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Algorithms and Computation in Mathematics #11: Polynomialsby Victor V. Prasolov
Synopses & ReviewsPublisher Comments:This comprehensive book covers both longstanding 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, integervalue 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 nonnegative 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 fullyear 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 nonexperts already at an undergraduate level and, because of its contents, to experts as well." G.Kowol, Monatshefte für Mathematik 146, Issue 4, 2005
Synopsis:Covers its topic in greater depth than the typical standard books on polynomial algebra
Table of ContentsForeword 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. Integervalued 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 LenstraLenstraLovasz algorithm Bibliography
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