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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
Ideal, intersection of      184 377 462
Ideal, irreducible      207
Ideal, maximal      198
Ideal, maximum principle      260
Ideal, monomial      67ff 429ff 447 448 452 467
Ideal, P-primary      207
Ideal, primary      207
Ideal, prime      195 215 254 338 378 463 469
Ideal, principal      44 177 179
Ideal, principal ideal      40 78
Ideal, product      182 377 462
Ideal, projective elimination      388 389 396
Ideal, proper      198
Ideal, quotient      191 388
Ideal, radical      35 36 173 176 177 236 237 376 378
Ideal, radical of      174 373
Ideal, saturation      194 195
Ideal, standard basis of      see "Basis standard"
Ideal, sum of      181 377
Ideal, syzygy      339
Ideal, weighted homogeneous      398
Ideal-variety correspondence, affine      175 237
Ideal-variety correspondence, projective      372 376
Implicit representation      16
Implicitization      17 47 51 96 521
Inclusion-Exclusion Principle      440 444
Index of regularity      449
Inflection point      see "Point inflection"
Integral domain      see "Ring integral
Invariance under a group      324
Inverse kinematic problem      see "Kinematics problem of robotics inverse"
Inverse lexicographic order      see "Monomial ordering"
Irreducibility question      205
Irredundant, intersection of ideals      204
Irredundant, union of varieties      203
Isomorphic rings      222
Isomorphic varieties      217 237 343 468
Isotropy subgroup      347
Jenks, R.      507
Joint space      see "Space joint
Joints (of robots), ball      263 266
Joints (of robots), helical      263 266
Joints (of robots), prismatic      262
Joints (of robots), revolute      262
Joints (of robots), spin      274
Jouanolou, J.      154
k(V)      463 473
Kapranov, M.      154
Kendig, K.      463 480 481
Kinematic redundancy      286
Kinematic singularities      278—280 286
Kinematics problem of robotics, forward      264
Kinematics problem of robotics, inverse      264
Kirwan, F.      423 424
Klein, F.      323
Knoerrer, H.      424 428 520
Kredel, H.      513 521
k[V]      214 219 221 246 463 466 469
Lagrange multipliers      99
Lang, S.      127 471
Lazard, D.      109 110 118 519
Leading coefficient      57
Leading monomial      57
Leading term      37 57
Leading terms, ideal of      see "Ideal of
Least common multiple (LCM)      81 186
Level set      217
Lexicographic order      see "Monomial ordering"
Lin, A.      510
Line at infinity      353
Line, affine      3 353
Line, limit of      487ff
Line, projective      350 352 363 406
Line, secant      487ff
Line, tangent      134 136ff
Little, J.      118 233 521 522
Local property      423 474
Locally constant      423
Loustaunau, P.      206 510 517 522
Macaulay (program)      see "Computer algebra systems"
Macaulay, F.S.      154 447
MacDonald, I.G.      209
Macsyma      see "Computer algebra systems"
Magma      see "Computer algebra systems"
Manifold      481
Manocha, D.      118 131 154 521
Maple      see "Computer algebra systems"
Mapping      406
Mapping, dominating      472
Mapping, polynomial      213
Mapping, projection      120 213 385 387 469
Mapping, pullback      239
Mapping, rational      248
Mapping, regular      213
Mapping, Segre      384
Mapping, stereographic projection      252
MAs      see "Computer algebra systems"
Mathematica      see "Computer algebra systems"
Matrix group      321
Matrix permutation      322
Matrix, Jacobian      278 480
Matrix, row-reduced echelon      see "Echelon matrix"
Matrix, Sylvester      151
Matsumura, H.      491
Mayr, E.      109
Melenk, H.      512 514
Meyer, A.      109
Mignotte, M.      118 149
Mines, R.      149 176 206
Mishra, B.      118 309 520
Mixed order      see "Monomial ordering"
Module      522
Moeller, H.M.      108 512 514
Monomial      1
Monomial ordering      53ff 70 340 394 485
Monomial ordering, elimination      72 118 119
Monomial ordering, graded      379 382 448 462 466
Monomial ordering, graded lexicographic (grlex)      55 486
Monomial ordering, graded reverse lexicographic (grlex)      56
Monomial ordering, inverse lexicographic (invlex)      58
Monomial ordering, lexicographic (lex)      54 94ff 113 114 297 316 388 486
Monomial ordering, mixed      72
Monomial ordering, product      72
Monomial ordering, weight      72
Mora, T.      108 519 521
Multidegree (multideg)      57
Multinomial coefficient      336
Multiplicity of root      45 135 143 155
Multiplicity, intersection      135 414 419ff
Mumford, D.      109 480 490 493
Neun, W.      512
Newton identities      317 320
Newton polygon      520
Newton’s method      519
Niesi, G.      108
Nilpotent      223 225 226
Noether, E.      331
Normal form      80
Nullstellensatz      4 34 36 45 122 170 191 199 231 233 297 382 425 451 461
Nullstellensatz, Hilbert’s      168 170 191
Nullstellensatz, in k[V]      237
Nullstellensatz, Projective Strong      375 453
Nullstellensatz, Projective Weak      374 389
Nullstellensatz, strong      174 199 292 374
Nullstellensatz, weak      168 199 230 374
Numerical solutions      99 118
octahedron      328
