Changeset 4f461c in git


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Timestamp:
Mar 5, 2010, 2:28:28 PM (14 years ago)
Author:
Hans Schönemann <hannes@…>
Branches:
(u'fieker-DuVal', '117eb8c30fc9e991c4decca4832b1d19036c4c65')(u'spielwiese', '45e267b9942dec9429fe249ce3e5e44ab80a6a3a')
Children:
24c3680c7cb6e7c29e4766bc33d10684a0c2e1f2
Parents:
8a7e767173900998af700bab01a16d332d191b3d
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format

git-svn-id: file:///usr/local/Singular/svn/trunk@12599 2c84dea3-7e68-4137-9b89-c4e89433aadc
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  • Singular/LIB/dmodvar.lib

    r8a7e76 r4f461c  
    44info="
    55LIBRARY: dmodvar.lib     Algebraic D-modules for varieties
     6
    67AUTHORS: Daniel Andres,          daniel.andres@math.rwth-aachen.de
    7 @*       Viktor Levandovskyy,    levandov@math.rwth-aachen.de
    8 @*       Jorge Martin-Morales,   jorge@unizar.es
     8       Viktor Levandovskyy,    levandov@math.rwth-aachen.de
     9       Jorge Martin-Morales,   jorge@unizar.es
    910
    1011THEORY: Let K be a field of characteristic 0. Given a polynomial ring
    11 @*      R = K[x_1,...,x_n] and a set of polynomial f_1,..., f_r in R, define
    12 @*      F = f_1 * ... * f_r and F^s:=f_1^s_1*...*f_r^s_r for symbolic discrete
    13 @*      (that is shiftable) variables s_1,..., s_r.
    14 @*      The module R[1/F]*F^s has a structure of a D<S>-module, where
    15 @*      D<S> := D(R) tensored with S over K, where
    16 @*      - D(R) is an n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1>
    17 @*      - S is the universal enveloping algebra of gl_r, generated by s_{ij}, where s_{ii}=s_i.
    18 @* One is interested in the following data:
    19 @* - the left ideal Ann F^s in D<S>, usually denoted by LD in the output
    20 @* - global Bernstein polynomial in one variable s = s_1 + ...+ s_r, denoted by bs,
    21 @* - its minimal integer root s0, the list of all roots of bs, which are known
    22 @*   to be rational, with their multiplicities, which is denoted by BS
    23 @* - an r-tuple of operators in D<S>, denoted by PS, such that the functional equality
    24 @*     sum(k=1 to k=r) P_k*f_k*F^s = bs*F^s holds in R[1/F]*F^s.
     12      R = K[x_1,...,x_n] and a set of polynomial f_1,..., f_r in R, define
     13      F = f_1 * ... * f_r and F^s:=f_1^s_1*...*f_r^s_r for symbolic discrete
     14      (that is shiftable) variables s_1,..., s_r.
     15      The module R[1/F]*F^s has a structure of a D<S>-module, where
     16      D<S> := D(R) tensored with S over K, where
     17      - D(R) is an n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1>
     18      - S is the universal enveloping algebra of gl_r, generated by s_{ij}, where s_{ii}=s_i.
     19 One is interested in the following data:
     20 - the left ideal Ann F^s in D<S>, usually denoted by LD in the output
     21 - global Bernstein polynomial in one variable s = s_1 + ...+ s_r, 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 - an r-tuple of operators in D<S>, denoted by PS, such that the functional equality
     25     sum(k=1 to k=r) P_k*f_k*F^s = bs*F^s holds in R[1/F]*F^s.
    2526
    2627REFERENCES:
    27 @*      (BMS06) Budur, Mustata, Saito: Bernstein-Sato polynomials of arbitrary varieties (2006).
    28 @*      (ALM09) Andres, Levandovskyy, Martin-Morales : Principal Intersection and Bernstein-Sato Polynomial of an Affine Variety (2009).
     28  (BMS06) Budur, Mustata, Saito: Bernstein-Sato polynomials of arbitrary varieties (2006).
     29  (ALM09) Andres, Levandovskyy, Martin-Morales : Principal Intersection and Bernstein-Sato Polynomial of an Affine Variety (2009).
    2930
    3031MAIN PROCEDURES:
     
    166167          // the case: given ORD, given engine
    167168          eng = int(#[2]);
    168         } 
     169        }
    169170        else
    170171        {
     
    197198  int ppl = printlevel-voice+2;
    198199  // returns a list with a ring and an ideal LD in it
    199   // save, N, P and the indices are already defined 
     200  // save, N, P and the indices are already defined
    200201  int Nnew = 2*N+P+P^2;
    201202  list RL = ringlist(basering);
     
    328329  intvec iv = P+1..Nnew;
    329330  tmpM = imap(@R@,@D);
    330   kill @R@; 
     331  kill @R@;
    331332  LordM = submat(tmpM,iv,iv);
    332333  matrix @D2 = LordM;
     
    548549  // Name, Dname will be used further
    549550  kill NName, TName, Name, DTName, DName;
    550   // ORD already set, default ord dp; 
     551  // ORD already set, default ord dp;
    551552  L[3] = ORDstr2list(ORD,Nnew);
    552553  // we are done with the list
     
    605606@*       Varnames of the basering do not include t(1),...,t(r) and
    606607@*       Dt(1),...,Dt(r), where r is the number of entries of the input ideal.
    607 BACKGROUND:  In this proc, the initial ideal of the multivariate Malgrange ideal 
     608BACKGROUND:  In this proc, the initial ideal of the multivariate Malgrange ideal
    608609@*       defined by I is computed and then a system of linear equations is solved
    609610@*       by linear reductions following the ideas by Noro.
     
    619620@*       time.
    620621@*       If b<>0, @code{std} is used for GB computations in characteristic 0,
    621 @*       otherwise, and by default, @code{slimgb} is used. 
     622@*       otherwise, and by default, @code{slimgb} is used.
    622623@*       If c<>0, a matrix ordering is used for GB computations, otherwise,
    623624@*       and by default, a block ordering is used.
     
    838839  I = std(I);
    839840//ideal I = z(6)^2-z(3)*z(7), z(5)*z(6)-z(2)*z(7), z(5)^2-z(1)*z(7),
    840 //  z(4)*z(5)-z(3)*z(6), z(3)*z(5)-z(2)*z(6), z(2)*z(5)-z(1)*z(6), 
     841//  z(4)*z(5)-z(3)*z(6), z(3)*z(5)-z(2)*z(6), z(2)*z(5)-z(1)*z(6),
    841842//  z(3)^2-z(2)*z(4), z(2)*z(3)-z(1)*z(4), z(2)^2-z(1)*z(3);
    842843  bfctVarIn(I,1); // no result yet
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