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Bliss G.A. — Algebraic Functions
Bliss G.A. — Algebraic Functions



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Название: Algebraic Functions

Автор: Bliss G.A.

Аннотация:

This text surveys approaches and basic results of all three classes of algebraic functions: transcendental (function theoretic), algebraic-geometric, and arithmetic. Worked examples include both exercises and explanations of methods


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 240

Добавлена в каталог: 03.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel      iii iv 94 119
Abelian integrals, classification of      64 95
Abelian integrals, definition      64 93
Abelian integrals, for algebraic equations of the third degree in y      208ff (see also Elementary integrals)
Abelian integrals, for elliptic and hyperelliptic curves      191ff
Abelian integrals, inversion of      169ff
Abelian integrals, normal integrals      110
Abelian integrals, periods      96ff 105
Abelian integrals, singularities      94
Abel’s theorem      119ff
Abel’s Theorem, elementary applications of      121
Abel’s Theorem, for elementary integrals      129
Abel’s Theorem, for integrals of the first kind      116
Adjoined polynomials      153
Algebraic function, branch of      32
Algebraic function, continuations of      25
Algebraic function, defined by equation of the third or fourth degree in y      194ff
Algebraic function, definition of      24
Algebraic function, discriminant      24
Algebraic function, expansions      29 35
Algebraic function, genus of      58 90 147
Algebraic function, having genus $p=0$ or $p=1$      139
Algebraic function, hyperelliptic and elliptic cases      141 143
Algebraic function, integral algebraic function      195
Analytic functions      1
Arcs, regular      2
Arcs, regular arc on a Riemann surface      80
Arcs, simply closed      2
Barnum      123
Bases, complementary bases      59ff 200
Bases, for a divisor      48
Bases, for all rational functions      46
Bases, for special examples      196ff
Bases, for the divisor $Q=1$ and an equation of the third degree in y      196ff
Bases, normal at $x=\infty$      52
Bases, normal for the divisor $Q=1$ and an equation of the third degree in y      202
Bases, transformation of      51
Bauer      v 197 211 213
Bauer, his methods for equations of the third degree in y      194ff
Bauer, theorem of      204
Birational transformations      133ff
Birational transformations, for curves of the third degree in y      203
Birational transformations, in the function- theoretic plane      167
Birational transformations, projective      144
Birational transformations, reducing singularities to double points in the projective plane      163
Birational transformations, reducing singularities to multiple points with distinct tangents      156
Branch cycle      32 56
Branch cycle, in the function-theoretic plane      166
Branch cycle, linear branch in the projective plane      160
Branch cycle, of a rational function      63
Branch cycle, primitive branch      32 159 166
Branch of an algebraic function      32
Brill      iii
Canonical system of cuts      90
Canonical system of cuts, for elliptic or hyperelliptic curves      190
Cauchy      iv 2 7 84
Cauchy’s integral formula      8
Cauchy’s Theorem      3
Cauchy’s theorem, consequences of      5 7
Cauchy’s theorem, on a Riemann surface      84 92
Clebsch      iii
Column order of a rational function at $x=\infty$      52
Complementary bases      59ff 200
Connected regions      5 26
Connectivity of surfaces      82 87
Connectivity of surfaces, of a closed surface      88
Connectivity of surfaces, of a Riemann surface      86ff
Continuations for an algebraic function      25
Cremona      iv 156
Cremona transformations      133
Cremona transformations, reducing singularities to multiple points with dis-tinct tangents      156
Curves, in the function-theoretic plane with double points only      165
Curves, in the projective plane with double points only      159
Curves, of elliptic and hyperelliptic types      189
Curves, of genus not greater than 2      139
Curves, of the fourth degree in x and y      210
Curves, of the third degree in y      194ff
Cusps      41
Cuts      82
Cuts, canonical systems      90
Cuts, cross and loop cuts      82
Cycles for an algebraic function      32
Cycles for an algebraic function, branch cycles      56
Dedekind      iii iv
Determinant $\Delta$      48
Discriminant of a polynomial      18
Divisor      47
Divisor, basis for a divisor      48
Divisor, ideal norm of      55
Divisor, multiples of      51 54
Divisor, of a rational function      62
Divisor, of branch cycles      56 63
Divisor, order of a divisor      62
Divisor, special divisor      71
Elementary integrals      64 102
Elementary integrals, for curves of elliptic or hyperelliptic types      191-2
Elementary integrals, for equations of the third degree in y      205 209
Elementary integrals, of first kind      65 98 151
Elementary integrals, of second kind      66 102
Elementary integrals, of third kind      66 102
Elementary integrals, period relations      106
Elliptic curves      139
Elliptic functions, analogies with rational functions on a Riemann surface      174ff
Elliptic functions, further properties      177
Elliptic integrals      94
Elliptic integrals, addition formulas      123
