Cox D.A., Little J., O'Shea D. — Using Algebraic Geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Cox D.A., Little J., O'Shea D. — Using Algebraic Geometry

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Название: Using Algebraic Geometry

Авторы: Cox D.A., Little J., O'Shea D.

Аннотация:

This book illustrates the many uses of algebraic geometry, highlighting some of the more recent applications of Grobner bases and resultants. In order to do this, the authors provide an introduction to some algebraic objects and techniques which are more advanced than one typically encounters in a first course, but nonetheless of great utility. The book is written for nonspecialists and for readers with a diverse range of backgrounds. It assumes knowledge of the material covered in a standard undergraduate course in abstract algebra, and it would help to have some previous exposure to Grobner bases. The book does not assume the reader is familiar with more advanced concepts such as modules.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 499

Добавлена в каталог: 20.11.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Encoding, function      415 430
Encoding, systematic      see systematic encoder
Engineering      360
Entropy function      464
Equal Ideals Have Equal Varieties principle      20
Equivalent analytic functions      137 138
Equivalent codes      459 465
Equivalent matrices      197
Equivalent modules      218 221
Errata      x
Error      415 415 421 445
Error locations      436 439 446
Error locator polynomial      437 438 446 447
Error polynomial      436 437 447
Error values      436 446
Error-correcting      415 417-419 424 435 436 445-447 463
Error-correcting codes      407 415ff
Error-correcting, burst      426 435
Error-detecting      415-417 419 424
Euclidean algorithm      76 436 448
Euclidean geometry      170
Euler characteristic      404
Euler s formula      106 107
Evaluation mapping      449 450 452 457 458
Ewald, G.      316 320 323 346 470
Exact sequence      234ff 246 247 251 261 262 264 265 267 275277 285 383 403 404 453 456 467
Exact sequence, split      245
Exponent vector of a monomial      see monomial exponent
Extension theorem      25 27 28 57
Extraneous factor      99ff
Face of a polytope      292 293 318 333 344 345 349 389
Face ring      406
Facet of a polytope      293 294 309 316 318-320 324 326 332
Facet variable      309 312 313 316
Faires, J.      ix 32 55 469
Farin, G.      385 470
Faugere, J.      46 56 443 470
Feasible region      360 361 364 370 371 374 376
Fenchel, W.      ix 316 320 323 469
FGLM algorithm      see Gr $\ddot{o}$ bner basis conversion
Field, algebraically closed      see algebraically closed field
Field, automorphism      414
Field, finite      see finite field
Field, isomorphism of      408 411 413 414
Field, of rational functions      see rational function field
Field, prime      see prime field
Field, residue      see residue field
Finite element method      385
Finite field      104 208 220 407ff 415ff 424ff 436ff 449ff
Finite free resolution      239 242 244 245 247 249
Finite free resolution, graded      257 270
Finite free resolution, minimal      260 (see also free resolution)
Finite Generation of Invariants      281
Finite group      281 283 284 289 407
Finite group, abelian      412
Finite matrix group      see finite group
Finite-dimensional algebra      see algebra
Finitely generated module      183 192 196 226 228 230-232 237 245 246 256 257 260 266 267 269 270 282
Finiteness theorem      37 41 51 58 276
Finiteness Theorem, in module case      210 440
First fundamental theorem of invariant theory      92
First isomorphism theorem      see Fundamental Theorem of Homomorphisms
First syzygy module      189 239-241 248 266
Fitting, equivalent matrices      197
Fitting, ideal      see Fitting invariant
Fitting, invariant ( $F_{i}(M)$ )      230 232 233
Fitzpatrick, P.      x 436 440 443 444 470
Fixed Point Iteration      33
Fogarty, J.      312 470
Follow generically      173 (see also generic Formal Implicit)
