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Cox D., Little J., O'Shea D. — Ideals, varieties, and algorithms
Cox D., Little J., O'Shea D. — Ideals, varieties, and algorithms

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Название: Ideals, varieties, and algorithms

Авторы: Cox D., Little J., O'Shea D.

Аннотация:

Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.

The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered it the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric Theorem proving.

In preparing a new edition of "Ideals, Varieties and Algorithms" the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple, Mathematica and REDUCE.



Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-nd edition

Год издания: 2006

Количество страниц: 553

Добавлена в каталог: 08.12.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$k(t_{1},...,t_{m})$      15
$k[f_{1},...,f_{m}]$      330
$k[x_{1},...,x_{n}]$      2
Adams, W.      206 517 522
Admissible geometric theorem      289ff
Affine, cone over a projective variety      see "Cone affine"
Affine, Dimension Theorem      see "Theorem Dimension"
Agnesi, M.      24
Algebraic relation      338
Algebraically independent      294 466 469 470
Algorithm      37 142
Algorithm, algebra (subring) membership      334
Algorithm, associated primes      209
Algorithm, Buchberger’s      87 92 99 100 106 108 377 519 521
Algorithm, computation in $k[x_{1},...,x_{n}]/I$      230
Algorithm, consistency      170
Algorithm, dimension (affine variety)      451 468
Algorithm, dimension (projective variety)      453
Algorithm, division      37ff 157 220
Algorithm, division in $k[x_{1},...,x_{n}]$      33 59ff 93 194 226 227 377 518
Algorithm, division in k[x]      302 303
Algorithm, Euclidean      41 92 149 152 156 157 178 179
Algorithm, finiteness of solutions      230
Algorithm, Gaussian elimination (row reduction)      9 49 50 52 91 154 304
Algorithm, greatest common divisor      187
Algorithm, ideal equality      91
Algorithm, ideal intersection      186 194 202
Algorithm, ideal membership      34 43 93
Algorithm, ideal quotient      194 202
Algorithm, irreducibility      205
Algorithm, least common multiple      187
Algorithm, polynomial implicitization      127
Algorithm, primality      205
Algorithm, primary decomposition      209
Algorithm, projective closure      382
Algorithm, projective elimination      393
Algorithm, pseudodivision      303
Algorithm, radical      233
Algorithm, radical membership      176 292
Algorithm, rational implicitization      130
Algorithm, Ritt’s decomposition      304 309 520
Algorithm, tangent cone      485ff
Algorithm, Wu — Ritt      309 510
Altitude      299
Anderson, D.      131 521
artificial intelligence      286
Ascending chain condition (acc)      76 77 201 373 391
Associated primes question      209
Atiyah, M.      209
Automated geometric theorem proving      286ff
Automorphism of varieties      242
Axiom      see "Computer algebra systems"
Baillieul, J.      282
Bajaj, C.      154 521
Barrow, I.      24
Basis, minimal      35 71 72
Basis, of an ideal      see "Ideal basis
Basis, standard      74 76
Bayer, D.      72 109 119 519
Becker, J.      517
Becker, T.      80 108 176 186 206 233 278 522
Benson, C.T.      323 334 335 520
Bernoulli, J.      24
Bezier, P.      20
Billera, L.      520
Birationally equivalent varieties      250ff 470
Blow-up      494 495
Boege, W.      513 521
Brieskom, E.      424 428 520
Bruce, J.W.      134 139 143
Buchberger, B.      75 84 88 108 227 282
Canny, J.      131 154
Centroid      299
Chain, ascending, of ideals      76 78
Chain, descending, of varieties      79 201 373
Char, B.      509
Characteristic of a field      see "Field"
Characteristic sets      304 309 520
Chou, S.C.      309 520
Circumcenter      299
Cissoid of Diodes      see "Curve cissoid
Classification of varieties      217 252
Clemens, H.      424
Closure, projective      381 394 401 454 461
