Changeset 34cfbc in git


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Timestamp:
Sep 29, 2010, 5:46:53 PM (14 years ago)
Author:
Hans Schoenemann <hannes@…>
Branches:
(u'spielwiese', '17f1d200f27c5bd38f5dfc6e8a0879242279d1d8')
Children:
c6af37758b6eecc0501fc75ccf32f552185b2924
Parents:
e9259fee845ceb709118f8725a54e2dd0aa137d1
Message:
format fix

git-svn-id: file:///usr/local/Singular/svn/trunk@13333 2c84dea3-7e68-4137-9b89-c4e89433aadc
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  • Singular/LIB/dmod.lib

    re9259f r34cfbc  
    55LIBRARY: dmod.lib     Algorithms for algebraic D-modules
    66AUTHORS: Viktor Levandovskyy,     levandov@math.rwth-aachen.de
    7 @*             Jorge Martin Morales,    jorge@unizar.es
     7         Jorge Martin Morales,    jorge@unizar.es
    88
    99THEORY: Let K be a field of characteristic 0. Given a polynomial ring
    10 @*      R = K[x_1,...,x_n] and a polynomial F in R,
    11 @*      one is interested in the R[1/F]-module of rank one, generated by
    12 @*      the symbol F^s for a symbolic discrete variable s.
    13 @* In fact, the module R[1/F]*F^s has a structure of a D(R)[s]-module, where D(R)
    14 @* is an n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1> and
    15 @* D(R)[s] = D(R) tensored with K[s] over K.
    16 @* Constructively, one needs to find a left ideal I = I(F^s) in D(R), such
    17 @* that K[x_1,...,x_n,1/F]*F^s is isomorphic to D(R)/I as a D(R)-module.
    18 @* We often write just D for D(R) and D[s] for D(R)[s].
    19 @* One is interested in the following data:
    20 @* - Ann F^s = I = I(F^s) in D(R)[s], denoted by LD in the output
    21 @* - global Bernstein polynomial in K[s], denoted by bs,
    22 @* - its minimal integer root s0, the list of all roots of bs, which are known
    23 @*   to be rational, with their multiplicities, which is denoted by BS
    24 @* - Ann F^s0 = I(F^s0) in D(R), denoted by LD0 in the output
    25 @*   (LD0 is a holonomic ideal in D(R))
    26 @* - Ann^(1) F^s in D(R)[s], denoted by LD1 (logarithmic derivations)
    27 @* - an operator in D(R)[s], denoted by PS, such that the functional equality
    28 @*     PS*F^(s+1) = bs*F^s holds in K[x_1,...,x_n,1/F]*F^s.
     10      R = K[x_1,...,x_n] and a polynomial F in R,
     11      one is interested in the R[1/F]-module of rank one, generated by
     12      the symbol F^s for a symbolic discrete variable s.
     13In fact, the module R[1/F]*F^s has a structure of a D(R)[s]-module, where D(R)
     14is an n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1> and
     15D(R)[s] = D(R) tensored with K[s] over K.
     16Constructively, one needs to find a left ideal I = I(F^s) in D(R), such
     17that K[x_1,...,x_n,1/F]*F^s is isomorphic to D(R)/I as a D(R)-module.
     18We often write just D for D(R) and D[s] for D(R)[s].
     19One is interested in the following data:
     20- Ann F^s = I = I(F^s) in D(R)[s], denoted by LD in the output
     21- global Bernstein polynomial in K[s], denoted by bs,
     22- its minimal integer root s0, the list of all roots of bs, which are known
     23   to be rational, with their multiplicities, which is denoted by BS
     24- Ann F^s0 = I(F^s0) in D(R), denoted by LD0 in the output
     25   (LD0 is a holonomic ideal in D(R))
     26- Ann^(1) F^s in D(R)[s], denoted by LD1 (logarithmic derivations)
     27- an operator in D(R)[s], denoted by PS, such that the functional equality
     28     PS*F^(s+1) = bs*F^s holds in K[x_1,...,x_n,1/F]*F^s.
    2929
    3030REFERENCES:
    31 @* We provide the following implementations of algorithms:
    32 @* (OT) the classical Ann F^s algorithm from Oaku and Takayama (Journal of
    33 @* Pure and Applied Math., 1999),
    34 @* (LOT) Levandovskyy's modification of the Oaku-Takayama algorithm (ISSAC 2007)
    35 @* (BM) the Ann F^s algorithm by Briancon and Maisonobe (Remarques sur
    36 @*        l'ideal de Bernstein associe a des polynomes, preprint, 2002)
