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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.

Язык:

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

Inward pointing normal, primitive      294 309 319 326 331 332
Irreducible component of a variety      86 170 173
Irreducible factor      100 144
Irreducible polynomial      80 84 86 99 149 150 301 302 342 407-412 414 459
Irreducible variety      22 23 86 88
Isobaric polynomial      100 107 108
Isolated singularity      see singularity isolated
Isolated solution      139 (see also singularity isolated)
Isomorphic resolutions      259 262
Isomorphism field      see field isomorphism
Isomorphism, graded      see graded isomorphism
Isomorphism, module      see module isomorphism
Isomorphism, of resolutions      see isomorphic resolutions
Isomorphism, ring      see ring isomorphsim Second
Jacobian matrix or determinant      82 150
Jacobian variety      449
Jacobson, N.      430 471
Jouanolou, J.      89 90 92 94 95 471
Julia set      34
Kaltofen, E.      104 469
Kapranov      76 80 87 89 93 94 103 301 302 304 307 308 316 336 342 343 353 471
Kbasis (Maple procedure)      44 45 57
Kernel      180 181 185 188 192-194 2333235 237 239 240 243 259ff 3766379 394-396 404 405 414 456 457
Key equation      438 439 441-445 447
Khovanskii, A.G.      335 337 472
Kirwan, F.      145 472
Klein Quartic curve      454 466 467
Klimaszewski, K      274 474
Koblitz, N.      415 472
Koszul complex      252 275 285 286 456 467
Koszul relations      228
Kozen, D.      67 468
Kushnirenko, A.G.      335 472
L (d)      296 300 306 316 329 330 353 356
Lagrange s Theorem      412 427
Lakshman, Y.      104 469
Lattice      319 321
Lattice polytope      291 294-296 304 318-321 324
Laurent monomial      296 333 341
Laurent polynomial ( $k[x_{1}^{\pm 1}, ..., x^{\pm 1}_{n}]$ )      296 298 300 327 329-331 337 342 350 353 354 356 359 371 377
Lazard, D.      46 56 158 167 175 443 470
LCM      13 198-200 205 211 212 435
Leading coefficient (LC())      154 155 202
Leading monomial (LM())      154 155 202
Leading term (LT())      9 151 154-156 159ff 164 166 167 202ff 211 246 441
Least common multiple      see LCM
Leichtweiss, K.      316 320 323 472
Length of a finite free resolution      239
Level curve      362
Lexicographic order      see $lex$

