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| Mishra B. — Algorithmic algebra |
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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,  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  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,  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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