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Hirschfield J.W. Ч Projective Geometries over Finite Fields
Hirschfield J.W. Ч Projective Geometries over Finite Fields

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Ќазвание: Projective Geometries over Finite Fields

јвтор: Hirschfield J.W.


This book is an account of the combinatorics of projective spaces over a finite field, with special emphasis on one and two dimensions. With its successor volumes, Finite projective spaces over three dimensions (1985), which is devoted to three dimensions, and General Galois geometries
(1991), on a general dimension, it provides the only comprehensive treatise on this area of mathematics. The area is interesting in itself, but is important for its applications to coding theory and statistics, and its use of group theory, algebraic geometry, and number theory. This new edition is
a complete reworking, containing extensive revisions, particularly in the chapters on generalities, the geometry of arcs in ovals, the geometry of arcs of higher degree, and blocking sets. Part I gives a survey of finite fields and an outline of the fundamental properties of projective spaces and
their automorphisms; it includes the properties of algebraic varieties and curves used throughout the book and in the companion volumes. Part II covers, in an arbitrary dimension, the properties of subspaces, of partitions into both subspaces and subgeometries, and of quadrics and Hermitian
varieties, as well as polarities. Part III is a detailed account of the line and plane; with little reference to the generalities from Parts I and II, the author revisits fundamental properties of the plane and then describes the structure of arcs and their relation to curves. This part includes
chapters on blocking sets and on small planes (those with orders up to thirteen). With a comprehensive bibliography containing over 3,000 items, this volume will prove invaluable to researchers in finite geometry, coding theory and combinatorics. Ч This text refers to the

