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Autor: Victor V. Prasolov
ISBN-13: 9783540407140
Einband: Book
Seiten: 301
Gewicht: 599 g
Format: 244x165x24 mm
Sprache: Englisch

Polynomials

11, Algorithms and Computation in Mathematics
Originaltitel:Mnogochleny
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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. 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. Many of the modern results have only been published in journals so far.
ForewordNotational 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 norms3. 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
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.Thisbook 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.

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Autor: Victor V. Prasolov
ISBN-13 :: 9783540407140
ISBN: 3540407146
Erscheinungsjahr: 09.07.2004
Verlag: Springer-Verlag GmbH
Gewicht: 599g
Seiten: 301
Sprache: Englisch
Auflage 2004. 2004
Sonstiges: Buch, 244x165x24 mm, 8 schwarz-weiße Abbildungen, Bibliographie