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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: 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 Предметный указатель 15 330 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 230 Algorithm, consistency      170 Algorithm, dimension (affine variety)      451 468 Algorithm, dimension (projective variety)      453 Algorithm, division      37ff 157 220 Algorithm, division in 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 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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