Projective geometries over finite fields:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Clarendon Press
1998
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Oxford mathematical monographs
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 555 S. graph. Darst. |
| ISBN: | 0198502958 |
Internformat
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV011770235 | ||
| 003 | DE-604 | ||
| 005 | 19980805 | ||
| 007 | t | ||
| 008 | 980211s1998 d||| |||| 00||| eng d | ||
| 020 | |a 0198502958 |9 0-19-850295-8 | ||
| 035 | |a (OCoLC)439569654 | ||
| 035 | |a (DE-599)BVBBV011770235 | ||
| 040 | |a DE-604 |b ger |e rakddb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-703 |a DE-91G |a DE-20 |a DE-384 |a DE-355 |a DE-11 |a DE-29T | ||
| 080 | |a 514 | ||
| 084 | |a SK 380 |0 (DE-625)143235: |2 rvk | ||
| 084 | |a MAT 517f |2 stub | ||
| 084 | |a MAT 124f |2 stub | ||
| 100 | 1 | |a Hirschfeld, James W. P. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Projective geometries over finite fields |c J. W. P. Hirschfeld |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Oxford |b Clarendon Press |c 1998 | |
| 300 | |a XIV, 555 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Oxford mathematical monographs | |
| 650 | 0 | 7 | |a Projektive Geometrie |0 (DE-588)4047436-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Galois-Feld |0 (DE-588)4155896-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Endliche Geometrie |0 (DE-588)4014650-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Projektive Geometrie |0 (DE-588)4047436-7 |D s |
| 689 | 0 | 1 | |a Galois-Feld |0 (DE-588)4155896-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Endliche Geometrie |0 (DE-588)4014650-9 |D s |
| 689 | 1 | |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=007942698&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-007942698 | ||
Datensatz im Suchindex
| _version_ | 1804126312714993664 |
|---|---|
| adam_text | CONTENTS
I Introduction
1 Finite fields 3
1.1 Definitions and existence 3
1.2 Automorphisms 4
1.3 Functions 5
1.4 Quadratic equations 7
1.5 Roots of an element 9
1.6 Primitive and subprimitive polynomials 11
1.7 Notation for small finite fields 14
1.8 Invariants of polynomials 15
1.9 Some equations with few terms 18
1.10 Cubic equations 21
1.11 Quartic equations 24
1.12 Notes and references 29
2 Projective spaces and algebraic varieties 31
2.1 Projective space over a field 31
2.1.1 Definition of the space and its automorphisms 31
2.1.2 The Fundamental Theorem of Projective
Geometry 32
2.1.3 The Principle of Duality 33
2.1.4 Coordinate frames 33
2.1.5 Polarities 34
2.2 Affine space over a field 36
2.3 Incidence structures 38
2.4 Canonical forms for projectivities 41
2.5 Subprimitive characteristic polynomials 46
2.6 Orders of projectivities 47
2.7 Varieties 48
2.8 Plane curves 51
2.9 The number of points on an algebraic curve I: the Hasse
Weil theorem 55
2.10 Linear systems of primals 60
2.11 The number of points on an algebraic curve II: the Stohr
Voloch theorem 61
2.12 A generalization of the theorem of Menelaus 64
2.13 Some results from group theory 69
xii CONTENTS
2.14 Links with coding theory 72
2.14.1 Basic definitions 72
2.14.2 Projective systems and the weight hierarchy 73
2.14.3 Algebraic geometry codes 75
2.15 Notes and references 77
II Elementary General Properties
3 Subspaces 85
3.1 Numbers 85
3.2 Characterization of subspaces 87
3.3 Sets of subspaces 89
3.4 Notes and references 91
4 Partitions 93
4.1 Partitions by subspaces 93
4.2 Cyclic projectivities 95
4.3 Partitions by subgeometries 103
4.4 Notes and references 113
5 Canonical forms for varieties and polarities 115
5.1 Quadric and Hermitian varieties 115
5.2 Projective classification of quadrics 121
5.3 Polarities 126
5.4 Notes and references 131
III The Line and the Plane
6 The line 135
6.1 Harmonic tetrads 135
6.2 Quadric and Hermitian varieties 137
6.3 The projective group of the line 139
6.4 The line for small orders 143
6.5 Notes and references 151
7 First properties of the plane 153
7.1 Preliminaries 153
7.2 Conies 155
7.3 Hermitian curves 160
7.4 The projective group of the plane 162
7.5 Perspectivities 168
