The algorithmic resolution of diophantine equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
1998
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | London Mathematical Society: London Mathematical Society student texts
41 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 243 S. |
| ISBN: | 052164156X 0521646332 |
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Datensatz im Suchindex
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| adam_text | LONDON MATHEMATICAL SOCIETY STUDENT TEXTS 41 THE ALGORITHMIC RESOLUTION
OF DIOPHANTINE EQUATIONS NIGEL P. SMART HEWLETT-PACKARD LABORATORIES,
BRISTOL CAMBRIDGE UNIVERSITY PRESS CONTENTS PREFACE XI OUTLINE XII
COMPUTER PACKAGES XV NOTATION XV THANKS XVI CHAPTER I. INTRODUCTION 1
1.1. A BRIEF HISTORY 2 1.2. ALGORITHMS 7 1.3. WHAT IS A DIOPHANTINE
EQUATION? 9 1.4. AN ELLIPTIC CURVE 10 PART 1. BASIC SOLUTION TECHNIQUES
15 CHAPTER II. LOCAL METHODS 17 11.1. P-ADIC NUMBERS 17 11.2. P-ADIC
NUMERICAL ANALYSIS 23 11.3. EXERCISES 32 CHAPTER III. APPLICATIONS OF
LOCAL METHODS TO DIOPHANTINE EQUATIONS 33 111.1. APPLICATIONS OF
STRASSMANN S THEOREM 33 111.2. SKOLEM S METHOD 36 111.3. THE HASSE
PRINCIPLE 39 111.4. FINDING SMALL SOLUTIONS 40 III.5. EXERCISES 43
CHAPTER IV. TERNARY QUADRATIC FORMS 45 IV. 1. A NORMAL FORM 45 IV.2.
LOCAL SOLUBILITY ^ ; 46 IV.3. GLOBAL SOLUBILITY 49 IV.4. NEW SOLUTIONS
FOR OLD 53 IV. 5. EXERCISES 56 CHAPTER V. COMPUTATIONAL DIOPHANTINE
APPROXIMATION 59 V.I. CONTINUED FRACTIONS 59 V.2. APPROXIMATION LATTICES
64 V.3. LATTICES 65 V.4. THE LLL-ALGORITHM 71 VIII CONTENTS V.5.
EXERCISES 75 CHAPTER VI. APPLICATIONS OF THE LLL-ALGORITHM 77 VI. 1. A
FUN APPLICATION 77 VI.2. KNAPSACK PROBLEMS 79 VI.3. APPROXIMATING
LINEAR FORMS 82 VI.4. P-ADIC ANALOGUES 87 VI.5. EXERCISES 93 PART 2.
METHODS USING LINEAR FORMS IN LOGARITHMS 95 CHAPTER VII. THUE EQUATIONS
97 VII. 1. THUE EQUATIONS 97 VII.2. X 4 - 2Y 4 = 1 105 VII.3. THE
METHOD OF BILU AND HANROT 108 VII.4. INTEGRAL POINTS ON ELLIPTIC CURVES
(I) 111 VII.5. Y 2 = X 3 - 6X - 14 113 VII.6. EXERCISES 116 CHAPTER
VIII. THUE-MAHLER EQUATIONS 117 VIII.L. THUE-MAHLER EQUATIONS 117
VIII.2. THE PRIME IDEAL REMOVING LEMMA 118 VIII.3. THE METHOD 119
VIII.4. X 3 - X 2 Y + XY 2 + Y 3 = LL S 124 VIII.5. EXERCISES 132
CHAPTER IX. 5-UNIT EQUATIONS 133 IX. 1. 5-UNIT EQUATIONS 133 IX.2.
SIEVING 141 IX.3. AN 5-UNIT EQUATION IN A CYCLIC QUINTIC FIELD 146 IX.4.
INTEGRAL POINTS ON ELLIPTIC CURVES (II) 150 IX. 5. OTHER APPLICATIONS
151 IX.6. EXERCISE 152 CHAPTER X. TRIANGULARLY CONNECTED DECOMPOSABLE
FORM EQUATIONS 153 X.I. TRIANGULARLY CONNECTED LINEAR FORMS 153 X.2.
TCDF EQUATIONS - 155 X.3. SOLVING TCDF EQUATIONS 156 X.4. EXERCISES 163
CHAPTER XL DISCRIMINANT FORM EQUATIONS 165 XI.1. DISCRIMINANT AND INDEX
FORMS 165 XI.2. THE GENERAL CASE: DISCRIMINANT FORMS AS TCDFS 167 XI.3.