Closure, Zariski      122 190 191 197 254
CoCoa      see "Computer algebra systems"
Coefficient      2
Collinear      288 299
Comaximal ideals      189
Complement of a monomial ideal      433ff
Complete intersection      464
complexity      108 109 519
Computer algebra systems      37 39 42 56 65 66 90 98 100 133 149 164 203 206 209 234 285 293 303 455 505ff 518ff
Computer algebra systems, AXIOM      37 176 206 209 505ff
Computer algebra systems, CoCoA      455 516ff
Computer algebra systems, Macaulay      176 206 455 516ff
Computer algebra systems, MACSYMA      516
Computer algebra systems, Magma      517
Computer algebra systems, Maple      37 508ff
Computer algebra systems, MAS      517
Computer algebra systems, Mathematica      37 510ff
Computer algebra systems, PoSSo      517
Computer algebra systems, REDUCE      37 176 206 209 455 512ff 521
Computer algebra systems, SCRATCHPAD      505
Computer algebra systems, SINGULAR      517
Computer graphics      521
Computer-aided geometric design (CAGD)      20 22
Cone, affine      368 369 374 453 485
Cone, projectivized tangent      495
Cone, tangent      485ff
Configuration space      see "Space configuration
Congruence (mod I)      219
Conic section      27 133 351 358 399 401 415 426
Consistency question      11 45 170
Constructible set      123 259 260
Control points      21
Control polygon      21
Coordinate      235
Coordinate ring of a variety (k[V])      see "Ring coordinate
Coordinate subspace      430 435 467
Coordinate subspace, translate of      436 437 439
Coordinate, Pluecker      408ff
coordinates      235
Coordinates, homogeneous      352 360
Coordinates, Pluecker      412
Coset      347
Cox, D.      118 233 521 522
Coxeter, H.S.M.      323
Cramer’s Rule      390 500
Cross ratio      298
cube      322 327 328
Cubic, Bezier      20ff 27 520
Cubic, twisted      see "Curve twisted
Curve, cissoid of Diodes      25 26
Curve, dual      348
Curve, family of      138
Curve, folium of Descartes      133
Curve, four-leaved rose      12 13 144
Curve, rational normal      383 384
Curve, twisted cubic      7 8 19 32 66 84 97 125 127 131 172 195 212 217 244 245 364 366 368 370 379—382 450 464—466
Cuspidal edge      245
Czapor, S.      521
Davenport, J.H.      39 42 44 149 188
Decomposition, minimal, of a variety      203 373 476
Decomposition, minimal, of an ideal      204
Decomposition, primary      206ff
Decomposition, question      206
Degenerate case of a geometric configuration      294 307
Degeneration      428
Degree of a projective variety      465
Degree, total, of a monomial      2 438
Degree, total, of a polynomial      2
Degree, transcendence, of a field extension      470
Dehomogenization of a polynomial      363 486
Dehomogenization of an ideal      392
Derivative, formal      46 226 474
Descending chain condition (DCC)      79 201 259 373
Desingularization      495
Determinant      151 417 499
Determinant, Vandermonde      44
Dickson’s Lemma      69
Difference of varieties      14 191
DIMENSION      8ff 11 233 279 430 431 436 439 442 450 453 457ff 466 468—471 480 481
Dimension at a point      478 491
Dimension, question      11 426ff
Discriminant      155 319
dodecahedron      328
Dominating map      472
Dual curve      348
Dual projective plane      359 399
Dual projective space      369 407
Dual variety      348
Duality of polyhedra      328
Duality, projective principle of      351
Dube, T.      108
Echelon matrix      49 75 92 410 412
Eisenbud, D.      176 206 517 522
Elimination order      see "Monomial ordering elimination"
Elimination step      113
Elimination theory      17 112ff
Elimination theory, projective      384ff
Envelope      139ff
Equivalence, birational      250 470 473
Equivalence, projective      399 404
Euler line      299
Euler’s formula      369
Extension step      113
Factorization of polynomials      146ff 164
Family of curves      see "Curve family
Farin, G.      520
Faugere, J.      519
Feiner, S.      521
Fiber      259
Field      1 497 520