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1980

 оличество страниц: 488

ƒобавлена в каталог: 22.05.2008

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ѕредметный указатель
(k,l)-set      70Ч71
(k,l)-span      70
(k,l; r, s; n, q)-set      70
(k,r; n,q)-set      70
(k; r,s; n, q)-set      70
(k;n)-arc      320Ч364
(k;n)-arc of type $(m_1, \cdots, m_r, n)$      321
(k;n)-arc of type m      321Ч322
(k;n)-arc, complete      373
(k;n)-arc, deficiency of a      345Ч355
(k;n)-arc, i-secant of a      320Ч321
(k;n)-arc, maximal      324Ч335
(k;n)-arc, of type (0, 1, n)      342Ч343
(k;n)-arc, of type (m,n)      335Ч345
(k;n)-arc, t-maximal      335 338Ч341
(k;n)-arc, t-minimal      338Ч341
(k;r)-arc      70
(k;r)-cap      70
a-arc      186Ч199 235
Absolutely irreducible primal      50
Absolutely irreducible primal, variety      51
Acnode      223
Affine space      37 332Ч335
Affine space, axioms for      38Ч40
Arc, Herraitian      329 337Ч341 343Ч345
Arc, maximal      324Ч335
Associated point      128 132Ч133
Automorphism of a field      1 4
Automorphism of a space      30
Axioms for affine space      38Ч40
Axioms for projective space      38Ч39
B-point      393Ч395 397Ч404 408Ч414
Base point      55
Basis for a coordinate system      32
Bezout's theorem      223
Bisecant      70
Bisecant of a k-arc      165
Block      57
Blocking set      366Ч385
Blocking set, k-arc derived      366Ч367
Blocking set, minimal      366Ч368 373
Brianchon hexagon      393
Brianchon hexagon, point      393Ч395 397Ч404 408Ч414
Camot's theorem      57Ч58
Canonical form for a Hermitian variety      98Ч99
Canonical form for a polarity      110Ч115
Canonical form for a projectivity      40Ч46 124Ч128 149Ч157
Canonical form for a quadric      98Ч102 104Ч107
Category of a polyomial      13 19Ч20
Category of an element      4 13
Ceva's theorem      60
Character of a field      19
Characteristic of a field      1
Chord of a k-set      70Ч71
Class of a cubic curve      252
Collineation      30
Collineation, group      40Ч41 244 419Ч424
Committee      366 373
Companion matrix      44
Complete k-arc      163 232Ч239
Complete K-set      70
Complete series      54
Complex conjugate points      50
Complex inflexion      249 271
Complex point      50
Component of a curve      221
Component of a curve, irregular      221
Component of a curve, linear      221
Component of a curve, regular      221
Component of a primal      50
Component of a variety      51
Cone      51
Configuration, collineation group of a      244
Configuration, permutation group of a      244
Configuration, projective group of a      244
conic      107 140Ч146 235Ч239
Conic, nucleus of a      143
Conic, projective group of a      143Ч144 156Ч157
Conjugacy classes in GL(2,q)      124Ч125
Conjugacy classes in GL(3,q)      149Ч152
Conjugacy classes in PGL(2tq)      124Ч128
Conjugacy classes in PGL(5,q)      149Ч157
Conjugacy classes, number of      42 61Ч62
Conjugate points      33
Conjugate points, primes      33
Coordinate of a line      166
Coordinate system      31Ч32
Coordinate, non-homogeneous      54 119
Coordinates, prime      32Ч33
Coordinates, tangential      32Ч33
Correlation      33
Cross-axis of a projectivity      161
Cross-ratio      22Ч23 119Ч120
Crunode      223
Cubic curve with an isolated double point      226 259Ч268
Cubic curve with nine inflexions      278Ч280
Cubic curve with no inflexions      308Ч311
Cubic curve with one inflexion      282Ч307
Cubic curve with three inflexions      280Ч282
Cubic curve, equations      13 16 18Ч22
Cubic curves      225Ч226 241Ч318 357Ч358 361Ч363
Cubic curves equianharmonic      253 277
Cubic curves general      254 277
Cubic curves harmonic      253 277
Cubic curves nodal      226 254Ч256 267Ч268
Cubic curves non-singular      270Ч318
Cubic curves nucleus of      241
Cubic curves number of points on      311Ч313
Cubic curves singular      254Ч268
Cubic curves superharmonic      253
Cubic curves, classification of non-singular      313Ч318
Cubic curves, classification of singular      266Ч268
Cubic curves, cuspidal      226 256Ч259 267Ч268
Curve      49 51
Curve, genus of a      226Ч227
Curve, Hermitian      146Ч148 227 243
Curve, sub-Hermitian      148Ч149 227
Curves, cubic      225Ч226 241Ч318 361Ч363
Cusp      223
Cyclic projectivity      74Ч96 155
d-secant      70
Degree of a primal      49
Degree of of a variety      51
Desargues's theox-em      160Ч161
Design      37
Diagonal points of a pentastigm      139Ч140
Diagonal points of a tetrastigm      138
Diagonal points of an n-stigm      139Ч140
Dickson's criterion      3
Dimension of a projective space      29
Dimension of a variety      51
Dimension of a vector space      29
Dimension of an affine space      37
Discriminant      3 10
Dual space      31
Duality, principle of      31
Elation      153
Elementary divisor      43Ч46
Elementary divisor, operation      43
Elliptic quadric      106Ч111 114Ч115
Equations, cubic      13 16 18Ч22
Equations, equianharmonic quartic      22 25Ч27
Equations, harmonic quartic      22
Equations, quadratic      3 13
Equations, quartic      13 16Ч18 22Ч27
Equianharmonic quartic equations      22 25Ч27
