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Smart N.P. — The algorithmic resolution of Diophantine equations
Smart N.P. — The algorithmic resolution of Diophantine equations

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Название: The algorithmic resolution of Diophantine equations

Автор: Smart N.P.

Аннотация:

Beginning with a brief introduction to algorithms and diophantine equations, this volume provides a coherent modern account of the methods used to find all the solutions to certain diophantine equations, particularly those developed for use on a computer. The study is divided into three parts, emphasizing approaches with a wide range of applications. The first section considers basic techniques including local methods, sieving, descent arguments and the LLL algorithm. The second section explores problems that can be solved using Baker's theory of linear forms in logarithms. The final section looks at problems associated with curves, focusing on rational and integral points on elliptic curves. Each chapter concludes with a useful set of exercises. A detailed bibliography is included. This book will appeal to graduate students and research workers interested in solving diophantine equations using computational methods.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 259

Добавлена в каталог: 14.11.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$p$-adic elliptic logarithm      209
$p$-adic elliptic logarithm      209
$p$-adic exponential      31
$p$-adic exponential      31
$p$-adic integer      18
$p$-adic integer      18
$p$-adic logarithm      28
$p$-adic logarithm      28
$P$-adic metric      17
$P$-adic metric      17
$S$-integers      21
$S$-integers      21
$S$-unit equation      99 133-152 156 167 205
$S$-unit equation      99 133-152 156 167 205
$S$-units      21
$S$-units      21
ABC-Conjecture      222
Abel, N. H.      5
AGM      199 200
Agrawal, M.      124
Arithmetic-geometric mean      5 199
Bachet, C.      2
Baker — W$\ddot{u}$stholz theorem      102 107 123 137 225
Baker, A.      7 9 59 82 98 100 117 137 138 148 225
Bernstein, D.      117
Bhaskara      5
Bilu, Y.      105 108 110 124 197
Binary form      97 151 155 220 cubic quartic
Birch — Swinnerton — Dyer algorithm      187
Birch — Swinnerton — Dyer conjecture      193
Birch, B- J.      152
Bombieri, E.      105
Brahmagupta      5
Brahmagupta      5
Cantor’s algorithm      216
Cassels, J. W. S.      viii 17 183 222
Cassels, J. W. S.      viii 17 183 222
Catalan, E.      222
Catalan, E.      222
Catalan’s equation      222
Catalan’s equation      222
Chabauty, C.      217
cholesky      66
cholesky      66
Coates, J.      117 124 134
Coates, J.      117 124 134
Coghian, F. B.      206
Coghian, F. B.      206
Cohen, H.      viii 198
Cohen, H.      viii 198
Coleman, R. F.      218
Coleman, R. F.      218
complexity      8 42 50
Continued fraction      4 59 105
Continued fraction, convergents      60
Continued fraction, convergents      60
Continued fraction, partial quotient      60
Coombes, K. R.      219
Cremona, J. E.      xii 190 210
Cremona, J. E.      xii 190 210
Crouch, S.      xii
Crouch, S.      xii
Cubic form      151
Cubic form      151
Davenport, H      7 59 82
Davenport, H      7 59 82
David, S.      197 205 207 227
David, S.      197 205 207 227
David’s theorem      205 207 227
David’s theorem      205 207 227
de Weger, B. M. M.      vii xii 6 7 31 73 82 97 98 103 105 117 206 229
de Weger, B. M. M.      vii xii 6 7 31 73 82 97 98 103 105 117 206 229
Decomposable form      155
Decomposable form      155
Dem’janenko, V.      218
Dem’janenko, V.      218
Descent      2 52
Descent      2 52
Descent, 2-descent      183
Descent, 2-descent      183
Descent, higher      185
Descent, higher      185
Descent, infinite      190
Descent, infinite      190
Descent, via 2-isogeny      183
Descent, via 2-isogeny      183
Diophantos      2
Diophantos      2
Discriminant form      165
Discriminant form      165
Discriminant form equation      45 165-167 169-172 174
Discriminant form equation      45 165-167 169-172 174
Divisor      213
Divisor      213
Elliptic curve      1 50 124 177
Elliptic curve      1 50 124 177
Elliptic curve, canonical height      180 181 191 192 210
Elliptic curve, canonical height      180 181 191 192 210
Elliptic curve, conditional algorithm      192
Elliptic curve, conditional algorithm      192
Elliptic curve, conductor      124 193
Elliptic curve, group law      179
Elliptic curve, group law      179
Elliptic curve, height      226
Elliptic curve, height      226
Elliptic curve, height pairing matrix      182
Elliptic curve, height pairing matrix      182
Elliptic curve, integral points      10-12 45 111-115 150 151 202-210
Elliptic curve, L-series      193
Elliptic curve, L-series      193
Elliptic curve, minimal model      178
Elliptic curve, naive height      180 191
Elliptic curve, Neron — Tate height      181 (see also Elliptic curve canonical
Elliptic curve, Neron — Tate height      181 (see also Elliptic curve canonical
Elliptic curve, regulator      182
Elliptic curve, regulator      182
Elliptic function      198
Elliptic function      198
