Kolchin E.R. — Differential algebra and algebraic groups :: Электронная библиотека попечительского совета мехмата МГУ

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Kolchin E.R. — Differential algebra and algebraic groups

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Название: Differential algebra and algebraic groups

Автор: Kolchin E.R.

Аннотация:

It is common knowledge that algebra, including algebraic geometry, historically grew out of the study of algebraic equations with numerical coefficients. In much the same way, differential algebra sprang from the classical study of algebraic differential equations with coefficients that are meromorphic functions in a region of some complex space C^m. As a consequence, differential algebra bears a considerable resemblance to the elementary parts of algebraic geometry. Indeed, since an algebraic equation can be considered as a differential equation in which derivatives do not occur, it is possible to consider algebraic geometry as a special case of differential algebra.

Язык:

Рубрика: Математика/Алгебра/Дифференциальная алгебра/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1973

Количество страниц: 468

Добавлена в каталог: 12.03.2005

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Предметный указатель

$\mathfrak{G}$ -Basis      11
$\mathfrak{G}$ -Component of perfect $\mathfrak{G}$ -ideal      13
$\mathfrak{G}$ -Ideal      10
$\mathscr{F}$ -Closed set in $mathscr{U}^n$       148
$\mathscr{F}$ -Topology of $\mathscr{U}^n$       149
$\mathscr{L}$ -Generated      11
$\Sigma$ -Prime ideal      7
(L,K)-Homomorphism of homogeneous L-space into homogeneous K-space      230
(L,K)-Homomorphism of L-group into K-group      230
A-Permissible homomorphism      174
Abelian extension      396
Abelian function      382
Abelian K-group      377
Additive polynomial      360
Affine K-group      355
Algebraic codimension      4
Algebraic element of pre-K-set      215—216
Algebraic group      212
Algebraically dependent (or independent) derivative operators      96
Algebraically dependent (or independent) over constants      93
Artin — Rees lemma      39
Autoreduced set      77
Basis Theorem      126
Basis theorem, historical remark      128—129
Bessel differential polynomial      417
Bicompatible isomorphisms      218
Bidefined      303
Birational correspondence      17
Birationally equivalent ideals      17
Canonical coordinate functions on K-sub-group of GL(n)      355
Characteristic set      82
Choice function for characteristic set      183
Closed graph      304
Closed image      298
Closed set in $\mathscr{U}^n$       146
Closed set in homogeneous K-space      240
Coboundary, $G(\mathscr{G} / \mathscr{F})$ into G      422
Coboundary, $\mathfrak{g}(L/K)$ into G      275
Cocycle, $G(\mathscr{G} / \mathscr{F})$ into G      422
Cocycle, $\mathfrak{g}(L/K)$ into G      274
Coherent autoreduced set      136 167
Coherent autoreduced set, t-Coherent autoreduced set      135—136
Cohomologous cocycles, $G(\mathscr{G}/ \mathscr{F})$ into G      422
Cohomologous cocycles, $\mathfrak{g}(L/K)$ into G      275
Cohomology set, $G(\mathscr{G} / \mathscr{F})$ into G      422
Cohomology set, $\mathfrak{g}(L/K)$ into G      275
Commutator      264—265
Commutator group      265
Compatible homomorphisms      218
Complete differential ring of quotients      64
Complete K-set      376
Complete ring of quotients      7
Complex multiplication      381
Component of 1 of K-group      232
Component of closed subset of homogeneous K-space      243
Component of perfect ideal      14
Component theorem      185
Conjugates of algebraic element of pre-K-set      216
Conservative mapping      11
Conservative system      10
Constant      60—66
Constrained family (or element)      142
Constraint      142
Cotangent space      335
Cotangent vector      335
Crossed k-homomorphism of K-group into K-group      343
Crossed k-homomorphism of K-space into K-space      343
Defining differential ideal      71
Degenerate Abelian function field      382
Denomination      72
Dependent (or independent) derivative operators      97
Derivation on irreducible K-set      322
Derivation operator      58
Derivative      59
Derivative operator      59
Diagonal group      213
Difference transcendence polynomial      117
Differential affine space      145
Differential algebra      69
Differential basis      108
Differential conservative system      121
Differential dimension of differential vector space      108
Differential dimension of irreducible closed set in $\mathscr{U}^n$       148
Differential dimension of prime differential polynomial ideal      130
Differential dimension polynomial of irreducible closed set in $\mathscr{U}^n$       148
Differential dimension polynomial of prime differential polynomial ideal      130
Differential field      58
Differential field extension      59—60
Differential field of definition of closed set in $\mathscr{U}^n$       149
Differential field of definition of differential polynomial ideal      125
Differential field of quotients      64
Differential grading      73
Differential ideal      61
Differential indeterminates      69
Differential inseparability basis      105
Differential inseparability degree of extension      107
Differential inseparability degree of prime differential polynomial ideal      129
Differential inseparability polynomial of finite family      117
Differential inseparability polynomial of prime differential polynomial ideal      129
Differential integral domain      58
Differential module      66
Differential monomiat      70
Differential of rational function      323
Differential on irreducible K-set      323
Differential overfield      59—60
Differential overring      59
Differential polynomial      70
Differential polynomial algebra      70
Differential polynomial function      95
Differential power series      85
Differential power series algebra      85
Differential quotient module      66
Differential rational fraction      71
Differential residue ring      61
Differential ring      58
Differential ring of quotients      64
Differential specialization of differential integral domain      138—139
Differential specialization of family of elements      139
Differential subfield      59
Differential submodule      66
Differential subring      59
Differential subspace      66
Differential transcendence basis      108
Differential transcendence degree      109
Differential type of finitely generated extension      118
Differential type of irreducible closed set in $\mathscr{U}^n$       148
Differential type of prime differential polynomial ideal      129
Differential vector space      66
Differential Zariski topology      146
Differential Zariski topology, relative to $\mathscr{F}$       149
Differentially algebraic closure in extension (characteristic 0)      102
Differentially algebraic element      69
Differentially algebraic extension      100
Differentially algebraically dependent (or independent)      69
Differentially homogeneous      71
Differentially inseparable      99—100
Differentially linear      104
Differentially linearly independent      108
Differentially perfect      92
Differentially quasi-perfect      92
Differentially separable closure in extension      102
Differentially separable element      99—100
Differentially separable extension      100
Differentially separably dependent (or independent)      99
Differentially transcendental      69
Dimension of element of a pre-K-set, pre-K-set      216
Dimension of prime polynomial ideal      20
Direct product of homogeneous K-spaces      258
Direct product of K-groups      257—258
Divisible conservative system      12

