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Mishra B. — Algorithmic algebra
Mishra B. — Algorithmic algebra



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Название: Algorithmic algebra

Автор: Mishra B.

Аннотация:

Four main topics are covered: Gr<:;o>bner bases, characteristic sets, resultants and semialgebraic sets. The text is written for theoretical computer science students who would like to do research or understand the algorithmic underpinning of various commercial symbolic computations systems such as Mathematica, Maple, or Axiom. The book is self-contained, and focuses on very basic material. Problems with selected solutions are included.


Язык: en

Рубрика: Математика/Алгебра/Вычислительная алгебра/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 416

Добавлена в каталог: 12.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Matrix, cofactor      386
Matrix, identity matrix      386
Matrix, submatrix      385
Maximal common divisor      204
Mechanical theorem proving      167
Minimal ascending set      179—180
Minimal common multiplier      204
Minimal polynomial      319
Modular law      34
Module      23 50 69
Module basis      52
Module examples      50
Module homomorphism      50
Module of fractions      50
Module, free      52
Module, module of fractions      50
Module, Noetherian      53
Module, quotient submodule      51
Module, submodule      51
Module, syzygy      23 54
Monic polynomial      205
Monogenic submodule      52
Monomial      36
Monomial degree      36
Monomial ideal      37
Monomial ideal, head monomial ideal      44
Monomial, head monomial      43
Multiple      199
Multiple, common multiple      200
Multiple, minimal common multiple      200
Multiplication algorithm for algebraic numbers      333
MultiplicativeInverse algorithm for algebraic numbers      331
muMATH      9
NewGroebner algorithm      90
NewHeadReduction algorithm      88 90
NewOveHeadReduction algorithm      88
Nilpotent      29
Noetherianness      6
Noncommutative ring      69
Normal form      80
Normalize algorithm for algebraic numbers      329
Nullstellensatz      13 134 142—143 182 226
Offset surface      11
OneHeadReduction algorithm      83
OneHeadReduction algorithm, modified      88
Order isomorphism      301
Order representation      327
Ordered field      298
Ordered field, Archimedean      301
Ordered field, induced ordering      301
Ordering, $\prec$      171
Parallelization      7
Path connected      336
Pilot ACE      3
Pivoting      133
PM      8
Polynomial      35 36
Polynomial degree      35 36
Polynomial length      36
Polynomial rank      172
Polynomial remainder sequence, PRS      226 247—249 266 271
Polynomial remainder sequence, PRS, Euclidean polynomial remainder sequence, EPRS      248
Polynomial remainder sequence, PRS, primitive polynomial remainder sequence, PPRS      248
Polynomial, multivariate      35
Polynomial, ordering      172
Polynomial, repeated factor      239
Polynomial, ring      35
Polynomial, similarity      247
Polynomial, square-free,      39
Polynomial, univaxiate      35
Power product      36
Power product, admissible ordering      39
Power product, divisibility      36
Power product, greatest common divisor      36
Power product, least common multiple      36
Power product, multiple      36
Power product, semiadmissible ordering      39
Power product, total degree      36
Prenex form      356
Prenex form, matrix      356
Prenex form, prefix      356
Primality testing      197
Prime element      200
Prime element, relatively prime      205
Prime field      29
Primitive polynomial      205—206
Primitive polynomial remainder sequence, PPRS      248
Principal ideal domain, PID      199 207 209
Principal subresultant coefficient, principal subresultant coefficient chain      266
Principal subresultant coefficient, PSC      252 266
Product of ideals      71
PROLOG      9
Proof by example      186
Propositional algebraic sentences      335
PseudeDivisionRec algorithm      170
Pseudodivision      168 169 173 226 244
Pseudodivision, quotient      169
Pseudodivision, reduced      169
Pseudodivision, remainder      169
PseudoDivisionIt algorithm      170
Pseudoquotient      169 245
Pseudoremainder      169 245
Pseudoremainder chain      175
Pseudoremainder, homomorphism      246
Quantifier elimination      335
QUEUE      14
Quotient field      30
Quotient group      26
Quotient of ideals      71
Quotient, ring      30—31
Quotient, submodule      51
Randomization      7
Real algebra      301
Real algebraic geometry      20
Real algebraic integer      298 316
Real algebraic number      298 316 347
Real algebraic number, addition      332
Real algebraic number, additive inverse      331
Real algebraic number, arithmetic operations      331
Real algebraic number, conversion      330
Real algebraic number, degree      319
Real algebraic number, interval representation      320 327
Real algebraic number, minimal polynomial      319—320
Real algebraic number, multiplication      333
Real algebraic number, multiplicative inverse      331
Real algebraic number, normalization      328—329
Real algebraic number, order representation      320 327
Real algebraic number, polynomial      319
Real algebraic number, refinement      328—329
Real algebraic number, representation      327
Real algebraic number, sign evaluation      328 330
Real algebraic number, sign representation      320 327