Formal infinite product      410
Formal power series      288 403 409 437
Formal power series, ring ( $k[[x_{1},... , x_{n}]]$ )      133ff 146 156 162 165 175 176 222 226
Forney formula      439 446
Free module ( $R^_{m}$ )      180ff 192 197ff 231-233 236ff 245ff 395 396 400 401 403 404
Free module ( $R^_{m}$ ), graded      see graded free module
Free module ( $R^_{m}$ ), over a local ring      222ff
Free module ( $R^_{m}$ ), twisted      see graded free module
Free resolution      viii 208 234 239ff 245ff 252 284 285 383 467
Free resolution, finite      see finite free resolution
Free resolution, graded      see graded resolution
Free resolution, isomorphism of      see isomorphic resolutions
Free resolution, partial      see partial graded resolution
Free resolution, trivial      see trivial resolution
Friedberg, S.      149 471
Frobenius automorphism      415 435 461
Fulton, W.      307 316 323 336 346 375 471
Function field      see rational function field
Function Theorem      136
Fundamental lattice polytope      319 320 323 325
Fundamental theorem of algebra      331
Fundamental Theorem of Homomorphisms      52 141 414
Fundamental Theorem on Discrete Subgroups of Euclidean Space      319
Galois group      415
Galois theory      407 415 435
Galois, E.      27
Garcia, A.      464 471
Garrity, T.      x 102 468
Gatermann, K.      340 475
Gauss - Jordan elimination      422
GCD      39 40 46 69 101 198 273 275 276 279 285 286 289 348 352 357 436 448
Gelfand, I.      76 80 87 89 94 103 301 302 304 307 308 316 336 342 343 353 471
Generalized characteristic polynomial      105 115 356
Generalized characteristic polynomial, toric      356
Generalized companion matrix      128
Generalized eigenspace      see eigenspace generalized
Generalized eigenvector      see eigenvector generalized
Generating function      288 409 410 456
Generator, matrix (of a linear code)      416-419 421 424-426 432 450 463 466
Generator, polynomial (of a cyclic code)      425-428 431-436 447 459462 464
GENERIC      108-110 116 118 121 122 125 279 328-330 335-337 340 342 346 354 401
Generic linear subspace      270
Generic number of solutions      332
Generic polynomial in L(A)      296
Generic system of equations      330 331 334 337 353
Genus      452-454 462 467
Geometric Goppa code      407 448ff
Geometric Goppa code, one-point      466
Geometric modeling      305
Geometric series, finite summation formula      411
Geometric series, formal summation formula      134 403 409 410 437
Geometry Center at the University of Minnesota      348
Georg, K.      338 468
Germ of an analytic function      138
Getform (Maple procedure)      68 69
Getmatrix (Maple procedure)      63 68 73 92
Gianni, P.      46 56 443 470
Gilbert - Varshamov bound      420 463 464
GL(, R)      197 281 284
GL(two, R)      282 283 289
Goppa code      see geometric Goppa code
Goppa, V.      448 449 452 463
Gorenstein codimension      3
Gorenstein ideal      288
Gr $\ddot{a}$ be, H.      158 173 471
Gr $\ddot{o}$ bner basis      vii-ix 12 15 16 25 27 28 34 36 40 48 49 56 58 60 68 76 81 121 138 151 163 165-167 169 176 179 208 238 240 274 278 328-330 338 375 377-380 382 384 4299431 457 467
Gr $\ddot{o}$ bner basis conversion      46ff 56 443 444
Gr $\ddot{o}$ bner basis conversion, Main Loop      46ff
Gr $\ddot{o}$ bner basis conversion, Next Monomial      47ff
Gr $\ddot{o}$ bner basis conversion, Termination Test      47ff
Gr $\ddot{o}$ bner basis, for modules      197 200 204ff 210ff 245ff 257 385 386 394-398 401 436 439-444 446-448 466
Gr $\ddot{o}$ bner basis, for modules over local rings      222ff
Gr $\ddot{o}$ bner basis, in integer programming      359 365ff
Gr $\ddot{o}$ bner basis, monic      15 51 204 207 250
Gr $\ddot{o}$ bner basis, reduced      15 37 42 62 204 207 214 247 253 254 328 369 430 432 441 445
Gr $\ddot{o}$ bner basis, software for linear programming      373
Gr $\ddot{o}$ bner basis, specialization of      328 (see also standard basis)
Graded      254
Graded free module      254ff 404
Graded free module, standard structure ( $R^{m}$ )      254