A DISCRIMINANT FORM EQUATION IN A CYCLIC QUINTIC FIELD 169 XI.4. SPECIAL
CASES 170 XL 5. EXERCISES 174 CONTENTS IX PART 3. INTEGRAL AND RATIONAL
POINTS ON CURVES 175 CHAPTER XII. RATIONAL POINTS ON ELLIPTIC CURVES 177
XII.1. BASICS ON ELLIPTIC CURVES 177 XII.2. THEWEAK MORDELL-WEIL THEOREM
182 XII.3. THE MORDELL-WEIL THEOREM 190 XII.4. A CONDITIONAL ALGORITHM
192 XII.5. EXERCISES 194 CHAPTER XIII. INTEGRAL POINTS ON ELLIPTIC
CURVES 197 XIII. 1. ELLIPTIC LOGARITHMS 197 XIII.2. ELLIPTIC INTEGRALS
AND THE AGM 198 XIII.3. INTEGRAL POINTS 202 XIII.4. INTEGRAL POINTS ON
THE CURVE Y 2 = X 3 - 2 206 XIII.5. 5-INTEGRAL POINTS 207 XIII.6. OTHER
METHODS AND PROBLEMS 210 XIII.7. EXERCISES 211 CHAPTER XIV. CURVES OF
GENUS GREATER THAN ONE 213 XIV. 1. CURVES AND THEIR JACOBIANS 213 XIV.2.
HYPERELLIPTIC CURVES AND THEIR JACOBIANS 215 XIV.3. RATIONAL POINTS ON
CURVES OF GENUS GREATER THAN ONE 217 XIV.4. INTEGRAL POINTS ON
HYPERELLIPTIC AND SUPERELLIPTIC CURVES 219 XIV.5. FERMAT CURVES 221
XIV.6. CATALAN S EQUATION 222 XIV.7. EXERCISES 224 APPENDIX A. LINEAR
FORMS IN LOGARITHMS 225 A.I. LINEAR FORMS IN COMPLEX LOGARITHMS 225 A.2.
LINEAR FORMS IN P-ADIC LOGARITHMS 225 A.3. LINEAR FORMS IN ELLIPTIC
LOGARITHMS 226 APPENDIX B. TWO USEFUL LEMMATA 229 REFERENCES 231 INDEX
241
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| indexdate | 2024-07-09T18:26:44Z |
| institution | BVB |
| isbn | 052164156X 0521646332 |
| language | English |
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| physical | XVI, 243 S. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Cambridge Univ. Press |
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| series | London Mathematical Society: London Mathematical Society student texts |
| series2 | London Mathematical Society: London Mathematical Society student texts |
| spelling | Smart, Nigel P. 1967- Verfasser (DE-588)173244815 aut The algorithmic resolution of diophantine equations Nigel P. Smart 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1998 XVI, 243 S. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society: London Mathematical Society student texts 41 Diophantine equations Diophantische Gleichung (DE-588)4012386-8 gnd rswk-swf Algorithmus (DE-588)4001183-5 gnd rswk-swf Diophantische Gleichung (DE-588)4012386-8 s Algorithmus (DE-588)4001183-5 s DE-604 London Mathematical Society: London Mathematical Society student texts 41 (DE-604)BV000841726 41 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008404229&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Smart, Nigel P. 1967- The algorithmic resolution of diophantine equations London Mathematical Society: London Mathematical Society student texts Diophantine equations Diophantische Gleichung (DE-588)4012386-8 gnd Algorithmus (DE-588)4001183-5 gnd |
| subject_GND | (DE-588)4012386-8 (DE-588)4001183-5 |
| title | The algorithmic resolution of diophantine equations |
| title_auth | The algorithmic resolution of diophantine equations |
| title_exact_search | The algorithmic resolution of diophantine equations |
| title_full | The algorithmic resolution of diophantine equations Nigel P. Smart |
| title_fullStr | The algorithmic resolution of diophantine equations Nigel P. Smart |
| title_full_unstemmed | The algorithmic resolution of diophantine equations Nigel P. Smart |
| title_short | The algorithmic resolution of diophantine equations |
| title_sort | the algorithmic resolution of diophantine equations |
| topic | Diophantine equations Diophantische Gleichung (DE-588)4012386-8 gnd Algorithmus (DE-588)4001183-5 gnd |
| topic_facet | Diophantine equations Diophantische Gleichung Algorithmus |
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