Li      337 472
Lift of a polytope Q ( $\hat{Q}$ )      346 347
Line at infinity      see hyperplane at
Linear code      407 416424 427 429 430 432 435 436 449
Linear programming      359 362 373
Linear system      449 466
Linearly in dependent elements in a module      185 186
Little, J.      vii-ix 4 9 10 12-15 20 21 23 25 34 35 37 39 49 52 72 73 81 86 145 149 166 167 1699171 173 174 176 177 199 202 203 212 221 253 268 270 273 282 284 285 289 308 328 340 365 366 368 369 375 381 382 402 414 425 429 431 448 453 455 456 458 467 470 471
Local Division Algorithm      see Division Algorithm in
Local Elimination Theorem      174
Local intersection multiplicity      139
Local order      152ff 168 169 173
Local order, for a module over a local ring      223
Local ring      viii 130 132ff 138ff 164ff 169 170 175 222ff 249
localization      131ff 139 146 156 175 182 222 223 234 247
Localization, at a point       222 223
Localization, of a homomorphism      251
Localization, of a module      222 223 250 251
Localization, of an exact sequence      251
Localization, with respect to (Loc ${}_{>}(k[x_{1}, ..., x_{n}])$ )      154ff 159 162 165 166
Logar, A.      187 472
Logarithmic derivative      410
Loustaunau, P.      vii 4 9 12-15 21 25 37 174 201 202 359 468
Lower facet      346 347 350
M $\ddot{o}$ ller, H.      55 472
M-dimensional cyclic code      see cyclic m-dimensional
Macaulay      39 2066209 219 220 240 256 258 366 373 379 403 432
Macaulay $\verb”<ring”$ script      208
Macaulay commands      208
Macaulay hilb      403
Macaulay mat      208
Macaulay nres      258
Macaulay pres      240
Macaulay putmat      220
Macaulay putstd      209
Macaulay res      240 258
Macaulay ring      208
Macaulay std      209
Macaulay syz      219
Macaulay, F.      ix 101 102 472
MacMahon, P.      384
MacWilliams, F.      415 427 472
Magic squares      313 375ff
Magic squares, symmetric      385
Manocha, D.      104-106 114 115 119 128 129 356 469 472
Maple      10 14 15 18 21 25 28 36 38-40 44 54 56658 63 68 73 108 114 120 144 167 206 208 328 373 412 413 422 431 432 445
Maple, alias      413 422 431
Maple, array      422
Maple, charpoly      68
Maple, collect      431
Maple, Domains package      431
Maple, eigenvals      58
Maple, expand      30 45
Maple, factor      26 27 108 144
Maple, finduni      39 44 45 46
Maple, fsolve      28 30 31 114 120
Maple, Gauss j ord      422
Maple, gbasis      14 15 25 27 44 57
Maple, grobner package      10 14 39 432
Maple, implicitplot3d      18 21
Maple, minpoly      54
Maple, mod      413 422 446
Maple, normal      413
Maple, normalf      10 36 45 68
Maple, Rem      431 445
Maple, resultant      73
Maple, RootOf      27 108 413 422 431
Maple, simplify      40 46
Maple, solve      27
Maple, subs      26 28 445 446 getmatrix kbasis zdimradical)
Mariner Mars exploration craft      450
Martin, B.      158 471
Mathematica      56 167 206 373
Maximal ideal      5 131ff 139 142 225 226 228 231 407
Maximum distance separable code      427
MDS code      see maximum distance separable code
Melancholia      375
Metric      417
Meyer, F.      280 472
Milnor number ( $\mu$ )      138 147 150 168 170 177 178
Milnor, J.      147 472
Minimal element (with respect to a minimal monomial order)      4422445 448
Minimal graded resolution      258ff 283
Minimal polynomial (of a linear map)      53 54 61
Minimal polynomial (of a primitive element)      414 435 459 465
Minimal presentation      227-230
Minimal resolution      see minimal graded
Minimal set of generators of a submodule      188 224-230 234 258 259
Minimal set of generators of a submonoid      377
Minimum distance (of a code)      418-420 422 424 427-429 434-436 450 455 458 459 461-464
Minimum distance (of a code), relative      464
Minkowski sum of polytopes (P Q)      317ff 331 343ff
Minkowski sum of polytopes (P Q), coherent      349 350 357
Minor of a matrix      229 230 233 248 249 266 279 287 288 314 434
Mishra, B.      63 472
Mixed cell of a mixed subdivision      345 346 351 352 354
Mixed elimination-weight monomial order      373
Mixed monomial basis      see monomial basis mixed
Mixed order      154 155 172
Mixed sparse resultant      see resultant mixed