Graded free module, twisted structure ( $R(d_{1}\oplus\cdot\cdot\oplus R(d_{m})$ )      255ff 268ff 283ff
Graded free resolution      see graded resolution
Graded Hilbert Syzygy Theorem      see Hilbert Syzygy Theorem Graded
Graded homomorphism      255ff 268 281
Graded isomorphism      259 262
Graded matrix      256
Graded minimal resolution      see minimal graded
Graded module      208 209 241 253ff 266ff 283 402-404
Graded resolution      252 257ff 266 268ff 285-288
Graded reverse lexicographic order      see grevlex
Graded submodule      254 261
Graded subring      281
Graded twisted module      see twist of a graded module M
Graded, graded resolution      252 257ff 266 268ff 2855288
Graph      398
Graph, connected      398 400
Graph, dual      398 400
Graph, edge of      see edge of
Graph, oriented edge of      see edge oriented
Graph, vertex of      see vertex of
Grassmann, H.      158 471
Greatest common divisor      see GCD
Green      273
Greuel, G. - M.      x 158 471
grevlex      8 10 15 36 38 42 46 47 56 57 68 205 208-210 241 242 247 249
grlex      209
Group, affine      see affine group
Group, cyclic      see cyclic group
Group, finite      see finite group
Group, finite matrix      see finite group
Group, Galois      see Galois group
Group, invariant theory of      see invariant theory of
Group, multiplicative      see multiplicative group
Gusein - Zade, S.      178 468
H ${\o}$ holt      449 471
Half-space      361
Hall s marriage theorem      385
Hamming code      418 419 421-424
Hamming distance      417 464
Hasse - Weil bound      454 455
Hasse - Weil Theorem      452 453
Heegard, C.      436 449 458 471
Heptagon      316 324
Hereditary complex      389-391 393 395 398-401
Hermitian code      463
Hermitian curve      454 457
Herstein, I.      27 54 65 412 471
Herzog, J.      285 469
Hidden variables      115ff 120 121 128 145
Higher syzygies      239 240 243
Hilbert - Burch Theorem      248 252 264 271 279 280 286
Hilbert basis      377-380 384 385
Hilbert basis theorem      4 12 165 175
Hilbert function      viii ix 175 208 266ff 282 285-288 375 381 382 404 455 457 467
Hilbert function, affine      455-457 467
Hilbert polynomial      264 266 270ff 285-288 453 457
Hilbert series      282 288 289 403 404
Hilbert syzygy theorem      245 247 257
Hilbert Syzygy Theorem, Graded      257 259 260 263 280 284
Hilbert, D.      ix 245 263 280 471
Hilbert, Gr $\ddot{o}$ bner basis algorithm for      377 378
Hironaka, H.      169 471
Hold generically      329 (see also generic)
Homogeneous coordinates      85 88 90 402 463
Homogeneous element of a graded module      253 256
Homogeneous ideal      240 252 253 266 270 276 287 288 369 381 451 453 467
Homogeneous polynomial      2 74 78ff 89ff 120 159 164 166 176 177 208 209 253 254 256 263 267 276 277 281 283 286 302 314 321 322 347 401 402
Homogeneous polynomial homogeneous polynomial, with respect to nonstandard degrees      369 370
Homogeneous polynomial, weighted      see weighted
Homogeneous syzygy      211 221 283
Homogenize      158 160 166 176 276 279 286 299 343 402 467
Homogenize, a resolution      258 263
Homogenize, in the sparse or toric context      309ff
Homomorphism ring      see ring homormorphism
Homomorphism, Fundamental Theorem of      see Fundamental Theorem of Homomorphisms
Homomorphism, graded      see graded homomorphism
Homomorphism, localization of      see localization of
Homomorphism, module      see module homomorphism