7.6 Pencils of quadrics 171
7.7 Notes and references 172
CONTENTS xiii
8 Ovals 176
8.1 Arcs 176
8.2 Ovals for fields of odd order 178
8.3 Polarities 180
8.4 Ovals in a plane of even order 184
8.5 Translation ovals 193
8.6 Almost ovals 197
8.7 Notes and references 198
9 Arithmetic of arcs of degree two 201
9.1 The basic equations 201
9.2 Small arcs 206
9.3 Examples of complete arcs 209
9.4 Notes and references 213
10 Arcs in ovals 216
10.1 The connection between arcs and curves 216
10.2 The number of points on an algebraic curve III: further
results 218
10.3 Arcs in ovals in planes of even order 222
10.4 Arcs in ovals in planes of odd order 226
10.5 The second largest complete arc: further results 229
10.6 Arcs and Hermitian curves 233
10.7 Notes and references 234
11 Cubic curves 237
11.1 Configurations 237
11.2 Double points, points of inflexion, and the group law 240
11.3 Nuclei of a cubic in characteristic three 248
11.4 Classification of singular cubics 250
11.4.1 Cubics with a node: type I 250
11.4.2 Cubics with a cusp: type II 252
11.4.3 Cubics with an isolated double point: type III 254
11.5 Properties of non singular cubics 262
11.6 Non singular cubics with nine rational inflexions 267
11.7 Non singular cubics with three rational inflexions 269
11.8 Non singular cubics with one rational inflexion 270
11.9 Non singular cubics with no rational inflexions 287
11.10 Number of rational points on a cubic 289
11.11 Classification of non singular cubics 291
11.12 Notes and references 296
12 Arcs of higher degree 301
12.1 Definitions and basic consequences 301
12.2 Maximal arcs in planes of even order 304
12.3 Arcs with two intersection numbers 305
xiv CONTENTS
12.4 Arcs with small deficiency 312
12.5 Further upper bounds for the size of an arc 320
12.6 Arcs in small planes 321
12.7 Maximal arcs in planes of odd order 323
12.8 Arcs and algebraic curves 326
12.9 Notes and references 331
13 Blocking sets 334
13.1 Definitions and examples 334
13.2 Bounds on the size of blocking sets 338
13.3 Lacunary polynomials and lower bounds 341
13.4 Blocking sets of Redei type 344
13.5 Blocking sets in small planes 345
13.6 Multiple blocking sets 349
13.7 Nuclei 354
13.8 Blocking and multiple blocking sets in affine planes 358
13.9 Notes and references 361
14 Small planes 364
14.1 The plane of order two 364
14.2 The plane of order three 366
14.3 The plane of order four 368
14.4 The plane of order five 372
14.5 The plane of order seven 373
14.6 The plane of order eight 376
14.7 The plane of order nine 379
14.8 The plane of order eleven 385
14.9 The plane of order thirteen 386
14.10 Notes and references 387
Appendix I. Orders of and isomorphisms among the
semi linear groups 390
AI.l Definitions 390
AI.2 Isomorphisms proved in the text 391
AI.3 Table of group orders 391
AI.4 Notes and references 394
Appendix II. Errata for General Galois Geometries 395
Notation 397
References 404
Index 547
|
| any_adam_object | 1 |
| author | Hirschfeld, James W. P. |
| author_facet | Hirschfeld, James W. P. |
| author_role | aut |
| author_sort | Hirschfeld, James W. P. |
| author_variant | j w p h jwp jwph |
| building | Verbundindex |
| bvnumber | BV011770235 |
| classification_rvk | SK 380 |
| classification_tum | MAT 517f MAT 124f |
| ctrlnum | (OCoLC)439569654 (DE-599)BVBBV011770235 |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01627nam a2200433 c 4500</leader><controlfield tag="001">BV011770235</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">19980805 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">980211s1998 d||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">0198502958</subfield><subfield code="9">0-19-850295-8</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)439569654</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV011770235</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakddb</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-91G</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-384</subfield><subfield code="a">DE-355</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-29T</subfield></datafield><datafield tag="080" ind1=" " ind2=" "><subfield code="a">514</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 380</subfield><subfield code="0">(DE-625)143235:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 517f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 124f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Hirschfeld, James W. P.