Polynomials:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English Russian |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2010
|
| Schriftenreihe: | Algorithms and computation in mathematics
11 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | Literaturangaben |
| Beschreibung: | XIII, 301 S. graph. Darst. 24 cm |
| ISBN: | 9783642039799 9783540407140 |
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| 100 | 1 | |a Prasolov, Viktor V. |d 1956- |e Verfasser |0 (DE-588)129271977 |4 aut | |
| 245 | 1 | 0 | |a Polynomials |c Victor V. Prasolov. Transl. from the Russ. by Dimitry Leites |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2010 | |
| 300 | |a XIII, 301 S. |b graph. Darst. |c 24 cm | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Algorithms and computation in mathematics |v 11 | |
| 500 | |a Literaturangaben | ||
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Datensatz im Suchindex
| _version_ | 1807954450150588416 |
|---|---|
| adam_text |
CONTENTS ROOTS OF POLYNOMIALS 1 1.1 INEQUALITIES FOR ROOTS 1 1.1.1 THE
FUNDAMENTAL THEOREM OF ALGEBRA 1 1.1.2 CAUCHY'S THEOREM 2 1.1.3
LAGUERRE'S THEOREM 5 1.1.4 APOLAR POLYNOMIALS 7 1.1.5 THE ROUTH-HURWITZ
PROBLEM 11 1.2 THE ROOTS OF A GIVEN POLYNOMIAL AND OF ITS DERIVATIVE 12
1.2.1 THE GAUSS-LUCAS THEOREM 12 1.2.2 THE ROOTS OF THE DERIVATIVE AND
THE FOCAL POINTS OF AN ELLIPSE 14 1.2.3 LOCALIZATION OF THE ROOTS OF THE
DERIVATIVE 15 1.2.4 THE SENDOV-ILIEFF CONJECTURE 18 1.2.5 POLYNOMIALS
WHOSE ROOTS COINCIDE WITH THE ROOTS OF THEIR DERIVATIVES 20 1.3 THE
RESULTANT AND THE DISCRIMINANT 20 1.3.1 THE RESULTANT 20 1.3.2 THE
DISCRIMINANT 23 1.3.3 COMPUTING CERTAIN RESULTANTS AND DISCRIMINANTS 25
1.4 SEPARATION OF ROOTS 27 1.4.1 THE FOURIER-BUDAN THEOREM 27 1.4.2
STURM'S THEOREM 30 1.4.3 SYLVESTER'S THEOREM 31 1.4.4 SEPARATION OF
COMPLEX ROOTS 35 1.5 LAGRANGE'S SERIES AND ESTIMATES OF THE ROOTS OF A
GIVEN POLYNOMIAL 37 1.5.1 THE LAGRANGE-BIIRMANN SERIES 37 1.5.2
LAGRANGE'S SERIES AND ESTIMATION OF ROOTS 40 1.6 PROBLEMS TO CHAPTER 1
41 1.7 SOLUTIONS OF SELECTED PROBLEMS 42 BIBLIOGRAFISCHE INFORMATIONEN
HTTP://D-NB.INFO/996064540 DIGITALISIERT DURCH X CONTENTS 2 IRREDUCIBLE
POLYNOMIALS 47 2.1 MAIN PROPERTIES OF IRREDUCIBLE POLYNOMIALS 47 2.1.1
FACTORIZATION OF POLYNOMIALS INTO IRREDUCIBLE FACTORS. 47 2.1.2
EISENSTEIN'S CRITERION 50 2.1.3 IRREDUCIBILITY MODULO P 51 2.2
IRREDUCIBILITY CRITERIA 52 2.2.1 DUMAS'S CRITERION 52 2.2.2 POLYNOMIALS
WITH A DOMINANT COEFFICIENT 56 2.2.3 IRREDUCIBILITY OF POLYNOMIALS
ATTAINING SMALL VALUES . 58 2.3 IRREDUCIBILITY OF TRINOMIALS AND
FOURNOMIALS 59 2.3.1 IRREDUCIBILITY OF POLYNOMIALS OF THE FORM X N X M
X P L 59 2.3.2 IRREDUCIBILITY OF CERTAIN TRINOMIALS 63 2.4 HUBERT'S
IRREDUCIBILITY THEOREM 65 2.5 ALGORITHMS FOR FACTORIZATION INTO
