Fundamental problems of algorithmic algebra:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Oxford Univ. Press
2000
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 511 S. |
| ISBN: | 0195125169 |
Staff View
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV013174401 | ||
| 003 | DE-604 | ||
| 005 | 20060803 | ||
| 007 | t | ||
| 008 | 000525s2000 |||| 00||| eng d | ||
| 020 | |a 0195125169 |9 0-19-512516-9 | ||
| 035 | |a (OCoLC)247815438 | ||
| 035 | |a (DE-599)BVBBV013174401 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-703 |a DE-739 |a DE-91G |a DE-634 |a DE-11 |a DE-83 | ||
| 050 | 0 | |a QA155.7.E4 | |
| 082 | 0 | |a 512/.0285 |2 21 | |
| 084 | |a SK 200 |0 (DE-625)143223: |2 rvk | ||
| 084 | |a ST 600 |0 (DE-625)143681: |2 rvk | ||
| 084 | |a 13P10 |2 msc | ||
| 084 | |a 12Y05 |2 msc | ||
| 084 | |a 68W30 |2 msc | ||
| 084 | |a DAT 702f |2 stub | ||
| 100 | 1 | |a Yap, Chee-Keng |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Fundamental problems of algorithmic algebra |c Chee Keng Yap |
| 264 | 1 | |a New York [u.a.] |b Oxford Univ. Press |c 2000 | |
| 300 | |a XV, 511 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Algebra |2 gtt | |
| 650 | 4 | |a Algorithms | |
| 650 | 7 | |a Algoritmen |2 gtt | |
| 650 | 7 | |a Álgebra computacional |2 larpcal | |
| 650 | 7 | |a Álgebra |2 larpcal | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Algebra |x Data processing | |
| 650 | 0 | 7 | |a Computeralgebra |0 (DE-588)4010449-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Computeralgebra |0 (DE-588)4010449-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008975943&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-008975943 | ||
Record in the Search Index
| _version_ | 1804127874433679360 |
|---|---|
| adam_text | CONTENTS
Preface xiii
Current Textbooks in Algorithmic Algebra xv
0 Introduction 1
0.1 FUNDAMENTAL PROBLEM OF ALGEBRA 2
0.2 FUNDAMENTAL PROBLEM OF CLASSICAL ALGEBRAIC
GEOMETRY 4
0.3 FUNDAMENTAL PROBLEM OF IDEAL THEORY 6
0.4 REPRESENTATION AND SIZE 9
0.5 COMPUTATIONAL MODELS 10
0.6 ASYMPTOTIC NOTATIONS 13
0.7 COMPLEXITY OF MULTIPLICATION 15
0.8 ON BIT VERSUS ALGEBRAIC COMPLEXITY 18
0.9 MISCELLANY 20
0.10 COMPUTER ALGEBRA SYSTEMS 26
1 Arithmetic 27
1.1 THE DISCRETE FOURIER TRANSFORM 28
1.2 POLYNOMIAL MULTIPLICATION 32
1.3 MODULAR FAST FOURIER TRANSFORM 34
1.4 FAST INTEGER MULTIPLICATION 37
1.5 MATRIX MULTIPLICATION 41
2 The Greatest Common Denominator 43
2.1 UNIQUE FACTORIZATION DOMAIN 44
2.2 EUCLID S ALGORITHM 47
2.3 EUCLIDEAN RING 50
2.4 THE HALF GREATEST COMMON DENOMINATOR
PROBLEM 54
2.5 PROPERTIES OF THE NORM 57
2.6 POLYNOMIAL HALF GCD 61
APPENDIX A: Integer Half GCD 67
3 Sub resultants 77
3.1 PRIMITIVE FACTORIZATION 78
3.2 PSEUDOREMAINDERS AND POLYNOMIAL REMAINDER
SEQUENCE 82
3.3 DETERMINANTAL POLYNOMIALS 84
3.4 POLYNOMIAL PSEUDOQUOTIENT 87
3.5 THE SUBRESULTANT POLYNOMIAL REMAINDER
SEQUENCE 89
3.6 SUBRESULTANTS 90
3.7 PSEUDOSUBRESULTANTS 92
3.8 SUBRESULTANT THEOREM 97
3.9 CORRECTNESS OF THE SUBRESULTANT POLYNOMIAL
REMAINDER SEQUENCE ALGORITHM 101
4 Modular Techniques 104
4.1 CHINESE REMAINDER THEOREM 104
4.2 EVALUATION AND INTERPOLATION 107
4.3 FINDING PRIME MODULI 111
4.4 LUCKY HOMOMORPHISMS FOR THE GCD 113
4.5 COEFFICIENT BOUNDS FOR FACTORS 116
4.6 A MODULAR GREATEST COMMON DENOMINATOR
ALGORITHM 120
4.7 WHAT ELSE IN GCD COMPUTATION? 122
5 Fundamental Theorem of Algebra 124
5.1 ELEMENTS OF FIELD THEORY 125
5.2 ORDERED RINGS 129
5.3 FORMALLY REAL RINGS 130
5.4 CONSTRUCTIBLE EXTENSIONS 132
5.5 REAL CLOSED FIELDS 135
5.6 FUNDAMENTAL THEOREM OF ALGEBRA 138
CONTENTS ix
6 Roots of Polynomials 141
6.1 ELEMENTARY PROPERTIES OF POLYNOMIAL ROOTS 142
6.2 ROOT BOUNDS 147
6.3 ALGEBRAIC NUMBERS 151
6.4 RESULTANTS 155
6.5 SYMMETRIC FUNCTIONS 161
6.6 DISCRIMINANT 167
6.7 ROOT SEPARATION 169
6.8 A GENERALIZED HADAMARD BOUND 173
6.9 ISOLATING INTERVALS 178
6.10 ON NEWTON S METHOD 180
6.11 GUARANTEED CONVERGENCE OF NEWTON
ITERATION 182
7 Sturm Theory 186
7.1 STURM SEQUENCES FROM POLYNOMIAL REMAINDER
SEQUENCES 187
7.2 A GENERALIZED STURM THEOREM 190
7.3 COROLLARIES AND APPLICATIONS 195
7.4 INTEGER AND COMPLEX ROOTS 201