Domain of bidefinition      303
Dominate      178
Dominate, factorially      179
Dominate, strongly      179
Domination lemma      181—182
Essential order      83
Exponential      404
Extension (of differential field)      59—60
Field of constants      60
Field of definition of polynomial ideal      125
Field of definition of subspace of vector space      67
Fundamental system of zeros      151
G-extension      396
G-primitive      419
G-Primitive extension      419
Galois cohomology set      276
Galois group of linear differential ideal      411
Galois group of set of linear differential polynomials      411
Galois group of strongly normal extension      396
General component      157
General irreducible component      157
General linear group      213
General solution      157
Generic composite of K-mappings      299
Generic differential specialization      139
Generic inverse      301
Generic point      150
Generic specialization of element of pre-K-set      216
Generic specialization of family of elements of a field extension      33
Generic specialization of family of isomorphisms      386
Generic zero of prime differential polynomial ideal      146
Generic zero of prime polynomial ideal      19
Generically invertible      301
Generically surjective      301
Habitat      294
Holomorphic at a specialization      288
Holomorphic at an element of a K-set, derivation      338
Holomorphic at an element of a K-set, differential      338
Holomorphic at an element of a K-set, K-function      317
Holomorphic at an element of a K-set, K-mapping      318
Homogeneous K-space      220—221
Homogeneous K-space quotient      267
Homogeneous part      29
Homogeneous space      219
Homomorphism of differential modules      66
Homomorphism of differential rings      61
Homomorphism of pointed sets      275
Implicit function theorem      31
Independent elements of pre-K-sets      217
Induced homogeneous L-space      231
Induced L-group      230
Induced pre-K-mapping      217
Initial      75
Inseparability basis      4
Inseparability degree      4
Integrated ranking      75
Invariant derivation      325
Invariant differential      325
Irreducible closed set in homogeneous K-space      243
Irreducible component      147
Irreducible topological space      147
Isobaric      73
Isolated isomorphism      386
Isotropy group      257
K-Affine coordinates      331—332
K-Affine subset of K-set      307
K-closed set in homogeneous K-space      240
K-Coboundary      319
K-Cocycle      318
K-Cohomologous K-cocycles      319
K-Cohomology set      319
K-component of pre-K-set      216
K-Derivation on irreducible K-set      322
K-Differential on irreducible K-set      323
K-Equivalent pre-K-mappings      294
k-function      306
K-Generic element      216
k-group      218—219
K-Group quotient      267
K-Homomorphism of (M,G) into (M',G')      342
K-Homomorphism of homogeneous K-spaces      226
K-Homomorphism of K-groups      226
K-Homomorphism of K-spaces      341
K-Homomorphism of pointed pre-K-sets      277
K-Irreducible      216
K-isomorphism of homogeneous K-spaces      226
K-isomorphism of K-groups      226
K-Liouvillian K-group      374
K-Liouvillian K-group of type (i)      374
K-Mapping      295
K-Minimal      294
K-Operation of K-group on K-group      342
K-Operation of K-group on K-space      342—343
k-set      227
K-Simple K-group      375
k-space      341
K-Splits      321
K-Subgroup      226
k-subset      227
K-Subspace      342
K-Topology of homogeneous K-space      240
Krull topology      274
Krull's theorem      39—40
Lattice      382
Leader      75
Leading coefficient theorem      172
Levi's lemma      177
Lexicographic order      49
Lie algebra of homogeneous K-space      325
Linear differential polynomial ideal      150
Linear dimension      151
Linear extension      396
Linear K-group      355
Linearly dependent (or independent) derivative operators      96
Linearly dependent (or independent) over constants      88
Liouvillian extension      408
Liouvillian extension of type (i)      408
Liouvillian K-group      374
Liouvillian K-group of type (i)      374
Local component of derivation      338
Local component of invariant differential      339
Local derivation      334—335
Local ring on Vat $\nu$       331
Localization at prime ideal      7
Locus      216
Logarithmic derivation      350
Logarithmic derivative      349
Low power theorem      187
Multiplicity of differential polynomial at point      164
Multiplicity of zero of differential polynomial      164
Nakayama s lemma      34
Noetherian conservative system      13
Noetherian topological space      147
Nonsingular zero or solution      155
Normalization lemma      43
Numerical polynomial      50
Opposite K-group      223
Order of derivative      59
Order of derivative operator      59
Order of differential polynomial      70
Orderly ranking      75
Orthogonal group      213
Partial remainder      77 78
Partially pseudo-reduced      83
Partially reduced      77
Perfect conservative system      12
Perfect ideal      7
Permissible grading      73
Picard — Vessiot element      415
Picard — Vessiot extension      410

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