Field of characteristic zero      180 482
Field of finite (positive) characteristic      180 402 404 410 482
Field of fractions of a domain      254
Field of rational functions (k(V))      147 246 463 469 473
Field, algebraically closed      4 34 124 163 168 170 172 174 175 192 196 199 200 204 210 230 231 254 374—376 382 389 394 395 404 405 417 451 453 458—462 469 486
Field, finite      3—5 36
Field, fractions of a domain      245
Field, infinite      3 4 32 126 130 172 196 342 372
Final remainder (in Wu’s method)      306
Finite Generation of Invariants      327 333 336
Finiteness question      11 230 235 460
Finkbeiner, D.T.      156 166 404 469 499 500
Foley, J.      521
Folium of Descartes      see "Curve folium
Follows generically from      295
Follows strictly from      292
Forward kinematic problem      see "Kinematics problem of robotics forward"
Fulton, W.      423 424
Function field      see "Field of
Function, algebraic      119
Function, coordinate      235
Function, polynomial      3 213
Function, rational      15 119 245 473
Garrity, T.      154 520 521
Gauss, C.F      314
Gaussian elimination      see "Algorithm Gaussian
Gebauer, R.      108 513 521
Gelfand, I.      154
Gianni, P.      176 206 519
Giblin, P.J.      134 139 143
Giovini, A.      108 521
Giusti, M.      109
GL(n,k)      see "Group general
Goldman, R.      131 521
Goldstine, S.      511
Grabe, H.G.      516
Graded lexicographic order      see "Monomial ordering"
Graded monomial order      see "Monomial ordering"
Graded reverse lexicographic order      see "Monomial ordering"
Gradient      10 136 137
Graph      6 126
Grassmannian      409
greatest common divisor (GCD)      40ff 178 187
Greuel, G.-M.      517
Griffiths, P.      424
Gritzmann, P.      109
Groebner basis      31 44 74ff 113ff 127 128 130 160 170 176 186 194 226ff 244 275ff 284 293 302 309 314 316 334ff 340ff 382 388 394 485ff 519ff
Groebner basis, comprehensive      278 522
Groebner basis, conversion      519
Groebner basis, criterion for      82 104
Groebner basis, minimal      89 92
Groebner basis, reduced      90 92 170 176 292 296 374
Groebner basis, specialization of      276 278 283ff
Groebner basis, universal      522
Groebner, W.      75
Group      498
Group of symmetries of a cube      322 327 347
Group of symmetries of a tetrahedron      328
Group, cyclic      322
Group, finite matrix      321ff
Group, general linear (GL(n,k))      321 410 418 498
Group, generators for      325
Group, Klein four-      326
Group, orbit of a point under      343
Group, permutation      322 499
Group, projective general linear (PGL(n,k))      410
Group, subgroup of      499
Grove, L.C.      323 334 335 520
Heintz, J.      109
Hermann, G.      176 206
Herstein, I.N.      318 521
Hilbert function      452 452 459
Hilbert function, affine      447 467
Hilbert, D.      74 168 311 335 433
Hironaka, H.      76
Hodge, W.V.D.      410
Hoffmann, C.      519 520
Homogenization of a polynomial      172 364
Homogenization of an ideal      378 453 486
Hughes, J.      521
Huneke, C.      176 206
Hyperboloid      247
Hyperplane      363 400
Hyperplane at infinity      361 461
Hypersurface      363 458 461
Hypersurface, cubic      363
Hypersurface, nonsingular quadric      405
Hypersurface, quadric      363 401ff 404
Hypersurface, quartic      363
Hypersurface, quintic      363
icosahedron      328
Ideal      29 498
Ideal description question      34 47 73
Ideal membership question      34 44 47 65 70 80 93 519
Ideal of a variety (I(V))      31 372
Ideal of leading terms $(\langle LT(I)\rangle)$      73
Ideal of relations      339
Ideal, basis of      30 35
Ideal, colon      191
Ideal, complete intersection      464
Ideal, determinantal      110
Ideal, elimination      113 340 394
Ideal, generated by a set of polynomials      29
Ideal, Groebner basis of      see "Groebner basis"
Ideal, homogeneous      363 371 377
Ideal, in a ring      223
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