IRREDUCIBLE FACTORS 68 2.5.1 BERLEKAMP'S ALGORITHM 68 2.5.2
FACTORIZATION WITH THE HELP OF HENSEL'S LEMMA 71 2.6 PROBLEMS TO CHAPTER
2 73 2.7 SOLUTIONS OF SELECTED PROBLEMS 74 3 POLYNOMIALS OF A PARTICULAR
FORM 77 3.1 SYMMETRIC POLYNOMIALS 77 3.1.1 EXAMPLES OF SYMMETRIC
POLYNOMIALS 77 3.1.2 MAIN THEOREM ON SYMMETRIC POLYNOMIALS 79 3.1.3
MUIRHEAD'S INEQUALITIES 81 3.1.4 THE SCHUR FUNCTIONS 83 3.2
INTEGER-VALUED POLYNOMIALS 85 3.2.1 A BASIS IN THE SPACE OF
INTEGER-VALUED POLYNOMIALS 85 3.2.2 INTEGER-VALUED POLYNOMIALS IN
SEVERAL VARIABLES 87 3.2. CONTENTS XI 3.5 BERNOULLI POLYNOMIALS 112
3.5.1 DEFINITION OF BERNOULLI POLYNOMIALS 112 3.5.2 THEOREMS OF
COMPLEMENT, ADDITION OF ARGUMENTS AND MULTIPLICATION 115 3.5.3 EULER'S
FORMULA 116 3.5.4 THE FAULHABER-JACOBI THEOREM 117 3.5.5 ARITHMETIC
PROPERTIES OF BERNOULLI NUMBERS AND BERNOULLI POLYNOMIALS 120 3.6
PROBLEMS TO CHAPTER 3 125 3.6.1 SYMMETRIC POLYNOMIALS 125 3.6.2
INTEGER-VALUED POLYNOMIALS 126 3.6.3 CHEBYSHEV POLYNOMIALS 126 3.7
SOLUTION OF SELECTED PROBLEMS 127 CERTAIN PROPERTIES OF POLYNOMIALS 133
4.1 POLYNOMIALS WITH PRESCRIBED VALUES 133 4.1.1 LAGRANGE'S
INTERPOLATION POLYNOMIAL 133 4.1.2 HERMITE'S INTERPOLATION POLYNOMIAL
136 4.1.3 THE POLYNOMIAL WITH PRESCRIBED VALUES AT THE ZEROS OF ITS
DERIVATIVE 137 4.2 THE HEIGHT OF A POLYNOMIAL AND OTHER NORMS 139 4.2.1
GAUSS'S LEMMA 139 4.2.2 POLYNOMIALS IN ONE VARIABLE 141 4.2.3 THE
MAXIMUM OF THE ABSOLUTE VALUE AND S. BERNSTEIN'S INEQUALITY 145 4.2.4
POLYNOMIALS IN SEVERAL VARIABLES 148 4.2.5 AN INEQUALITY FOR A PAIR OF
RELATIVELY PRIME POLYNOMIALS 151 4.2.6 MIGNOTTE'S INEQUALITY 152 4.3
EQUATIONS FOR POLYNOMIALS 154 4.3.1 DIOPHANTINE EQUATIONS FOR
POLYNOMIALS 154 4.3.2 FUNCTIONAL EQUATIONS FOR POLYNOMIALS 162 4.4
TRANSFORMATIONS OF POLYNOMIALS 166 4.4.1 TCHIRNHAUS'S TRANSFORMATION 166
4.4.2 5TH DEGREE EQUATION IN BRING'S FORM 168 4.4.3 REPRESENTATION OF
POLYNOMIALS AS SUMS OF POWERS OF LINEA XII CONTENTS 5 GALOIS THEORY 181
5.1 LAGRANGE'S THEOREM AND THE GALOIS RESOLVENT 181 5.1.1 LAGRANGE'S
THEOREM 181 5.1.2 THE GALOIS RESOLVENT 185 5.1.3 THEOREM ON A PRIMITIVE
ELEMENT 189 5.2 BASIC GALOIS THEORY 191 5.2.1 THE GALOIS CORRESPONDENCE
191 5.2.2 A POLYNOMIAL WITH THE GALOIS GROUP SR, 195 5.2.3 SIMPLE
RADICAL EXTENSIONS 196 5.2.4 THE CYCLIC EXTENSIONS 197 5.3 HOW TO SOLVE
EQUATIONS BY RADICALS 199 5.3.1 SOLVABLE GROUPS 199 5.3.2 EQUATIONS WITH
SOLVABLE GALOIS GROUP 200 5.3.3 EQUATIONS SOLVABLE BY RADICALS 201 5.3.4
ABELIAN EQUATIONS 204 5.3.5 THE ABEL-GALOIS CRITERION FOR SOLVABILITY OF
EQUATIONS OF PRIME DEGREE 208 5.4 CALCULATION OF THE GALOIS GROUPS 212
5.4.1 THE DISCRIMINANT AND THE GALOIS GROUP 212 5.4.2 RESOLVENT
POLYNOMIALS 213 5.4.3 THE GALOIS GROUP MODULO P 216 6 IDEALS IN
POLYNOMIAL RINGS 219 6.1 HUBERT'S BASIS THEOREM AND HUBERT'S THEOREM ON
ZEROS 219 6.1.1 HILBERT'S BASIS THEOREM 219 6.1.2 HILBERT'S THEOREM ON