Mixed subdivision      324 342 344ff
Mixed subdivision, coherent      347-352 354 358
Mixed volume      viii 316 322-324 326 327 330-332 334-337 340-343 345 346 351-355 357
Mixed volume, computing      324 345 346
Mixed volume, normalized      323 332 334 341
Mobius inversion formula      410
Modular curve      464
Module      viii ix 179ff 234ff 245ff 252 281 282 383 394 399 400 439 440 445 448 453 446
Module free      see free module
Module homomorphism      183 184 192 194 195 197 234 251 265 394 399 400
Module homomorphism, graded      see graded homomorphism
Module homomorphism, hom(, N)      192
Module homomorphism, localization of      see localization of
Module isomorphism      185 190 192 195 217 218 229 235 262 265 282 286 399
Module isomorphism, graded      see graded isomorphism
Module, basis of      186 188 395
Module, direct sum of      see direct sum of modules
Module, equivalent      see equivalent modules
Module, finitely generated      see finitely module generated
Module, Gr $\ddot{o}$ nber basis of      see Gr $\ddot{o}$ bner basis for
Module, graded      see graded module
Module, graded free      see graded free module
Module, homomorphism of      see module homomorphism
Module, isomorphism      see module isomorphism
Module, localization of      see localization of
Module, of leading terms ( $\langle$ LT(M) $\rangle$ )      204 210 404 440 441
Module, over a local ring      222ff 241
Module, presentation of      see presentation of a module
Module, projective      see projective module
Module, quotient      see quotient module
Module, quotient of      see quotient of modules
Module, rank of      see rank of
Module, syzygy      see syzygy module
Module, twisted free      see graded free module twisted
Molecular structure      305
Molien s Theorem      284
Molien series      281 283 284 289
Monic polynomial      409411 414
Monoid      377
Monomial      1 406 452 457
Monomial basis      138 426
Monomial basis, determined by a monomial order      see basis monomials
Monomial basis, for generic polynomials      122 126
Monomial basis, mixed      354-356
Monomial ideal      12 151 199
Monomial in a free module      198ff
Monomial in a free module, belongs to $f\in R$       198
Monomial in a free module, contains a basis vector      198
Monomial in a free module, quotient of monomials      198
Monomial order      7 12 56 159 164 365 367 368 370 372 382
Monomial order, adapted      367 370 372 429 430 440-442 456
Monomial order, in $R^{m}$       197 200ff
Monomial submodule      199 206 207
Monomial, exponent vector of      290-292 350 364 368
Monomial, Laurent      see Laurent monomial
Monomial, relation to integer programming      364-366
Monomial, weighted      289
Mora      46 56 156 160 169 173 175 443 468 470 472
Mora Normal Form Algorithm      159ff 165 ff
Mora's algorithm for modules      224
Moreno, C.      449 453 472
Moving hyperplane      280 287
Moving line      274
Moving line basis      see $\mu$ -basis
Multidegree (multideg())      154 155
Multiplication map ( $m_{f}$ )      51ff 58ff 74 77 78 91 96 112 116-118 125 126 143 144 148 149 356 429
Multiplicative group      408 412 413
Multiplicity      42 65 66 112 115 119 130 138ff 168 170 177 249 329 330 340 353 354 463
Multipolynomial resultant      see resultant multipolynomial
Multivariable factorization      100 113
Multivariate splines      385 389
Mumford, D.      312 470
Nakayama s lemma      225 226
National Science Foundation      x
Nearest neighbor decoding function      418 419 421 463
Neumann, W.      158 471
Newton - Raphson method      28 29 32234 56 63 339
Newton - Raphson method numerical methods, root-finding      327
Newton polytope of (NP())      295 296 298 316-318 320 321 324 329 330 332 333 336 337 340 343 353
Newton's method      see Newton - Raphson method
Noetherian ring      189 196 197
Non-generic equations      114 115 128 129
Non-mixed cell in a mixed subdivision      346 (see also mixed cell)
Noncollinear points      271 272 278 279
Nonstandard monomial      383 430 431
Nontrivial solution      75 79 80 85 87 309
Nontrivial solution, in sparse context      311 312
Nontrivial splines      391 397
Normal form      13 (see also remainder)
Normal vector      362 (see also inward pointing normal)
Normalized volume      see volume of a polytope normalized
NP-complete decision problem      347 363
Nullstellensatz      see Strong Nullstellensatz
Number of generators of a module ( $\mu(M)$ )      224ff
Number theory      464
Numerical methods      28 32 56 60 338 339
Numerical methods, accumulated errors      57
Numerical methods, homotopy continuation      see homotopy continuation method
Numerical methods, Newton-Raphson      see
octahedron      362 374 403
OFF      234 236ff 271 272 277 280 282 283 395 398 399
One variable splines      see univariate splines
One-point code      458 467 one-point)
One-to-one      see injective
Onto      see surjective
Operations research      360
Optimization      361
Order of a pole      see pole order
Order of a polynomial g (ord())      163
Ordering monomials, $>_{r}$       440-446 448
Ordering monomials,       see $alex$

Ordering monomials, see $arevlex$

Ordering monomials, see $grevlex$

Ordering monomials, see $lex$

Ordering monomials, adapted      see monomial order adapted
Ordering monomials, anti-graded      see degree- anticompatible order
Ordering monomials, degree-anticompatible      see degree- anticompatible order
Ordering monomials, in $R^{m}$       see monomial order in
Ordering monomials, local      see local order
Ordering monomials, mixed      see mixed order
Ordering monomials, monomial      see monomial order
Ordering monomials, POT      see POT order
Ordering monomials, product      see product order
Ordering monomials, semigroup      see semigroup order
Ordering monomials, TOP      see TOP order
Ordering monomials, weight      see weight order
Ordinary differential equations (odes)      339
Ordinary double point      see singularity ordinary
Oriented edge      see edge oriented
Orthogonal subspace      423
Oshea, D.      vii-ix 4 9 10 12-15 20 21 23 25 34 35 37 39 49 52 72 73 81 86 145 149 166 167 169171 173 174 176 177 199 202 203 212 221 253 268 270 273 282 284 285 289 308 328 340 365 366 368 369 375 381 382 402 414 425 429 431 448 453 455 456 467 470
Ostebee, A.      x
Outward normal      294 (see also inward pointing normal)
Over a local ring      223
p-constant      178 (see also Milnor number)
Pallet      359 360
Parallelogram      170 171
Parameters of a code ([n, , d])      419 420 423 434 448 455 463 467
Parametrization      76 80 81 87 132 133 266 273ff 285 286 299 300 305
Parametrized solution      335 338 339
Parity check, matrix      416 417 419 422-424 434
Parity check, position      419 430 431
Park, H.      187 472
Partial differential equations      385
Partial graded resolution      263
Partial solution      25
Partition (of an interval)      388 389
Pedersen, P.      x 63 64 66 67 122 304 347 353 473
Peitgen, H. - O.      32 34 473
Pellikaan, R.      449 471

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