    37 @* (LM08) V. Levandovskyy and J. Martin-Morales, ISSAC 2008
    38 @* (C) Countinho, A Primer of Algebraic D-Modules,
    39 @* (SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric
    40 @*         Differential Equations', Springer, 2000
     31We provide the following implementations of algorithms:
     32@*(OT) the classical Ann F^s algorithm from Oaku and Takayama (Journal of
     33Pure and Applied Math., 1999),
     34@*(LOT) Levandovskyy's modification of the Oaku-Takayama algorithm (ISSAC 2007)
     35@*(BM) the Ann F^s algorithm by Briancon and Maisonobe (Remarques sur
     36l'ideal de Bernstein associe a des polynomes, preprint, 2002)
     37@*(LM08) V. Levandovskyy and J. Martin-Morales, ISSAC 2008
     38@*(C) Countinho, A Primer of Algebraic D-Modules,
     39@*(SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric
     40Differential Equations', Springer, 2000
    4141
    4242
    4343GUIDE:
    44 @* - Ann F^s = I(F^s) = LD in D(R)[s] can be computed by Sannfs [BM, OT, LOT]
    45 @* - Ann^(1) F^s in D(R)[s] can be computed by Sannfslog
    46 @* - global Bernstein polynomial bs in K[s] can be computed by bernsteinBM
    47 @* - Ann F^s0 = I(F^s0) = LD0 in D(R) can be computed by annfs0, annfs, annfsBM,
    48 @*    annfsOT, annfsLOT, annfs2, annfsRB etc.
    49 @* - all the relevant data to F^s (LD, LD0, bs, PS) are computed by operatorBM
    50 @* - operator PS can be computed via operatorModulo or operatorBM
    51 @*
    52 @* - annihilator of F^{s1} for a number s1 is computed with annfspecial
    53 @* - annihilator of F_1^s_1 * ... * F_p^s_p is computed with annfsBMI
    54 @* - computing the multiplicity of a rational number r in the Bernstein poly
    55 @*   of a given ideal goes with checkRoot
    56 @* - check, whether a given univariate polynomial divides the Bernstein poly
    57 @*   goes with checkFactor
    58 
    59 
    60 MAIN PROCEDURES:
     44- Ann F^s = I(F^s) = LD in D(R)[s] can be computed by Sannfs [BM, OT, LOT]
     45- Ann^(1) F^s in D(R)[s] can be computed by Sannfslog
     46- global Bernstein polynomial bs in K[s] can be computed by bernsteinBM
     47- Ann F^s0 = I(F^s0) = LD0 in D(R) can be computed by annfs0, annfs, annfsBM,
     48    annfsOT, annfsLOT, annfs2, annfsRB etc.
     49- all the relevant data to F^s (LD, LD0, bs, PS) are computed by operatorBM
     50- operator PS can be computed via operatorModulo or operatorBM
     51
     52- annihilator of F^{s1} for a number s1 is computed with annfspecial
     53- annihilator of F_1^s_1 * ... * F_p^s_p is computed with annfsBMI
     54- computing the multiplicity of a rational number r in the Bernstein poly
     55   of a given ideal goes with checkRoot
     56- check, whether a given univariate polynomial divides the Bernstein poly
     57   goes with checkFactor
     58
     59
     60PROCEDURES:
    6161
    6262annfs(F[,S,eng]);       compute Ann F^s0 in D and Bernstein polynomial for a poly F
    … …  
    7777checkFactor(I,F,q[,eng]); check whether a polynomial q in K[s] is a factor of the global Bernstein polynomial of F from the known Ann F^s in D[s]
    7878
    79 AUXILIARY PROCEDURES:
    80 
    8179arrange(p);           create a poly, describing a full hyperplane arrangement
    8280reiffen(p,q);         create a poly, describing a Reiffen curve
    … …  
    9088
    9189KEYWORDS: D-module; D-module structure; left annihilator ideal; Bernstein-Sato polynomial; global Bernstein-Sato polynomial;
    92 Weyl algebra; Bernstein operator; logarithmic annihilator ideal; parametric annihilator; root of Bernstein-Sato polynomial; 
     90Weyl algebra; Bernstein operator; logarithmic annihilator ideal; parametric annihilator; root of Bernstein-Sato polynomial;
    9391hyperplane arrangement; Oaku-Takayama algorithm; Briancon-Maisonobe algorithm; LOT algorithm
    9492";
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