7.5 THE ROUTH HURWITZ THEOREM 204
7.6 SIGN ENCODING OF ALGEBRAIC NUMBERS: THOM S
LEMMA 209
7.7 PROBLEM OF RELATIVE SIGN CONDITIONS 211
7.8 THE BKR ALGORITHM 214
8 Gaussian Lattice Reduction 219
8.1 LATTICES 219
8.2 SHORTEST VECTORS IN PLANAR LATTICES 224
8.3 COHERENT REMAINDER SEQUENCES 227
9 Lattice Reduction and Applications 234
9.1 GRAM SCHMIDT ORTHOGONALIZATION 235
9.2 MINKOWSKI S CONVEX BODY THEOREM 239
9.3 WEAKLY REDUCED BASES 242
9.4 REDUCED BASES AND THE LLL ALGORITHM 243
9.5 SHORT VECTORS 247
9.6 FACTORIZATION VIA RECONSTRUCTION OF MINIMAL POLY¬
NOMIALS 251
10 Linear Systems 258
10.1 SYLVESTER S IDENTITY 259
10.2 FRACTION FREE DETERMINANT COMPUTATION 262
10.3 MATRIX INVERSION 269
10.4 HERMITE NORMAL FORM 271
10.5 A MULTIPLE GCD BOUND AND ALGORITHM 275
10.6 HERMITE REDUCTION STEP 280
10.7 BACHEM KANNAN ALGORITHM 286
10.8 SMITH NORMAL FORM 292
10.9 FURTHER APPLICATIONS 296
11 Elimination Theory 300
11.1 HILBERT BASIS THEOREM 301
11.2 HILBERT NULLSTELLENSATZ 304
11.3 SPECIALIZATIONS 308
11.4 RESULTANT SYSTEMS 313
11.5 SYLVESTER RESULTANT REVISITED 318
11.6 INERTIAL IDEAL 321
11.7 THE MACAULAY RESULTANT 327
11.8 [/ RESULTANT 334
11.9 GENERALIZED CHARACTERISTIC POLYNOMIAL 337
11.10 GENERALIZED [/ RESULTANT 342
11.11 A MULTIVARIATE ROOT BOUND 350
APPENDIX A: Power Series 355
APPENDIX B: Counting Irreducible Polynomias 359
12 Grobner Bases 363
12.1 ADMISSIBLE ORDERINGS 364
12.2 NORMAL FORM ALGORITHM 372
12.3 CHARACTERIZATIONS OF GROBNER BASES 378
12.4 BUCHBERGER S ALGORITHM 382
12.5 UNIQUENESS 384
12.6 ELIMINATION PROPERTIES 386
12.7 COMPUTING IN QUOTIENT RINGS 391
CONTENTS xi
12.8 SYZYGIES 393
13 Bounds in Polynomial Ideal Theory 398
13.1 SOME BOUNDS IN POLYNOMIAL IDEAL THEORY 399
13.2 THE HILBERT SERRE THEOREM 401
13.3 HOMOGENEOUS SETS 407
13.4 CONE DECOMPOSITION 412
13.5 EXACT DECOMPOSITION OF NF(/) 416
13.6 EXACT DECOMPOSITION OF IDEALS 423
13.7 BOUNDING THE MACAULAY CONSTANTS 424
13.8 TERM REWRITING SYSTEMS 428
13.9 A QUADRATIC COUNTER 432
13.10 UNIQUENESS PROPERTY 436
13.11 LOWER BOUNDS 438
APPENDIX A: Properties of 50 442
14 Continued Fractions 446
14.1 INTRODUCTION 447
14.2 EXTENDED NUMBERS 449
14.3 GENERAL TERMINOLOGY 451
14.4 ORDINARY CONTINUED FRACTIONS 455
14.5 CONTINUED FRACTIONS AS MOBIUS TRANSFORMATIONS
460
14.6 CONVERGENCE PROPERTIES 465
14.7 REAL MOBIUS TRANSFORMATIONS 470
14.8 CONTINUED FRACTIONS OF ROOTS 474
14.9 ARITHMETIC OPERATIONS 478
References 485
Index 495
Index to Symbols 508
|
| any_adam_object | 1 |
| author | Yap, Chee-Keng |
| author_facet | Yap, Chee-Keng |
| author_role | aut |
| author_sort | Yap, Chee-Keng |
| author_variant | c k y cky |
| building | Verbundindex |
| bvnumber | BV013174401 |
| callnumber-first | Q - Science |
| callnumber-label | QA155 |
| callnumber-raw | QA155.7.E4 |
| callnumber-search | QA155.7.E4 |
| callnumber-sort | QA 3155.7 E4 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 200 ST 600 |
| classification_tum | DAT 702f |
| ctrlnum | (OCoLC)247815438 (DE-599)BVBBV013174401 |
| dewey-full | 512/.0285 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.0285 |
| dewey-search | 512/.0285 |
| dewey-sort | 3512 3285 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01638nam a2200481 c 4500</leader><controlfield tag="001">BV013174401</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20060803 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">000525s2000 |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">0195125169</subfield><subfield code="9">0-19-512516-9</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)247815438</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV013174401</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-703</subfield><subfield code="a">DE-739</subfield><subfield code="a">DE-91G</subfield><subfield code="a">DE-634</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-83</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA155.7.E4</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">512/.0285</subfield><subfield code="2">21</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 200</subfield><subfield