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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

#P-complete enumerative problem      347
$(\mathbb{C}^{*})^{n}$       300-303 306 307 313 328 329 331 332 334 336 337 341 342 351 353 354
      152ff 168 172 176 177
$C^{r}$ function      387 388 390 404
$C^{r}$ spline      387 388
$C^{\infty}$ function      386
$K^{n}$       see affine n-dimensional
      8 10 15 16 25 27 38 46ff 56 57 62 121 138 151 202-204 209 214 238 328 366-369 457
      see free module
$\acute{e}$ cart      158 163 164
$\mathbb{C}^{n}$       see affine n-dimensional
$\mathbb{F}_{q}^{n}$       see affine n-dimensional
$\mathbb{P}^{n}$       see projective n-dimensional space
$\mathbb{R}^{n}$       see affine n-dimensional
$\mu^{*}$ -constant      178 (see also Tessier $\mu^{*}$ -invariant)
(Non-standard) degree      369
Abel, N.      27
Acton, F.      32 55 468
Adams, W.      vii 4 9 12-15 21 25 37 174 201 202 359 468
Adapted order      see monomial order adapted
Adjacent cells or polytopes in a polyhedral complex      389 391-394 402
Affine group (Aff(, $\mathbb{F}_{q}$ ))      465
Affine half-space      389
Affine Hilbert function      see Hilbert function affine
Affine hyperplane      293 361 391 394 398 399 402
Affine linear space      see affine subspace
Affine n-dimensional space over ( $k^{n}$ )      16 227
Affine n-dimensional space over $\mathbb{C}$ ( $\mathbb{C}^{n}$ )      85 90
Affine n-dimensional space over $\mathbb{F}_{q}$ ( $\mathbb{F}_{q}^{n}$ )      415ff 424ff 449ff
Affine n-dimensional space over $\mathh{R}$ ( $\mathbb{R}^{n}$ )      63ff 290ff 359ff
Affine subspace      297 325 370
Affine variety      18
Alexandrov - Fenchel inequality      346
Algebra over a field       35
Algebra over a field , finite-dimensional algebra      34ff 41 51ff 90 434
Algebraic closure of a field      451
Algebraic coding theory      viii-ix 407 414 415ff
Algebraic curve      449 451 452 454 457 462 466
Algebraic decomposition      see subdivision algebraic
Algebraic number      453
Algebraically closed field      21 39 90 140 142 143 248 465
Allgower, E.      338 468
Alonso      169 173 175 468
Alphabet (of a code)      415 427 435
Analytic function      see convergent power series ring
Annihilator of (ann())      193
Anti-graded lex order      see ales
Anti-graded order      see degree-anticompatible order
Anti-graded revlex order      see arevlex
Arevlex      152ff
Arithmetic genus      454
Arnold, V.      178 468
Artin      231 468
Ascending chain condition      176 206 207
Automorphism      see field automorphism
Auzinger, W.      129 468
Axiom      39 432
B $\acute{e}$ out's Theorem      91 108ff 177 329 330 337 463
B $\acute{e}$ zout bound      331 339
B $\acute{e}$ zout Formula      94
B-basis      275 276 278 279 283 289
Bajaj      102 468
Basic Question of resultants      79 80
Basis monomials      36 38 42 45 46 49 51 58 457 458
Basis of a free module      see module basis
Basis of an ideal      4
Batyrev, V.      312 313 468
BCH code      427 434 435 450
BCH code, designed distance of      435
Becker, T.      vii 4 9 12-15 25 37 51 63 69 174 468
Ben – Or, M.      67 468
Berlenkamp - Massey algorithm      435 443
Berlenkamp - Welch algorithm      447
Bernstein s Theorem      327 329 331ff 353
Bernstein, D.      331 335 337 468
Betke, U.      346 468
Bihomogeneous polynomial      315
Bijective mapping      433
Bilinear mapping      423
Bilinear surface parametrization      298
Billera, L.      ix 347 385 386 395 401 404 406 469
Binomial (= difference of two monomials)      373 379 383 ideal)
Binomial coefficient      98 106 241 251 267 269 271 285 287 288 308 383 384 396-398 401 453 456
Binomial theorem      54 78
Bivariate splines      401
BKK bound      336 337-339 341
Blahut, R.      ix 415 435 469
Block length (of a code)      415 419 420 423 426 427 432-434 449 450 458 459 462 465
Block matrix      216 217
Bonnesen, T.      ix 316 320 323 469
Bottom down order      209
Bracket ( $[ii...i_{n}]$ )      92-94 304
Brian $\k{c}$ on, J.      178
Bruns, W.      285 469
Bryant, B.      385 469
Buchberger's algorithm      14 46 160 163 165ff 208 366 404
Buchberger's Algorithm, in module case      200 204 206-208 211 214 432 443 448
Buchberger's Criterion      13 160 166 175
