Changeset cad507 in git for Singular/LIB/dmod.lib
 Timestamp:
 Mar 6, 2009, 10:05:54 PM (15 years ago)
 Branches:
 (u'spielwiese', '8e0ad00ce244dfd0756200662572aef8402f13d5')
 Children:
 55a8282afcb99a7a85304b3f8ecf99062c4dd376
 Parents:
 fda698647a5a06bd263906206d3ce89a87b42c0d
 File:

 1 edited
Legend:
 Unmodified
 Added
 Removed

Singular/LIB/dmod.lib
rfda6986 rcad507 1 1 ////////////////////////////////////////////////////////////////////////////// 2 version="$Id: dmod.lib,v 1.3 5 20090212 20:25:22levandov Exp $";2 version="$Id: dmod.lib,v 1.36 20090306 21:05:54 levandov Exp $"; 3 3 category="Noncommutative"; 4 4 info=" … … 7 7 @* Jorge Martin Morales, jorge@unizar.es 8 8 9 THEORY: Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R, 10 @* one is interested in the R[1/F]module of rank one, generated by F^s 11 @* for a natural number s. 9 THEORY: 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. 12 13 @* In fact, the module R[1/F]*F^s has a structure of a D(R)[s]module, where D(R) 13 14 @* is an nth Weyl algebra K<x_1,...,x_n,d_1,...,d_n  d_j x_j = x_j d_j +1> and … … 18 19 @* One is interested in the following data: 19 20 @*  Ann F^s = I = I(F^s) in D(R)[s], denoted by LD in the output 20 @*  global Bernstein polynomial in K[s], denoted by bs, its minimal integer root s0 and 21 @* the list of all roots of bs, which are rational, with their multiplicities is denoted by BS 22 @*  Ann F^s0 = I(F^s0) in D(R), denoted by LD0 in the output (LD0 is a holonomic ideal in D(R)) 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)) 23 26 @*  Ann^(1) F^s in D(R)[s], denoted by LD1 (logarithmic derivations) 24 @*  an operator in D(R)[s], denoted by PS, such that PS*F^(s+1) = bs*F^s holds in K[x_1,...,x_n,1/F^s]. 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. 25 29 26 30 @* We provide the following implementations: 27 @* OT) the classical Ann F^s algorithm from Oaku and Takayama (J . Pure28 31 @* OT) the classical Ann F^s algorithm from Oaku and Takayama (Journal of 32 @* Pure and Applied Math., 1999), 29 33 @* LOT) Levandovskyy's modification of the OakuTakayama algorithm (ISSAC 2007) 30 34 @* BM) the Ann F^s algorithm by Briancon and Maisonobe (Remarques sur 31 l'ideal de Bernstein associe a des polynomes, preprint, 2002)35 @* l'ideal de Bernstein associe a des polynomes, preprint, 2002) 32 36 33 37 GUIDE: 34 @*  Ann F^s = I = I(F^s) = LD in D(R)[s] can be computed by SannfsBM, SannfsOT, SannfsLOT38 @*  Ann F^s = I(F^s) = LD in D(R)[s] can be computed by Sannfs [BM, OT, LOT] 35 39 @*  Ann^(1) F^s in D(R)[s] can be computed by Sannfslog 36 @*  global Bernstein polynomial bs resp. BS in K[s] can be computed by bernsteinBM 37 @*  Ann F^s0 = I(F^s0) = LD0 in D(R) can be computed by annfs0, annfsBM, annfsOT, annfsLOT, annfs2 38 @*  all the relevant data (LD, LD0, bs, PS) are computed by operatorBM 40 @*  global Bernstein polynomial bs in K[s] can be computed by bernsteinBM 41 @*  Ann F^s0 = I(F^s0) = LD0 in D(R) can be computed by annfs0, annfs, annfsBM, 42 @* annfsOT, annfsLOT, annfs2, annfsRB etc. 43 @*  all the relevant data to F^s (LD, LD0, bs, PS) are computed by operatorBM 44 @* 45 @*  annihilator of F^{s1} for a number s1 is computed with annfspecial 46 @*  annihilator of F_1^s_1 * ... * F_p^s_p is computed with annfsBMI 47 @*  computing the multiplicity of a rational number r in the Bernstein poly 48 @* of a given ideal goes with checkRoot 49 @*  check, whether a given univariate polynomial divides the Bernstein poly 50 @* goes with checkFactor 51 39 52 40 53 MAIN PROCEDURES: … … 78 91 LIB "matrix.lib"; // for submat 79 92 LIB "nctools.lib"; 80 LIB "elim.lib"; 93 LIB "elim.lib"; // for nselect 81 94 LIB "qhmoduli.lib"; // for Max 82 95 LIB "gkdim.lib"; … … 237 250 poly F = z*x^2+y^3; 238 251 def A = annfs(F); // here, the default BM algorithm will be used 239 setring A; 240 LD; 241 BS; 252 setring A; // the Weyl algebra in (x,y,z,Dx,Dy,Dz) 253 LD; //the annihilator of F^{1} over A 254 BS; // roots with multiplicities of BS polynomial 242 255 } 243 256
Note: See TracChangeset
for help on using the changeset viewer.