Operational space      see "Space configuration
Orbit of a point      343
Orbit, G      343
Orbit, space      343
Order (of a group)      321
Ordering      see "Monomial ordering"
orthocenter      299
O’Shea, D.      118 233 521 522
Parametric representation      125
Parametric representation, polynomial      16 196 237 342
Parametric representation, rational      15 129 197
Parametrization      17
Partial solution      114ff 120
Path connected      423 426
Paul, R.      282
Pedoe, D.      410
Pencil of hypersurfaces      369
Pencil of lines      359
Pencil of surfaces      238
Pencil of varieties      238 370
Permutation      499
Permutation, sign of      499
Perspective      350 354
PGL(n,k)      see "Group projective
Plane, affine      3
Plane, Euclidean      287
Plane, projective      349
Point of inflection      144
Point, critical      98
Point, nonsingular      136 244 427 474 478 491
Point, singular      7 134 136 244 411 478 484 519
Point, smooth      478
Point, Steiner      301
Point, vanishing      350
Polyhedron, duality      328
Polyhedron, regular      328
Polynomial      2
Polynomial ring $(k[x_{1},...,x_{1}])$      see "Ring polynomial"
Polynomial, affine Hilbert      449 455 467
Polynomial, elementary symmetric      312
Polynomial, Hilbert      453 455 458 459
Polynomial, homogeneous      317 362
Polynomial, homogeneous component of      317
Polynomial, integer      151
Polynomial, invariant      324
Polynomial, irreducible      146ff 177
Polynomial, linear part      474
Polynomial, Newton — Gregory interpolating      445
Polynomial, partially homogeneous      386
Polynomial, reduced      45 178 416 476
Polynomial, S-      81ff 86ff 100ff
Polynomial, square-free      45 178
Polynomial, symmetric      311
Polynomial, weighted homogeneous      396 398
POSSO      see "Computer algebra systems"
PostScript      22
Power sums      317
Primality question      205
Primary decomposition question      209
Principal ideal domain (PID)      40 224 522
Product order      see "Monomial ordering"
Pseudocode      37 501ff
Pseudodivision      255 302ff
Pseudodivision, successive      306
Pseudoquotient      303
Pseudoremainder      303
Puiseux expansions      520
Pyramid of rays      354
Quadric hypersurface      247 363 399ff 405
Quadric hypersurface, nonsingular      405
Quadric hypersurface, over $\mathbb{R}$      405
Quadric hypersurface, rank of      403
Quotient, field      see "Field of
Quotient, vector space      446
Quotients on division      59
R-sequence      464
Radical of an ideal      see "Ideal radical
Radical, generators of      176
Radical, membership      see "Algorithm radical
Rank of a matrix      279ff 481
Rank of a quadric      403
Rank, deficient      279
Rank, maximal      279
Real projective plane      349
Reduce      see "Computer algebra systems"
Reduction of a polynomial      45 178 476
Remainder on division      59 79 80 86ff 93 227ff
Resultant      131 151ff 157 158ff 416
Resultant, generalized      161
Resultant, multipolynomial      131 154 521
Reverse lexicographic order      see "Monomial ordering"
Reynolds operator      330
Richman, F.      149 176 206
Riemann sphere      361 367
Ring      324
Ring of invariants      324
Ring, commutative      2 215 497
Ring, coordinate, of a variety (k[V])      235ff 256 341 463 466 468 469
Ring, homomorphism      173 222
Ring, integral domain      215 235 252 256 463 498
Ring, isomorphism      222 339
Ring, polynomial $(k[x_{1}, ..., x_{n}])$      2
Ring, quotient $(k[x_{1}, ..., x_{n}]/I)$      220 254 339 459
Robbiano, L.      73 108 521
Robotics      10 13 14 261ff
Rose, L.      520
Roth, L.      410
Row-reduced echelon matrix      see "Echelon matrix"
Ruitenberg, W.      149 176 206
Sederberg, T.      131 521
Segre map      see "Mapping Segre"
Seidenberg, A.      176 206
Semple, J.G.      410
Shafarevich, I.R.      463 479 480
Singular (program)      see "Computer algebra systems"
Singular locus      479
Siret, Y.      39 42 44 149 188
Smith, L.      335
Solving equations      48 93 116 519
Space, affine      3
Space, configuration (of a robot)      264
Space, joint (of a robot)      264
Space, orbit      343
Space, projective      360
Space, quotient vector      446
Space, tangent      474 491
Stabilizer      347
Stillman, M.      72 109 119 519
Strophoid      24 25
Sturmfels, B.      109 297 334 335 338 520
Subgroup      499
Subring      324
Subvariety      236
Sugar      108 521
Surface, Enneper      132
Surface, hyperboloid of one sheet      247
Surface, intersection      520
Surface, ruled      98 407
Surface, tangent, to the twisted cubic      19 97 125 127 128 131 212 245
Surface, Veronese      218 384 395
Surface, Whitney umbrella      132
Sutor, R.      507
Syzygy      35 102ff 109 110
Syzygy, homogeneous      103 110
Syzygy, ideal      339
Tangent line to a curve      see "Line tangent"
Tangent space to a variety      see "Space tangent"
Taylor’s Formula      475 492
Term      2
tetrahedron      328
Theorem, Affine Dimension      451
Theorem, Bezout’s      412ff 420ff
Theorem, Circle, of Apollonius      290 297 306 307
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