Equianharmonic tetrad      121Ч122 130Ч131
Equivalence of matrices      43
Extended symbol of a linear transformation      45Ч46
Extended symbol of a projectivity      45Ч46
External line of a k-arc      163
External line of an oval (conic)      165Ч166 170Ч171
External line, point of a k-arc      201
Field      1
finite      1
Finite fields      1Ч27
Finite fields, addition tables for      425Ч429
First polar      245Ч246
FORM      49
Fundamental theorem of projective geometry      30Ч31
General linear group      41 419Ч424
Generalized quadrangle      37
Grade of a point      164
Group, collineation      40Ч41 244 419Ч424
Group, general linear      41 124Ч125 149Ч152 419Ч424
Group, isomorphisms      422Ч424
Group, order of a semi-linear      420Ч421
Group, permutation      244
Group, projective      40 124Ч128 149Ч157 244 419Ч424
Group, projective, orthogonal      110 143Ч144 156Ч157 419Ч424
Group, projective, pseudo symplectic      113 174 419Ч424
Group, projective, special linear      41 419Ч424
Group, projective, symplectic      113 419Ч424
Group, projective, unitary      110 147 419Ч424
Group, special linear      41 419Ч424
Harmonic      152
Harmonic homology      152
Harmonic homology, polar      248
Harmonic homology, quartic equation      22
Harmonic homology, separation      120
Harmonic homology, syntheme      129Ч130
Harmonic homology, tetrad      120Ч121 129Ч131 133Ч135 169Ч170
Harmonic homology, tetrad, negative      134Ч135
Harmonic homology, tetrad, positive      134Ч135
Hasse Ч Weil theorem      225Ч232
Hermitian arc      329 337Ч341 343Ч345
Hermitian curve      146Ч148 228Ч229 243
Hermitian polarity      35Ч36 110 147Ч148
Hermitian variety      97Ч100 102Ч103 110Ч111 115
Hermitian variety in the plane      146Ч148
Hermitian variety on a line      122Ч123
Hermitian variety, canonical form for a      98Ч99
Hermitian variety, degenerate      97
Hermitian variety, group of a      110
Hermitian variety, number of points on a      102Ч103
Hessian of a cubic curve      246
Hessian polynomial      18Ч22
Hessian, indeterminate      248
Hexagon      55
Hexagon, Brianchon      393
Hexagon, Pascal      394
Hexastigm      55
Hexastigms, number of, not on a conic      142Ч143
Homology      152
Hyperbolic quadric      106Ч111 114Ч115
Hypercompanion matrix      44
Hyperplane      29
Hypersurface      49
Ideal line      38 74
Ideal point      38 74
Imaginary point      50
Incidence of subspaces      29
Incidence structure      37Ч40 74
Index of a k-arc      201 205Ч206
Index of a point      164
Inflexion      222
Inflexion of a cubic      246Ч250
Inflexional line      277
Inflexional tangent      246
Inflexional triangle      271 277
Internal point of a conic      165Ч166 170Ч171
Internal point of a k-arc      201
Intersection cycle      253
Intersection multiplicity      52 223
Intersection of subspaces      30
Invariant factor      43Ч44 46
Invariant, absolute      278 308
Invariants of polynomials      10Ч27
Involution      126Ч128
Involution, elliptic      126Ч128
Involution, hyperbolic      126Ч128
Involution, improper      126Ч128
Involution, parabolic      126Ч128
Involution, proper      126Ч128
Irreducible primal      50
Irreducible variety      51
Isolated double point      222
Isomorphisms of groups      422Ч424
Join      30
k-arc      54Ч55 70 163Ч219 387Ч414
k-arc, bisecant of a      163
k-arc, complete      163 205Ч208 214Ч219 232Ч239
k-arc, external line of a      163
k-arc, external line of a, point of a      201
k-arc, grade of a point off a      164
k-arc, index of a      201 205Ч206
k-arc, index of a, of a point off a      164
k-arc, internal point of a      201
k-arc, rank of a      201 205Ч206
k-arc, rank of a, of a point off a      204
k-arc, symmetric      207
k-arc, uniform      212Ч214
k-arc, unisecant of a      163 166Ч168
k-arcs, number of      137
k-cap      70
k-fold spread      81
k-set      69Ч71
k-set, chord of a      70Ч71
k-set, complete      70
k-set, tangent to a      70
Lagrange's interpolation formula      2
Line      29 38 119Ч135
Line, Hermitian variety on a      122Ч123
Line, ideal      38 74
Line, inflexional      277
Line, polarity on a      123
Line, projectivity on a      119 124Ч128
Line, proper      74
Line, quadric on a      122
Line, tangent      52
Linear independence      29
Linear series      54
Linear subsystem      53
Linear system      53Ч60
Linear transformation, extended symbol of a      45Ч46
Linear transformation, order of a      48Ч49
Linear transformation, symbol of a      45Ч46
Lines, parallel      38 332Ч335
Meet of subspaces      30
Menelaus's theorem      57Ч58
Model point      225
Multiplicity of a point      52 222
n-arc      54Ч55
n-curve      359Ч364
n-curve, complete      360
n-curve, multiple point on an      359Ч361
n-curve, multiplicity of a point on an      359
n-curve, simple point on an      359
N-gram      54Ч55
n-point      54Ч55
n-side      54Ч55
n-stigm      54Ч55
n-stigms, number of      137
Net      53
Node      222
Non-Desarguesian plane      74
Non-homogeneous coordinate      54 119
Nucleus of a (q+1)-arc      164Ч165
Nucleus of a conic      143
1 2
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