examples      183ff
Examples, algebraic functions defined by equations of the third or fourth degree in y      194ff 211 213
Examples, elliptic and hyperelliptic curves      189
Examples, for expansions      183
Examples, for Riemann surfaces      186
Expansion theorems      17 25 29 32
Expansion theorems, at singular points of special types      40
Expansion theorems, Laurent expansion      9
Expansion theorems, Newton’s polygon method      35
Fields      iii
Function-theoretic plane      165
Functions, algebraic      24
Functions, analytic      1
Functions, holomorphic on a plane      1
Functions, inverse      171ff
Functions, on a Riemann surface      80
Functions, rational      12ff 43ff 70
Functions, special rational      71
Fundamental systems of integrals, construction of      107
Fundamental systems of integrals, for curves of elliptic or hyperelliptic type      193-4
Fundamental systems of integrals, of the first and second kinds      103
Fundamental systems of integrals, of the first kind      98
Fundamental theorem of algebra      15
Genus, invariant property      62
Genus, of a Riemann surface      88
Genus, of an algebraic function      58 90 147
Genus, of an equation of the third degree in y      207
Genus, of elliptic or hyperelliptic curves      191
Gordan      iii
Halphen      155
Hensel      iii iv v 156
Holomorphic functions, on a Riemann surface      80
Holomorphic functions, on a z-plane      1
Holomorphic functions, together forming an algebraic function      28
Hyperelliptic curves      139 189
Ideal norm of a divisor      55
Integral algebraic function      195
Integrals, associated with an algebraic function      64 93 Elliptic
Integrals, of a function f(z)      2
Inverse function, when $p=0$      171
Inverse function, when $p=1$      172
Inversion of Abelian integrals      169ff
Irreducible algebraic equations      78
jacobi      176
Klein      155
kronecker      iii v 155-6
Landsberg      iii iv v 156
Laurent expansion      9
Laurent expansion, principal part of      13 15
Leading coefficients of a rational function      74
Logarithm of a rational function      114
Multiple points with distinct tangents      40
Multiples of a divisor      47 48 51 54 72
Newton      35
Newton’s polygons      36
Noether      iii 155
Noether, Emmy      iii
Norm, ideal norm of a divisor      55
Norm, of a rational function      44
Normal basis for a divisor      52
Normal form of equation of third degree in y      194
Normal integrals      110
Ordinary point      9 11
Ordinary point, ordinary place      80
Periods, of an Abelian integral      96ff 101
Periods, period relations      100 105-6
Periods, primitive periods of an elliptic function      179
Picard      155
Place on a Riemann surface      34 77
Place on a Riemann surface, for an equation of the third degree in y      207
Place on a Riemann surface, singular places      80
Place on a Riemann surface, special set of places      71
Primitive branch      32
Principal parts      15
Projective intersections of two curves      150
Projective plane      159
Projective transformations      144ff
Rational function, analogies with elliptic functions      174ff
Rational function, conjugate values      44
Rational function, expressed in terms of integrals      113
Rational function, in the plane      12ff
Rational function, integral rational function      196
Rational function, on a Riemann surface      43ff
Rational function, special rational function      71 73
Rational function, with prescribed poles      14-15 70 72
Reducibility of polynomials      21ff
Reduction of singularities      155ff
Reduction of singularities, to double points in the function-theoretic plane      167
Reduction of singularities, to double points in the projective plane      163
Reduction of singularities, to multiple points with distinct tangents      156
references      23 42 74 92 118 132 154 168 182 213 214
Region      26
Region, connected      5
Region, simply connected      5 26
Regular arc      2
Regular arc, on a Riemann surface      80
Residues      7 12 44
Residues, of a function on a Riemann surface      81
Residues, theorems concerning      8 11 14 45
Resultant of two polynomials      18
Riemann      iii iv 67 75
Riemann surface T      75ff
Riemann surface T, construction of T, T’, and T’’      75 92 96
Riemann surface T, examples      186ff
Riemann surface T, for an irreducible algebraic equation      78
Riemann surface T, genus of      88 147
Riemann surface T, places on      77
Riemann — Roch theorem      67ff
Roch      iv 67
Severi      iii
Simart      155
Singular places, of Abelian integrals      94
Singular places, of an algebraic function      34
Singular places, of functions on a Riemann surface      80
Singular points      9 11
Singular points, for an algebraic function      25
Singular points, of special types      40
Singularities, reduction of      155ff (see also Reduction of singularities)
Special places and rational functions      71 73
Special places and rational functions, special divisor      71
Surface, connected or simply connected      82
Surface, connectivity      87
Trace of a rational function      44
Transformations, projective      144ff (see also Birational transformations Cremona
Unicursal curves      139ff
Weber      iii iv
Weierstrass      iii 176-7
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