Van der Put M., Singer M.F. — Galois Theory of Linear Differential Equations :: Электронная библиотека попечительского совета мехмата МГУ

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Van der Put M., Singer M.F. — Galois Theory of Linear Differential Equations

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Название: Galois Theory of Linear Differential Equations

Авторы: Van der Put M., Singer M.F.

Аннотация:

This book is an introduction to the algebraic, algorithmic and analytic aspects of the Galois theory of homogeneous linear differential equations. Although the Galois theory has its origins in the 19th Century and was put on a firm footing by Kolchin in the middle of the 20th Century, it has experienced a burst of activity in the last 30 years. In this book we present many of the recent results and new approaches to this classical field. We have attempted to make this subject accessible to anyone with a background in algebra and analysis at the level of a first year graduate student. Our hope is that this book will prepare and entice the reader to delve further.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 462

Добавлена в каталог: 08.04.2006

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Предметный указатель

$(y_{1},..., y_{n})$       9
      246
$Aut(\bar{k}/k)$       382
$Aut^{\amalg}_(\omega)$       387
$C ((z))_{1/k}$       198
$C^{\infty}$ -variety of dimension n      407
      207
$C_{1}$ -field      383
$Diff_{\hat{K}}$       81
$f^{*}G$       410
$f_{*}F$       409
$F_{p}$       149
$GCLD(L_{1}, L_{2})$       40
$GCRD(L_{1}, L_{2})$       40
      371
$Gr_{1}$       81
$Gr_{2}$       254
$Gr_{3}$       289
      371
      371
$Hom_{k[\partial]} (M_{1}, M_{2})$       45
$H^{1} (Aut(\bar{k}/k), G(\bar{k}))$       382
$H^{J}$       412
$H_{1} (S^{1}, STS)$       256
$i_{*}F$       410
$i_{*}G$       410
$k[\partial_{1},..., \partial_{r}]$       424
$K_{conv, m}$       87
$K_{conv}$       87
$k_{q}$       107
$k_{\infty}$       107
$LCLM(L_{1}, L_{2})$       40
$LCLM(R_{1},..., R_{s})$       59
$LCRM(L_{1}, L_{2})$       40
$L_{1} \otimes L_{2}$       51
$M^{alg}$       171
$M^{an}$       171
      46
$N^{alg}$       171
$O_{W,P}$       368
$O_{X} (A)$       404
$O_{\textbf{P}} (n)$       169
      386
      170
$RegSing(P^{1}, S)$       172
$RegSing(\textbf{P},S)$       172
      393
$Repr_{\pi_{1}}$       168
      80
$R_{1} \otimes_{k} R_{2}$       362
$R{{y_{1},..., y_{n}}}$       4
$S(a, b, \rho)$       153 194
$Soln_{K} (L_{i})$       40
      246
$st_{d}(\hat{v})$       239
$Sym^{d}(L)$       52
$sym^{d}(V)$       126
$S_{4}^{SL_{2}}$       133
$T_{W, P}$       369
$UnivF_{\hat{K}}$       77
$UnivR_{\hat{K}}$       75 77
$V \otimes_{k} W$       361
      364
      80
$V_{O}$       207
$wr(y_{1},..., y_{n})$       9
$X_{1} \times_{k} X_{2}$       363
$\breve{H}^{n} (\mathcal{U}, F)$       417
$\Delta$       423
$\delta$ -field      423
$\delta$ -ideal      423
$\delta$ -ring      423
$\Delta_{d,q}$       275
$\Delta_{d}$       275
$\emph{C}^{\infty}$       405
$\emph{C}_1$       383
$\hat{\mathcal{B}}_{k}$       216
$\Lambda^{d}_{\mathcal{I}} (L)$       55
$\Lambda_{d}(L)$       52
$\lim \limits_\leftarrow B_i$       385
$\lim \limits_\rightarrow B_h$       419
$\mathcal{A} (a,b)$       194
$\mathcal{A} (S (a, b, \rho))$       194
$\mathcal{A}^{0}$       194
$\mathcal{A}_d$       194 200
$\mathcal{A}_{1/k} (S)$       196
$\mathcal{A}_{1/k}^{0} (S)$       196
$\mathcal{A}_{4}^{SK_{2}}$       133
$\mathcal{A}_{5}^{SK_{2}}$       133
$\mathcal{A}_{d}^{0}$       194 200
$\mathcal{B}_{k}$       217
$\mathcal{D}$       39
$\mathcal{F} (\sum)$       200
$\mathcal{F}$       200
$\mathcal{H}$       221
$\mathcal{K}_{A}$       220
$\mathcal{K}_{B}$       220
$\mathcal{L}(D)$       170
$\mathcal{L}(G)$       284
$\mathcal{L}_{k, d}$       216
$\mathcal{M}(A)$       404
$\nabla$       165 427
$\nabla_{D}$       165
$\nabla_{\eta}$       173
$\omega$       414
$\Omega_{A/k}$       164
$\Omega_{X}$       167
$\Omega_{\textbf{P}} (D)$       170
$\pi_{1} (U,p)$       150
$\textbf{A}_{k}^{1}$       354
$\textbf{A}_{k}^{n}$       355
$\theta$       424
$\underline{k}$ -summable      225
$\underline{k}$ -summable in the direction d      225
$\varepsilon(L)$       47
$\varepsilon(L_{1}, L_{2})$       47
$\widehat{Lie{S}}$       277
1-coboundries      418
1-cocycle      418
Accessory parameter      140 190
Additive group      371
Adjoint action      379
Affine group scheme      391
Affine group scheme over k G-module      392
Affine group scheme, representation      392
Affine line      354
Affine scheme      390
Affine scheme over k      390
Affine scheme over k, morphism of      390
Affine scheme over k, product      391
Affine space      355 356
Affine variety      353
Affine variety, defined over k      365
Affine variety, dimension      365
Affine variety, irreducible components      359
Affine variety, reducible      359
Airy equation      83 99 238 250
Alien derivations      275
Analytic continuation      149 150
Associated operator      54
Asymptotic expansion      194
Asymptotic lift      196
B(L)      91
b(N(L))      91
Beke's Algorithm      118
Bilinear map      361
Birkhoff's Problem      312
Borel transform, formal, of order k      216
Borel transform, of order k      217
Borel — Ritt theorem      195