Homotopy continuation method      327 338-340
Homotopy system      338
Huber, B.      324 336 337 340 346 347 471
Huguet, F.      414 473
Huneke, C.      273 284 470
Hyperplane      319 323 325
Hyperplane, at infinity      90 454
Hypersurface      64
Ideal      3 182 425 426 429433 455 465 467
Ideal - Variety Correspondence      21-23
Ideal, basis of      see basis of an ideal
Ideal, comaximal      see comaximal
Ideal, generated by $f_{1}, ..., f_{s} (\langle f_{1}, ..., f_{s}\rangle)$       3
Ideal, Gorenstein ñodimension      3; see Gorenstein ñodimension three ideal
Ideal, homogeneous      see homogeneous ideal
Ideal, ideals elimination      see elimination ideal
Ideal, intersection      see intersection of
Ideal, maximal      see maximal ideal
Ideal, monomial      see monomial ideal
Ideal, of a variety (I(V))      20
Ideal, of ith minors ()      229 230 233 266
Ideal, of leading terms ( $\langle$ LT() $\rangle$ )      12 36 37 151 164 165 168 169 175 176 430 456 of
Ideal, primary      see primary ideal
Ideal, primary decomposition of      see ideal primary decomposition
Ideal, prime      see prime ideal
Ideal, principal      see principal ideal
Ideal, product      see product of
Ideal, quotient      see quotient ideal
Ideal, radical      see radical ideal
Ideal, radical of      see radical of
Ideal, sum      see sum of
Ideal, variety of      see variety of
Ideal, zero-dimensional      see zero-dimensional ideal
Image      180 185 190 192 234 235 237 240 243 258ff 404
Implicit function theorem      338
Implicitization      80 81 273 278 298 299 303 305
Inclusion-Exclusion Principle      456
Information positions      419 430 432
Information rate      420 464
Initial value problem      339
Injective      235 242 260 286 313 341 382 414 415
Inner product      423 433 461
Insel, A.      149 471
Integer Polynomial property of resultants      72 73
Integer programming      viii-ix 359ff 374 384
Integer programming problem      360ff 376
Integer programming problem, Gr $\ddot{o}$ bner basis algorithm for solving      368 372
Integer programming problem, standard form of      see standard form
Integer row operation      341
Integral domain      136
Interior cell of a polyhedral complex      see cell interior of
Intermediate Value Theorem      31
Interpolation      60 62 63 105 127
Interpolation, dense      106
Interpolation, Lagrange interpolation formula      41 44
Interpolation, probabilistic      106
Interpolation, sparse      106
Interpolation, Vandermonde      106
Intersection of ideals ( $I\cap J$ )      6 16 22 218-220
Intersection of submodules (N $\cap$ M)      191 218 219 242 243
Intersection of varieties (V $\cap$ W)      18 22
Interval arithmetic      63
Invariant of a group action      281-283
Invariant theory, First Fundamental Theorem of      see First Fundamental Theorem of Invariant Theory
Invariant theory, of finite groups      266 280ff 365
Invariant theory, relation to combinatorics      375
Invariant theory, Second Fundamental Theorem of      see Second Fundamental Theorem of Invariant Theory
Inversion problem      300
Invertible matrix with entries in a ring       195 (see also GL( $m</a></span> <span class=subjpages><a href=

$m</a></span> <span class=subjpages><a href=

R$"/>)
Inward normal      see inward pointing normal
Inward pointing facet normal      see inward pointing normal
Inward pointing normal      293 294 309311 316 322 324 325 333 336

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