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### Synopses & Reviews

#### Publisher Comments

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

#### Review

"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)

#### Review

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 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)

#### Synopsis

Covers its topic in greater depth than the typical standard books on polynomial algebra

### Table of Contents

Foreword Notational conventions Chapter 1. Roots of polynomials 1. Inequalities for roots 2. The roots of a polynomial and of its derivative 3. The resultant and the discriminant 4. Separation of roots 5. Lagrange's series and estimates of the roots of a polynomial 6. Problems to Chapter 1 7. Solutions of selected problems Chapter 2. Irreducible polynomials 1. Main properties of irreducible polynomials 2. Irreducibility criteria 3. Irreducibility of trinomials and fournomials 4. Hilbert's irreducibility theorem 5. Algorithms for factorization into irreducible factors 6. Problems to Chapter 2 7. Solutions of selected problems Chapter 3. Polynomials of a particular form 1. Symmetric polynomials 2. Integer-valued polynomials 3. Cyclotomic polynomials 4. Chebyshev polynomials 5. Bernoulli's polynomials 6. Problems to Chapter 3 7. Solutions of selected problems Chapter 4. Certain properties of polynomials 1. Polynomials with prescribed values 2. The height of a polynomial and other norms 3. Equations for polynomials 4. Transformations of polynomials 5. Algebraic numbers 6. Problems to Chapter 4 Chapter 5. Galois theory 1. Lagrange's theorem and the Galois resolvent 2. Basic Galois theory 3. How to solve equations by radicals 4. Calculations of the Galois groups Chapter 6. Ideals in polynomial rings 1. Hilbert's basis theorem and Hilbert's theorem on zeros 2. Gröbner bases Chapter 7. Hilbert's seventeenth problem 1. The sums of squares: introduction 2. Artin's theory 3. Pfister's theory Chapter 8. Appendix 1. The Lenstra-Lenstra-Lovasz algorithm Bibliography