Elliptic logarithm      198
Elliptic logarithm      198
Elliptic logarithm, $p$-adic      209
Elliptic logarithm, $p$-adic      209
Ellison, W. J.      7
Ellison, W. J.      7
Eratosthenes      40
Eratosthenes      40
Euler, L      3 4 221
Euler, L      3 4 221
Evertse, J. H.      134 152
Evertse, J. H.      134 152
Exponential time      8
Exponential time      8
Factoring      1 40 48 53 119 144
Factoring      1 40 48 53 119 144
Factoring, polynomial      75
Factoring, polynomial      75
Faltings’ theorem      9 152 213 217
Faltings’ theorem      9 152 213 217
Fermat, P.      2 52 221
Fermat, P.      2 52 221
Fermat’s Last Theorem      13 193 221
Fermat’s Last Theorem      13 193 221
Fermat’s Little Theorem      20 34
Fermat’s Little Theorem      20 34
Fincke — Pohst, algorithm      66
Fincke — Pohst, algorithm      66
Flynn, E. V.      218
Flynn, E. V.      218
FORM      see also Binary form Discriminant
FORM      see also Binary form Discriminant
Form, decomposable      155
Form, decomposable      155
Formal group      208
Formal group      208
Frey, G      221
Frey, G      221
Ftey curve      221
Ftey curve      221
Ga$\acute{a}$l, I.      165 171-173
Ga$\acute{a}$l, I.      165 171-173
Gauss, C. F.      4 5 199
Gauss, C. F.      4 5 199
Gram matrix      75
Gram matrix      75
Gram — Schmidt process      67
Gram — Schmidt process      67
Grant, D.      219
Grant, D.      219
GRH      9 50 98
GRH      9 50 98
Guy, R.      205
Guy, R.      205
Gy$\H{o}$ry, K., x      134 152 153 167
Gy$\H{o}$ry, K., x      134 152 153 167
Hanrot, G.      105 108 110 124 197
Hanrot, G.      105 108 110 124 197
Hardy, G. H.      59
Hardy, G. H.      59
Hasse principle      40 49 182 184
Hasse principle      40 49 182 184
Hasse, H.      5
Hasse, H.      5
Hasse’s theorem      40
Hasse’s theorem      40
Height      134 147 Canonical
Height      134 147 Canonical
Height, function      22
Height, function      22
Height, Mahler      22
Height, Mahler      22
Height, modified      106 126 148 225-227
Height, modified      106 126 148 225-227
Height, Weil      22
Height, Weil      22
Hensel, K.      5 23
Hensel, K.      5 23
Hensel’s Lemma      23 36 47
Hensel’s Lemma      23 36 47
Hermite, C.      151 190
Hermite, C.      151 190
Hermite’s theorem      66 192 194
Hermite’s theorem      66 192 194
Hilbert, problems      6
Hilbert, problems      6
Holzer’s theorem      49
Holzer’s theorem      49
Hunt, D.      124
Hunt, D.      124
Hyper-graph      154
Hyper-graph      154
Hyperelliptic curve      50 215
Hypergeometric functions      xi
Hypergeometric functions      xi
Index form      166
Integral point      10 (see also Elliptic curve integral
Integral point      10 (see also Elliptic curve integral
Invariant theory      187
Invariant theory      187
Jacobian      214
Jacobian      214
Julia, G.      190
Julia, G.      190
Knapsack problem      79
Knapsack problem      79
Koblitz, N.      17
Koblitz, N.      17
Lagrange, J. L      4 52 151 199
Lagrange, J. L      4 52 151 199
Lang, S.      23 197
Lang, S.      23 197
Lattice      65 (see also LLL)
Lattice      65 (see also LLL)
Lattice, approximation      64
Lattice, approximation      64
Lattice, basis      65
Lattice, basis      65
Lattice, determinant      66
Lattice, enlargement      191
Lattice, enlargement      191
Lattice, successive minima      66
Lattice, successive minima      66
Lebesgue, V. A.      222
Lebesgue, V. A.      222
Lenstra, A. K.      viii 7 67
Lenstra, A. K.      viii 7 67
Lenstra, H. W.      viii 7 67 152
Lenstra, H. W.      viii 7 67 152
Ljunggren, W.      206
Ljunggren, W.      206
LLL      36 67-75
LLL      36 67-75
LLL, de Weger’s variant      73
LLL, de Weger’s variant      73
LLL, reduced basis      67
LLL, reduced basis      67
Lov$\acute{a}$sz, L.      viii 7 67
Lov$\acute{a}$sz, L.      viii 7 67
Mahler, K      117
Mahler, K      117
Manin, J.      218
Manin, J.      218
Mansfield, E.      xii
Mansfield, E.      xii
Matijasevi$\check{c}$, J.      7
Matijasevi$\check{c}$, J.      7
Merriman, J. R.      152
Merriman, J. R.      152
Mordell — Weil group      50 177 179 197 215 218
Mordell — Weil group      50 177 179 197 215 218
Mordell — Weil theorem      180 190 216
Mordell — Weil theorem      180 190 216
Mordell, L. J.      vii 5 45 111 180 217
Mordell, L. J.      vii 5 45 111 180 217
Mordell’s equation      206
Mordell’s equation      206
Nagell, T.      6 222
Nagell, T.      6 222
Neron, A.      181
Neron, A.      181
Number field sieve      119
Number field sieve      119
Ochoa curve      205
Ochoa curve      205
Pad$\acute{e}$ approximations      xi
Pad$\acute{e}$ approximations      xi
Pell’s equation      4 62 243
Peth$\H{o}$ and de Weger’s lemma      103 122 123 137 141 229
Peth$\H{o}$, A.      xii 6 97 103 105 165 171 172 229
Picard group      214
Picard group      214
Pohst, M.      viii 165 171 172
Pohst, M.      viii 165 171 172
Poincar$\acute{e}$, J. H.      5 180
Poincar$\acute{e}$, J. H.      5 180
Polynomial time      8 73
Polynomial time      8 73
Prime ideal removing lemma      118
Prime ideal removing lemma      118
Product formula      21
Product formula      21
1 2
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