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Mitsumi S., Sturmfels B., Takayama N. Groebner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics, Volume 6
Mitsumi S., Sturmfels B., Takayama N.  Groebner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics, Volume 6







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: Groebner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics, Volume 6

: Mitsumi S., Sturmfels B., Takayama N.

:

In recent years, new algorithms for dealing with rings of differential operators have been discovered and implemented. A main tool is the theory of Groebner bases, which is reexamined here from the point of view of geometric deformations. Perturbation techniques have a long tradition in analysis; Groebner deformations of left ideals in the Weyl algebra are the algebraic analogue to classical perturbation techniques. The algorithmic methods introduced here are particularly useful for studying the systems of multidimensional hypergeometric PDEs introduced by Gelfand, Kapranov and Zelevinsky. The Groebner deformation of these GKZ hypergeometric systems reduces problems concerning hypergeometric functions to questions about commutative monomial ideals, and leads to an unexpected interplay between analysis and combinatorics. This book contains a number of original research results on holonomic systems and hypergeometric functions, and raises many open problems for future research in this area.


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ed2k: ed2k stats

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: 15.04.2005

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$(a)_k$      see Pochhammer symbol
$(I: J^{\infty})$      36
$Ann(f^\alpha)$      211
$B_d$, $\bar{\mathcal{B}}$      203 230
$C^*$      86 (see also Polar)
$C_I[(t, u, v)]$      56
$C_w(I)$      see Grobner cone
$D^{\pm}$      190
$D_{Y\leftarrow X}$      228
$D_{Y\rightarrow X}$      201
$E_S$      see Difference operator
$face_w(P_u)$      123
$F_k$ (filtration)      14
$f_{\sigma}$      see Primitive integral support function
$gr_{(u,v)}$      4 (see also Associated graded ring)
$Hom_D(D/I, \mathcal{O}^{an}(U))$      72
$H^p(T, C)$      219
$H_A$[s]$      105 175
$H_A(\beta)$      see GKZ hypergeometric system
$in.(\dot)$ for D      46
$in.(\dot)$ for formal series      8990
$ind_w(I)$      70 (see also Indicial ideal)
$in_w(I_A)$      109116 139
$in_{(0,e)}(I)$      see Characteristic ideal
$I^{(h)}$      15 (see also Homogenization)
$I_A$      see Toric ideal
$I_p$      140 (see also $\mathcal{U}_p$)
$k[x_1^{\pm,...,k_n^{\pm}}]$      69
$link_{\tau}(\Delta_w)$      145
$normalForm_{\prec}(f, G)$      7
$N_w$      see Nilsson ring
$ord_{(u,v)}$      14 203
$Q_{A,p,w}$      159
$Start_{\prec w}(I)$      92
$star_{supp(p)}(\Delta_w)$      140
$S_{\beta}(M)$      158
$V_{\tau}$      see Volume polynomial
$[\dot]^m$      195
$[\dot]_m$      45 68 127
$\bar{H}_A(\beta)$      108
$\beta^{(\delta^a,\sigma)}$      116
$\Delta_1\times\Delta_{m1}$      43
$\Delta_w$      see Regular triangulation
$\mathbb{R}_+$ (the non-negative reals)      30
$\mathcal{E}(A)$      see Hole
$\mathcal{L}_{x,\gamma}$      225 (see also Locally constant sheaf)
$\mathcal{O}^{an}$      2
$\mathcal{U}_p$      140 160
$\partial_i\dot f$      2
$\phi_v$, $\phi_a^{\sigma}$      127 129
$\Psi$      3
$\theta_i$      10
$\theta_x$      see Euler operator
$\tilde{fin}_w(H_A(\beta))$      see Fake indicial ideal
$\tilde{I}$      68 (see also Distraction)
$\tilde{topfin}_w(H_A(\beta))$      155
$_pF_{p1}$      see Generalized hypergeometric function
A-hypergeometric differential equation      see GKZ hypergeometric system
A-hypergeometric function of degree $\beta$      104
A-hypergeometric ideal      104
Additive monoid      45
Airy integral      223
Algebraic torus      66
Algorithm, Buchbergers algorithm      9 243
Algorithm, Computing $Ann(f^s)$      214 217
Algorithm, Computing $in_{(u,v)}(I)$      14
Algorithm, Computing $I_A$      49
Algorithm, Computing all monomials in a given polynomial ideal      175
Algorithm, Computing integral ideal      230
Algorithm, Computing integral module      230
Algorithm, Computing restriction ideal      207
Algorithm, Computing restriction module      205
Algorithm, Computing standard pairs      113
Algorithm, de Rham cohomology group      234
Algorithm, Distraction      68
Algorithm, Finding asymptotic expansion of hypergeometric integral      235
Algorithm, Finding b-function      196 215
Algorithm, From D-ideal to a Pfaffian      39
Algorithm, Holonomic rank      34 91
Algorithm, Is I holonomic or not?      66
Algorithm, Normal form      7
Algorithm, Series solution      9698 128 133 145
Algorithm, Solving Frobenius ideal      74
Annihilating ideal      211
Aomoto Gelfand system      139
Appell      40
Appell, hypergeometric differential equation      48 85
Appell, hypergeometric function      40 47 48 61
Arithmetic degree      112
Arithmetic volume      157
Artinian      29 69
Ass(I)      30 110
Associated graded ring      4 33
Associated prime      29 30
Associated standard pair      187
Asymptotic expansion      77 236
b-function      178
b-function, of polynomial f, global b-function      215
b-function, parametric      178
b-function, with respect to weight vector w      194
Bernstein inequality      66
Bernstein Sato polynomial      218
Bessel function      223
Canonical series solution      23 93 236
Cauchy Kowalevskii Kashiwara Theorem      37
ch(I), $ch($D^r/M)$      28 200
Chain      219
Characteristic ideal      28