ZEROS 221 6.1.3 HILBERT'S POLYNOMIAL 224 6.1.4 THE HOMOGENEOUS HILBERT'S
NULLSTELLENSATZ FOR P-FIELDS . 231 6.2 GROEBNER BASES 233 6.2.1
POLYNOMIALS IN ONE VARIABLE 234 6.2.2 DIVISION OF POLYNOMIALS IN SEVERAL
VARIABLES 235 6.2.3 DEFINITION OF GROEBNER BASES 235 6.2. CONTENTS XIII
7.2.3 HILBERT'S SEVENTEENTH PROBLEM 266 7.3 PFISTER'S THEORY 270 7.3.1
THE MULTIPLICATIVE QUADRATIC FORMS 270 7.3.2 CI-FIELDS 273 7.3.3
PFISTER'S THEOREM ON THE SUMS OF SQUARES OF RATIONAL FUNCTIONS 274 8
APPENDIX 279 8.1 THE LENSTRA-LENSTRA-LOVASZ ALGORITHM 279 8.1.1 THE
GENERAL DESCRIPTION OF THE ALGORITHM 279 8.1.2 A REDUCED BASIS OF THE
LATTICE 280 8.1.3 THE LATTICES AND FACTORIZATION OF POLYNOMIALS 283
REFERENCES 289 INDEX 297 |
| any_adam_object | 1 |
| author | Prasolov, Viktor V. 1956- |
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| author_facet | Prasolov, Viktor V. 1956- |
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| dewey-full | 515.5 512.9422 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis 512 - Algebra |
| dewey-raw | 515.5 512.9422 |
| dewey-search | 515.5 512.9422 |
| dewey-sort | 3515.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| physical | XIII, 301 S. graph. Darst. 24 cm |
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| series | Algorithms and computation in mathematics |
| series2 | Algorithms and computation in mathematics |
| spelling | Prasolov, Viktor V. 1956- Verfasser (DE-588)129271977 aut Polynomials Victor V. Prasolov. Transl. from the Russ. by Dimitry Leites Berlin [u.a.] Springer 2010 XIII, 301 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Algorithms and computation in mathematics 11 Literaturangaben Polynom (DE-588)4046711-9 gnd rswk-swf 1\p (DE-588)4143389-0 Aufgabensammlung gnd-content Polynom (DE-588)4046711-9 s DE-604 Algorithms and computation in mathematics 11 (DE-604)BV011131286 11 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3346546&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025428396&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Prasolov, Viktor V. 1956- Polynomials Algorithms and computation in mathematics Polynom (DE-588)4046711-9 gnd |
| subject_GND | (DE-588)4046711-9 (DE-588)4143389-0 |
| title | Polynomials |
| title_auth | Polynomials |
| title_exact_search | Polynomials |
| title_full | Polynomials Victor V. Prasolov. Transl. from the Russ. by Dimitry Leites |
| title_fullStr | Polynomials Victor V. Prasolov. Transl. from the Russ. by Dimitry Leites |
| title_full_unstemmed | Polynomials Victor V. Prasolov. Transl. from the Russ. by Dimitry Leites |
| title_short | Polynomials |
| title_sort | polynomials |
| topic | Polynom (DE-588)4046711-9 gnd |
| topic_facet | Polynom Aufgabensammlung |
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