Buchberger's Criterion, in module case      205 211 212 223
Buchsbaum, D.      ix 288 469
Burago, Yu      316 320 323 469
Burden, R.      ix 32 55 469
Burst error-correcting      see error-correcting burst
CALI      see REDUCE
Canny, J.      104-106 114 115 119 324 347 351-353 357 358 469 470
Canonical curve      273
Categorical quotient      312
Cayley - Hamilton theorem      53 54
Cell of a polyhedral complex      389394 396-398 402 406
Cell of a polyhedral or mixed subdivision      344-349
Cell, interior, of a polyhedral complex      394 396 398 399 401 402 404
Chain rule      339
Chaos      34
Characteristic function      280 (see also Hilbert function)
Characteristic polynomial      53 55 68 105 143
Chen, F.      274 278-280 470 474
Chinese remainder theorem      104
Chow form      308
Chow form, twisted      356
Closed ball      417
CoCoa      206 220 240
Code      416 418 426 435 447 448 459 461 464
Code, alphabet of      see alphabet of a code
Code, BCH      see BCH code
Code, block length of      see block length of a code
Code, codimension of      see ñodimension of a code
Code, cyclic      see cyclic code
Code, dimension of      see dimension of
Code, dual      see dual code
Code, equivalent      see equivalent codes
Code, error-correcting      see error-correcting code
Code, geometric Goppa      see geometric Goppa code
Code, Hamming      see Hamming code
Code, Hermetian      see Hermetian code
Code, linear      see linear code
Code, MDS      see MDS code
Code, minimum distance of      see minimum distance of code
Code, one-point      see one-point code
Code, parameters of      see parameters of a code
Code, perfect      see perfect code
Code, punctured      see punctured code
Code, Reed - Muller      see Reed - Muller code
Code, Reed - Solomon      see Reed - Solomon code
Codeword      415-417 419-421 423 424 426-432 4344436 445 447 449 450 452 458 459 461 463 464 466
Codimension of a code      435
Coding theory      see algebraic coding theory
Cohen - Macaulay ring      248 249 271 284
Coherent Minkowski sum      see Minkowski sum of polytopes coherent
Coherent mixed subdivision      see mixed subdivision coherent
Coherent subdivision      see mixed subdivision coherent

Cokemel      194 195 235
Colley, S.      x
Collinear points      271-273 278
Column operation      230
Column vector f (in a free module)      see vector f (in a free module)
Comaximal ideals      148
Combinatorial data      397 398 401
Combinatorial enumeration problem      374 (see also magic square)
Combinatorics      viii-ix 313 374 384
Common Factor property of resultants      72 75
Commutative algebra      viii-ix 207 374 384
Commutative diagram      118 120 143 148 259 265
Commutative ring      2 135 179ff
Compact disk      415 426 435
Companion matrix      77 459 465
Compatibility property (of a monomial order)      367 368 372
Complete intersection      272 273
COMPLEX      see polyhedral complex
Computational geometry      305
Computer science      360
Computer-aided geometric design      273
Computing determinants, numerically      104
Computing determinants, symbolically      104 105 113
Cone      377
Congruence      438 439 442 443 445 446
Conic section      266
Connected graph      see graph connected
Conti, P.      359 363 372 373 470
Continuation system      338
Contraction mapping theorem      33
Convergent power series ring ( $k\{x_{1}, ...,x_{n}\}$ )      133ff 137 146 156 165 222
Convex combinations      290
Convex hull of a set S (Conv( S))      290-292 295 297 298 301-304 307 310 318-320 325 330 332 342 343 347 348 353 385 389
Convex polygon      291
Convex polyhedral cone      377
Convex polytope      see polytope
Convex set      290 291
Cools, R.      340 475
Copulation      189
Coset ()      35 183 190 226 227 354 355 408 412 421 424-426
Coset () leader      421
Courant functions      405 406
Cox, D.      vii-ix 4 9 10 12215 20 21 23 25 34 35 37 39 49 52 72 73 81 86 145 149 166 167 169-171 173 174 176 177 199 202 203 212 221 253 268 270 273 278-280 282 284 285 289 308 312 313 328 340 365 366 368 369 375 381 382 402 414 425 429 431 448 453 455 456 467 468 470
Cramer's rule      95
Cryptography      415
Cube in $\mathbb{R}^{3}$       293
Cubic curve      463
Cubic spline      388
Curve, algebraic      see algebraic curve