Characteristic variety      28 200
CKKT      see Cauchy Kowalevskii Kashiwara Theorem
Cocycle      219
Cohen Macaulay      43 114 153
Cohen Macaulay complex      160
Comparison theorem      226
Compatibility condition      37
Complete fan      52
Cone      52
Cone, polyhedral      52
Cone, rational      52
Contiguity relation      184
Convex hull      53
Core      150
CYCLE      219
D, $D_n$      see Weyl algebra
D-ideal      4
D-submodule      199
De Rham cohomology group      219
de Rham cohomology group, twisted      226
Degree      29 112 152
Degree(I)      see Degree
Degree, w-degree, (u, v)-degree      see Weight
Dehomogenization      11
Depth      114
Difference operator      212
dim(I)      see (Krull) dimension
DIMENSION      29
Dimension, Krull      29 63 152
Distraction      68 107
Divisible      7
e = (l, l, ..., l)      28
Elimination      10 36
Elimination term order      5 33 64
Embedded      187
Embedded, associated prime      30
Euler operator      17 82
Euler Jacobi cone      181
Exceptional arrangement      181
Exceptional set      181
EXPONENT      18 92
Face      52 113
Facet      52
Fake characteristic ideal      172
Fake exponent      108 154
Fake indicial ideal      108 154
Fake initial ideal      44 106 170
Fake rank      159
Fan      52
feasible      131
Fourier transform      82 229
Free resolution      153
Frobenius ideal      69
Frobenius method      18 22 85
FTAA      see Fundamental theorem of algebraic analysis
Fuchs condition      35
Fuchsian      78
Fundamental theorem of algebraic analysis      30
Gale duality      121
Gauss hypergeometric equation      16
Gauss hypergeometric series      18
Generalized hypergeometric function      224
Generic A-hypergeometric series      129
Generic weight vector      68
GKZ hypergeometric differential equation      48
GKZ hypergeometric integral      220
GKZ hypergeometric system      26 42 48 104
Gorenstein      76
Graded reverse lexicographic order      5
Grobner basis      5 6 8 12 33
Grobner cone      93
Grobner deformation      26 44 66
Grobner duality      73
Grobner fan      53 54 62 120 122
Grobner walk      62
H(p), H(I)      11 see
h.s.o.p.      see Homogeneous system of parameters
Harmonic polynomial      76
Hilbert polynomial      29 152
Hilbert series      152
hole      162
Holonomic      31 200
Holonomic rank      31
Homogeneity assumption      104
Homogeneous system of parameters (h.s.o.p.)      152
Homogenization      11
Homogenization, map      11
Homogenization, variable      11
Homogenized Weyl algebra      11
Hypergeometric fan      120 124
Hypergeometric integral      220
Hypergeometric polynomial      61 131
Indicial equation      18
Indicial ideal      70 91 107
Indicial polynomial      19 26 71 194
Induced order $\prec'$      13 70
Initial form      4 67 200
Initial ideal      5 6 68
Initial module      200
Initial monomial      6 90
Initial series      89
Initial system      44
Initial term      33
Integer programming problem      131 174
integral      229
Integral, ideal      228
Integral, module      229
Involutive bases      16
Koszul complex      167 209
l.s.o.p.      see Linear system of parameters
Lattice ideal      121
Legendre polynomial      27
Leibnitz formula      3
Lexicographic order      5 89
Linear system of parameters (l.s.o.p.)      153
Local b-function      218
Locally constant sheaf      225
Logarithm-free hypergeometric series      127
Logarithmic solutions      17
Mapping cone      209
Maximally degenerate      150 239
Mellin transform      213
Minimal free resolution      153
Minimal negative support      133
Minimal prime      30
Minkowski sum      52
mono(J)      175
Monodromic      82
Monodromy matrix      84
Monomial      3 5 88 200
Multiplicative monomial order      5 11
Multiplicity of root      75
N = {0, 1, 2, ...}      3
Negative support      132
New(f)      see Newton polytope
Newton, polyhedron      130
Newton, polytope      57
Nilsson ring      95 236
Non-term order      5
Nonresonant      181
normal      153 156
Normal cone      52
Normal fan      52
Normal form      8
Normalized volume      50 147
Normally ordered expression      3
nsupp(v)      132
Optimal solution      174
Optimal value      174
Orthogonal complement      72
Parameter shift      184
Partial term order      89
Period map      21
Pfaffian system      37
Pochhammer symbol      18
Polar, dual      52 86
Polyhedral fan      52
Polyhedron      52
Polynomial growth condition      35
Polytope      53
pos(B) (positive cone)      121
Primary      29 30 71
Primary ideal decomposition      29 40
Primitive integral support function      181
Principal $\mathcal{A}$-determinant      40
R      see Ring of differential operators
Radical      29
Rank      see Holonomic rank
rank(I)      34 (see also Holonomic rank)
Reduced Grobner basis      10
refinement      52
Reflexive polytope      150
Regular holonomic      78 80 200
Regular sequence      153
Regular singular      17 36 77
Regular triangulation      49 114
Restriction      79 201
Restriction, ideal      202
Restriction, module      201
Reverse lexicographic order      5
Ring of differential operators      2
Ring of differential operators, with rational function coefficients      33
S(M)      see Standard pair
s-pair      7
S-pair, criterion      9 106 212
Saturation      36 113
Secondary fan      120 121
Selberg type integral      225
Semi-nonresonant      181
Sheaf of differential operators      28 78
Shellable complex      143
Shelling basis      144
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