange's theorem and the Galois resolvent
Basic Galois theory
How to solve equations by radicals
Calculations of the Galois groups
Ideals in polynomial rings
Hilbert's basis theorem and Hilbert's theorem on zeros
Grobner bases
Hilbert's seventeenth problem
The sums of squares: introduction
Artin's theory
Pfister's theory
The Lenstra-Lenstra-Lovasz algorithm
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540407140
ISBN-10: 3540407146
Series: Algorithms and Computation in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 301
Published: 9th July 2004
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.88  x 2.54
Weight (kg): 0.59