Changeset 6a07eb in git for Singular/LIB/dmodvar.lib
 Timestamp:
 Jul 22, 2010, 10:29:11 PM (13 years ago)
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 (u'spielwiese', '0d6b7fcd9813a1ca1ed4220cfa2b104b97a0a003')
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 40e810dce492e702bf0e751809ae911eaf670a12
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 237b3e4eb8747703701bf0ed61e0bbd289339b4a
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Singular/LIB/dmodvar.lib
r237b3e4 r6a07eb 9 9 Jorge MartinMorales, jorge@unizar.es 10 10 11 THEORY: Let K be a field of characteristic 0. Given a polynomial ring 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 nth 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 rtuple 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. 11 THEORY: Let K be a field of characteristic 0. Given a polynomial ring R = K[x_1,...,x_n] and 12 @* a set of polynomial f_1,..., f_r in R, define F = f_1 * ... * f_r and F^s:=f_1^s_1*...*f_r^s_r 13 @* for symbolic discrete (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 D<S> := D(R) tensored with S over K, where 15 @*  D(R) is an nth Weyl algebra K<x_1,...,x_n,d_1,...,d_n  d_j x_j = x_j d_j +1> 16 @*  S is the universal enveloping algebra of gl_r, generated by s_{ij}, where s_{ii}=s_i. 17 @* One is interested in the following data: 18 @*  the left ideal Ann F^s in D<S>, usually denoted by LD in the output 19 @*  global Bernstein polynomial in one variable s = s_1 + ...+ s_r, denoted by bs, 20 @*  its minimal integer root s0, the list of all roots of bs, which are known 21 @* to be negative rational numbers, with their multiplicities, which is denoted by BS 22 @*  an rtuple of operators in D<S>, denoted by PS, such that the functional equality 23 @* sum(k=1 to k=r) P_k*f_k*F^s = bs*F^s holds in R[1/F]*F^s. 26 24 27 25 REFERENCES: … … 36 34 AUXILIARY PROCEDURES: 37 35 makeIF(F[,ORD]); create the Malgrange ideal, associated with F = F[1],..,F[P] 36 37 SEE ALSO: bfun_lib, dmod_lib, dmodapp_lib, gmssing_lib 38 39 KEYWORDS: Dmodule; Dmodule structure; BernsteinSato polynomial for variety; global BernsteinSato polynomial for variety; 40 Weyl algebra; parametric annihilator for variety; BudurMustataSaito approach; initial ideal approach; 38 41 "; 39 42 … … 362 365 363 366 proc bfctVarAnn (ideal F, list #) 364 "USAGE: bfctVarAnn(F[,gid,eng]); F an ideal, gid,eng optional ints 365 RETURN: list of ideal and intvec 366 PURPOSE: computes the roots of the BernsteinSato polynomial and their 367 @* multiplicities for an affine algebraic variety defined by 368 @* F = F[1],..,F[r]. 367 "USAGE: bfctVarAnn(F[,gid,eng]); F an ideal, gid,eng optional ints 368 RETURN: list of an ideal and an intvec 369 PURPOSE: computes the roots of the BernsteinSato polynomial and their multiplicities 370 @* for an affine algebraic variety defined by F = F[1],..,F[r]. 369 371 ASSUME: The basering is a commutative polynomial ring in char 0. 370 372 BACKGROUND: In this proc, the annihilator of f^s in D[s] is computed and then a 371 373 @* system of linear equations is solved by linear reductions in order to 372 @* find the minimal polynomial of S = s(1)(1) + ... + s(P)(P), following 373 @* the ideas by Noro. 374 NOTE: In the output list, the ideal contains all the roots and the invec their 375 @* multiplicities. 374 @* find the minimal polynomial of S = s(1)(1) + ... + s(P)(P) 375 NOTE: In the output list, the ideal contains all the roots and the intvec their multiplicities. 376 376 @* If gid<>0, the ideal is used as given. Otherwise, and by default, a 377 377 @* heuristically better suited generating set is used.
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