</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Projective geometries over finite fields</subfield><subfield code="c">J. W. P. Hirschfeld</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">2. ed.</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Oxford</subfield><subfield code="b">Clarendon Press</subfield><subfield code="c">1998</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XIV, 555 S.</subfield><subfield code="b">graph. Darst.</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="490" ind1="0" ind2=" "><subfield code="a">Oxford mathematical monographs</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Projektive Geometrie</subfield><subfield code="0">(DE-588)4047436-7</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Galois-Feld</subfield><subfield code="0">(DE-588)4155896-0</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Endliche Geometrie</subfield><subfield code="0">(DE-588)4014650-9</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Projektive Geometrie</subfield><subfield code="0">(DE-588)4047436-7</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Galois-Feld</subfield><subfield code="0">(DE-588)4155896-0</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="689" ind1="1" ind2="0"><subfield code="a">Endliche Geometrie</subfield><subfield code="0">(DE-588)4014650-9</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" 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=007942698&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:BVB01-007942698</subfield></datafield></record></collection> |
| id | DE-604.BV011770235 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:15:27Z |
| institution | BVB |
| isbn | 0198502958 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007942698 |
| oclc_num | 439569654 |
| open_access_boolean | |
| owner | DE-703 DE-91G DE-BY-TUM DE-20 DE-384 DE-355 DE-BY-UBR DE-11 DE-29T |
| owner_facet | DE-703 DE-91G DE-BY-TUM DE-20 DE-384 DE-355 DE-BY-UBR DE-11 DE-29T |
| physical | XIV, 555 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Clarendon Press |
| record_format | marc |
| series2 | Oxford mathematical monographs |
| spelling | Hirschfeld, James W. P. Verfasser aut Projective geometries over finite fields J. W. P. Hirschfeld 2. ed. Oxford Clarendon Press 1998 XIV, 555 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford mathematical monographs Projektive Geometrie (DE-588)4047436-7 gnd rswk-swf Galois-Feld (DE-588)4155896-0 gnd rswk-swf Endliche Geometrie (DE-588)4014650-9 gnd rswk-swf Projektive Geometrie (DE-588)4047436-7 s Galois-Feld (DE-588)4155896-0 s DE-604 Endliche Geometrie (DE-588)4014650-9 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007942698&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hirschfeld, James W. P. Projective geometries over finite fields Projektive Geometrie (DE-588)4047436-7 gnd Galois-Feld (DE-588)4155896-0 gnd Endliche Geometrie (DE-588)4014650-9 gnd |
| subject_GND | (DE-588)4047436-7 (DE-588)4155896-0 (DE-588)4014650-9 |
| title | Projective geometries over finite fields |
| title_auth | Projective geometries over finite fields |
| title_exact_search | Projective geometries over finite fields |
| title_full | Projective geometries over finite fields J. W. P. Hirschfeld |
| title_fullStr | Projective geometries over finite fields J. W. P. Hirschfeld |
| title_full_unstemmed | Projective geometries over finite fields J. W. P. Hirschfeld |
| title_short | Projective geometries over finite fields |
| title_sort | projective geometries over finite fields |
| topic | Projektive Geometrie (DE-588)4047436-7 gnd Galois-Feld (DE-588)4155896-0 gnd Endliche Geometrie (DE-588)4014650-9 gnd |
| topic_facet | Projektive Geometrie Galois-Feld Endliche Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007942698&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT hirschfeldjameswp projectivegeometriesoverfinitefields |