code="0">(DE-625)143223:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">ST 600</subfield><subfield code="0">(DE-625)143681:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">13P10</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">12Y05</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">68W30</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">DAT 702f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Yap, Chee-Keng</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Fundamental problems of algorithmic algebra</subfield><subfield code="c">Chee Keng Yap</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">New York [u.a.]</subfield><subfield code="b">Oxford Univ. Press</subfield><subfield code="c">2000</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XV, 511 S.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Algebra</subfield><subfield code="2">gtt</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Algorithms</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Algoritmen</subfield><subfield code="2">gtt</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Álgebra computacional</subfield><subfield code="2">larpcal</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Álgebra</subfield><subfield code="2">larpcal</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Datenverarbeitung</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Algebra</subfield><subfield code="x">Data processing</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Computeralgebra</subfield><subfield code="0">(DE-588)4010449-7</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Computeralgebra</subfield><subfield code="0">(DE-588)4010449-7</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HBZ Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008975943&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-008975943</subfield></datafield></record></collection> |
| id | DE-604.BV013174401 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:40:17Z |
| institution | BVB |
| isbn | 0195125169 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008975943 |
| oclc_num | 247815438 |
| open_access_boolean | |
| owner | DE-703 DE-739 DE-91G DE-BY-TUM DE-634 DE-11 DE-83 |
| owner_facet | DE-703 DE-739 DE-91G DE-BY-TUM DE-634 DE-11 DE-83 |
| physical | XV, 511 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| spelling | Yap, Chee-Keng Verfasser aut Fundamental problems of algorithmic algebra Chee Keng Yap New York [u.a.] Oxford Univ. Press 2000 XV, 511 S. txt rdacontent n rdamedia nc rdacarrier Algebra gtt Algorithms Algoritmen gtt Álgebra computacional larpcal Álgebra larpcal Datenverarbeitung Algebra Data processing Computeralgebra (DE-588)4010449-7 gnd rswk-swf Computeralgebra (DE-588)4010449-7 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008975943&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Yap, Chee-Keng Fundamental problems of algorithmic algebra Algebra gtt Algorithms Algoritmen gtt Álgebra computacional larpcal Álgebra larpcal Datenverarbeitung Algebra Data processing Computeralgebra (DE-588)4010449-7 gnd |
| subject_GND | (DE-588)4010449-7 |
| title | Fundamental problems of algorithmic algebra |
| title_auth | Fundamental problems of algorithmic algebra |
| title_exact_search | Fundamental problems of algorithmic algebra |
| title_full | Fundamental problems of algorithmic algebra Chee Keng Yap |
| title_fullStr | Fundamental problems of algorithmic algebra Chee Keng Yap |
| title_full_unstemmed | Fundamental problems of algorithmic algebra Chee Keng Yap |
| title_short | Fundamental problems of algorithmic algebra |
| title_sort | fundamental problems of algorithmic algebra |
| topic | Algebra gtt Algorithms Algoritmen gtt Álgebra computacional larpcal Álgebra larpcal Datenverarbeitung Algebra Data processing Computeralgebra (DE-588)4010449-7 gnd |
| topic_facet | Algebra Algorithms Algoritmen Álgebra computacional Álgebra Datenverarbeitung Algebra Data processing Computeralgebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008975943&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT yapcheekeng fundamentalproblemsofalgorithmicalgebra |