Curve, canonical      see canonical curve
Curve, cubic      see cubic curve
Curve, divisor on      see divisor (on an algebraic curve)
Curve, Hermitian      see Hermitian curve
Curve, Klein Quartic      see Klein Quartic curve
Curve, level      see level curve
Curve, modular      see modular curve
Curve, projective      see projective curve
Curve, rational normal      see rational normal curve
Curve, rational quartic      see rational quartic curve
Curve, singular      see singular curve
Curve, smooth      see smooth curve
Curve, with many automorphisms      458
Cycles (in a graph)      400
Cyclic code      407 424ff 458 459 462 465
Cyclic dual of      see dual code cyclic
Cyclic group      408 412 413
Cyclic m-dimensional code      428-430 432-434 436 458
Cyclic permutation      380 424 466
Cyclic shift      426 427 429 458
D-degree      315
D-homogeneous      315
D-resultant      see resultant sparse
Decoding      407 415 416 419-421 426 435 437 438 440-442 444 447 448 452
Decoding, function      416 436
Decoding, majority logic      450
Decoding, nearest neighbor      see nearest neighbor decoding function
Decoding, Reed - Solomon      see Reed - Solomon
Decoding, syndrome      see syndrome decoding
Degenerate special case      171
Degree of a polynomial      see total degree
Degree of a variety      270 271 286-288 453 455 467
Degree-anticompatible order      151ff
Dehomogenize      108 159 160 162 163 166 172-174 176 279 286
Dense polynomial      318
Dense resultant      see resultant dense
Dense start system      see start system dense
Dense system of equations      339
Descartes rule of signs      66 67 70
Descartes, R.      1
Designed distance      see BCH code designed
Devaney, R.      34 470
Devlin, K.      x
Diagonalizable matrix      58
Dickenstein, A.      x
Dickson s Lemma      12 49 199
Difference of varieties ( $V \ W$ )      23
Differentials      449
Diirer, A.      375
Dimca, A.      178 470
Dimension of a linear code      419 423 428 432 434-436 450 457 458 461 462 465
Dimension of a polyhedral complex      389 390 392
Dimension of a polytope      291 294 297 301 316 319 320 342-344
Dimension of a variety      270-272 286-288 451
Dimension of an affine subspace      297
Dimension theorem for a linear mapping      244 456
Direct sum of modules ( $M \oplus N$ )      182 194 217 221 230 242 243 245
Direct sum of modules ( $M \oplus N$ ), of graded modules      254ff 267 404
Discrete Fourier Transform      433
Discrete valuation      135 136
Discriminant of a polynomial / (Disc(f))      340
Distance function      417
Division Algorithm in       4 408 425 431 436 447 448
Division Algorithm in $k[x_{1},..., x_{n}]$       9 34 165 166 365 366 382 429 430 432 458 461
Division Algorithm in a free module      200 202-204 211 212 214 444
Division Algorithm in a free module over a local ring      223 224
Division Algorithm in a local ring      151 156ff 175
Divisors (on an algebraic curve)      449 466
Dixon resultant      306 316 343
Dixon, A      306
Doubly stochastic matrix      385
Drexler, F.      338 470
Dual basis      43
Dual code      423 432 433 449 452
Dual graph      see graph dual
Dual space of a vector space $V(V^{*})$       43
Eagon - Northcott complex      266
Edge of a graph      398
Edge of a polyhedral complex      389 390 393 404 406
Edge of a polytope      292 293 345 352
Edge, oriented      398 400
Eigenspaces      59
Eigenspaces, generalized      149
Eigenvalues      51 54ff 58ff 61 69 91 128 129 144 354
Eigenvalues, dominant      55 56
Eigenvalues, generalized      61 62 144 149
Eigenvalues, numerical techniques      55ff 127
Eigenvalues, real      65
Eigenvectors      58ff 126 128 129 354
Eigenvectors, left      58 61 62 127
Eigenvectors, right      58 59 61 127
Eisenbud, D.      ix 249 252 263 266 273 285 288 467 469 470
Elementary row operations      see row operations
Elementary symmetric functions      96
Elimination ideal      5 6 15 24 173 274
Elimination order ( $>_{elim}$ )      173 174 365 367-369 372 378
Elimination property of mixed orders      155
Elimination property of resultants      72 73
Elimination Theorem      15 16 25 37 173
Emiris, I.      x 122 124 129 305 324 346 347 351-353 355-358 469 470
Encoding      407 415 416 419 421 424 432

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