Real algebraic sets      337—338
Real algebraic sets, projection      339
Real closed field      189 297 301
Real geometry      297 334
Real root separation      320
Reduce      8 9
Reduction      71 133
Refine algorithm      329
Repeated factor      239
Representation      7
Residue class      26
Residue class of $\mathbb{Z}$ mod m      26
Residue class, ring      31
Resultant      225 227 235 296
Resultant, common divisor      261—262
Resultant, evaluation homomorphism      234
Resultant, homomorphism      232
Resultant, properties      228 230—231 260—262
Reverse lexicographic ordering      40—41
Ring      14 23 27 69
Ring examples      27
Ring homomorphism      31
Ring multiplication      27
Ring of fractions      30
Ring, addition      27
Ring, additive group of the ring      27
Ring, commutative      27
Ring, computable      72
Ring, detachable      72
Ring, full quotient ring      30
Ring, Noetherian      28
Ring, polynomial ring      35
Ring, quotient ring      30—31
Ring, reduced      29
Ring, residue class ring      31
Ring, residue classes mod m, $\mathbb{Z}_n$      27
Ring, strongly computable      71—72 102
Ring, subring      27—28
Ring, syzygy-solvable      72
RISC-LINZ, Research Institute for Symbolic Computation at the Johannes Kepler University, Linz, Austria      21
Ritt's principle      178
Robotics      9—10 297—298 334
Rolle's theorem      305
Root separation      315 320
RootIsolation algorithm      324
Rump's bound      321
S-polynomials      55 71 75 79 133
SAC-l      8
SAINT      8
SAME, Symbolic and Algebraic Manipulation in Europe      21
Sample point      348
Scratchpad      8 9
Sections      343
sectors      343
Sectors, intermediate      343
Sectors, lower semiinfinite      343
Sectors, upper semiinfinite      343
Segn algorithm for algebraic numbers      330
Semiadmissible ordering      39
Semiadmissible ordering, examples      40
Semiadmissible ordering, lexicographic      40
Semiadmissible ordering, reverse lexicographic      40 41
Semialgebraic cell-complex      337
Semialgebraic decomposition      336
Semialgebraic map      345
Semialgebraic set      298 334—335
Semialgebraically connected      336
Semialgebraically path connected      336
Semigroup      24
Set      14
Set deletion      15
Set difference      14
Set insertion      15
Set intersection      14
Set, choose      14
Set, empty set      14
Set, union      14
SETL      13
Sign, assignment      337
Sign, class      337
Sign, invariance      327
Sign, representation      327
Sign, variation      309
Similar polynomials      247
SMP      9
Solid modeling      297—298 334
Solvability      142 145 190
Solvability algorithm      145
Solvability, finite      145 149
Solving a system of polynomial equations      133 144
Square-free polynomial      239
STACK      14
Standard bases      70
Statement separator      15
Stone isomorphism lemma      154
Stratification      298
Strongly computable ring      71—72 102
Strongly computable ring, Euclidean domain      213
Strongly computable ring, example      73 76
Strongly triangular form      135—136
Sturm sequence      225
Sturm sequence, canonical      310
Sturm sequence, standard      310
Sturm sequence, suppressed      310
Sturm — Tarski theorem      309 314 330
Sturm's theorem      297 309 347
Subalgebra      69
Subfie]d      29
Subfie]d, examples      29
Subgroup      25
Subgroup, generated by a subset      25
Subgroup, normal      25
Subgroup, self-conjugate      25
Subideal      103—104
SubMatrix      385
Submodule      51
Submodule annihilator      52
Submodule product      52
Submodule quotient      52
Submodule sum      52
Submodule, cyclic      52
Submodule, finitely generated      52
Submodule, monogenic      52
Submodule, system of generators      52
Subresultant      225—226 250
Subresultant chain      266 271—272
Subresultant chain theorem      266 268—269 274 279 296
Subresultant chain, block structures      266—267
Subresultant chain, defective      266
Subresultant chain, nonzero block      267
Subresultant chain, regular      266
Subresultant chain, zero block      267
Subresultant polynomial remainder sequence, SPRS      249 271—272 296
Subresultant, defective      254
Subresultant, evaluation homomorphism      277 279
Subresultant, homomorphism      262—263 265
Subresultant, properties      256 258
Subresultant, regular      254
Subresultant, relation with determinant polynomial      254
Subring      27
Successive division      213
Successive pseudodivision      171
Successive pseudodivision lemma      175
Sycophante      9
Sylvester matrix      227
Sylvester's dialytic method of elimination      226 296
Symbal      9
Symmetric group      24
Symmetric polynomial      226
System of linear equations      388
System of linear equations, nontrivial solution      388
Syzygy      23 54 69
Syzygy basis      71
Syzygy computation      93—102
Syzygy condition      57
Syzygy solvability      71—72 93—102 213 215
Syzygy solvability, Euclidean domain      215
Syzygy, condition      57
Syzygy, S-polynomials      55 71 75 79 133
Tarski geometry      189 354
Tarski sentence      298 335 354
Tarski set      335
Tarski — Seidenberg theorem      345
Term ordering      69
Thom's lemma      315 320 325
Total degree, Tdeg      181
Total lexicographic ordering      42
Total reverse lexicographic ordering      42
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