Borel — Ritt Theorem, for $\textbf{C}((z))_{1/k}$       212
Boundary part      91
Brill equations      126
C({z})      87 406
Canonical form      78
CHARACTER      374
Chevalley module      300
Cocycle      382
Cohomology      382
Cohomology, Cech      417
Cohomology, groups of a complex      412
Cohomology, set      256
Companion matrix      8
Complementary submodule      59
Completely reducible      59
Completely reducible G-module      59
Completion at infinity      107
Composition series      60
Connection      427
Connection, for A/k      165
Connection, integrable      428
Connection, irreducible regular singular      178
Connection, irregular singular      172
Connection, regular      166
Connection, regular connection on a Riemann surface      167
Connection, regular singular      166 171
Constant      4 423
Convergent Laurent series      406
Coordinate ring      357
Covering, cyclic      263
Covering, m-periodic      262
Cyclic vector      42
C{z}      406
d(G)      285
Defect      285
Deltaideal, linear $\Delta$ -ideal      425
Deltaideal, linear dimension      425
Der(L)      128
Derivation      3
Derivative of L      128
Differential      164
Differential equation, eigenvalue      81
Differential equation, equivalent      7
Differential equation, Fuchsian      154 173
Differential equation, quasi-split      87 204
Differential equation, regular singular      68 150 154
Differential equation, split      87 204
Differential extension      3
Differential field      3
Differential Galois group      19
Differential homomorphism      4
Differential ideal      12
Differential module      6
Differential module, direct sum      44
Differential module, eigenvalue      81
Differential module, morphism      44
Differential module, quotient      44
Differential module, submodule      44
Differential module, tensor product      44
Differential operator, quasi-split      87
Differential operator, split      87
Differential polynomial      4
Differential polynomial, order      4
Differential ring      3
Differential, holomorphic      167
Differential, universal      164
Dimension, of a variety at a point      368
Dimension, of an affine variety      365
Direct limit      419
Direct system of abelian groups      419
Dual differential, module      45
E(G)      285
Eigenequations      121
Eigenring      47
Epsilon trick      370
Excess      285
Exponential growth of order $\leq k$ at $\infty$       216
Exponential solution      110
Exponential torus      80
Exponential torus, of N      82
Extension by exponentials      33
Extension by integrals      33
Exterior power      52
Exterior power, with respect to $\mathcal{I}$       55
Fine moduli space      309
Finite G-set      386
Finitely generated      353
Formal monodromy      79
Free associative algebra      276
Free Lie algebra      276
Frobenius      69
Fsets      387
Fuchsian      154 173
Fundamental matrix      8
Fundamental set of solutions      9
F{S}      276
GAGA      170
Gal(R/k)      19
Galois category      387
Galois correspondence      26
Generic fibre      173
Germ      405
Germ, flat      405
Gevrey function of order k      196
Gevrey series of order k      198
Hensel's Lemma, Classical      66
Hensel's Lemma, irregular singular modules      73
Hensel's Lemma, Regular Singular Modules      71
Hilbert Basissatz      354
Hilbert nullstellensatz      355
Holomorphic differentials      167
Homogeneous of order n      8
Hopf algebra      372 391
I(n)      123
Imprimitive      135
Indicial equation      112
Indicial polynomial      108
Inhomogeneous of order n      8
Integrability conditions      426
Internal horn      45
Inverse limit      386
Inverse system of abelian groups      385
Irreducible      359
Irregular singular      68
Irregularity      228
J(f)      196
J(n)      123
Jacobian criterion      369
Jordan — Holder theorem      60
Jordan's Theorem      123
k((z))      63
K(X)      359
k-algebra      353
k-summable      199
k-summable in the direction d      199
k-Summation Theorem      219 223
Kovacic Algorithm      131 134
Laplace transform of order k      216
Lattice      67
Laurent series, convergent      87
Laurent series, convergent in $z^{1/m}$       87
Laurent series, formal      63
Level      261
Lie algebra      6
Lie algebra, algebraic      379
Lie algebra, of a linear algebraic group      379
Lie{S}      276
Line bundle      166
Line bundle, of holomorphic differentials      167
Linear algebraic group      371

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