Changeset 6a07eb in git for Singular/LIB/dmodvar.lib

Timestamp:

Jul 22, 2010, 10:29:11 PM (13 years ago)

Author:

Viktor Levandovskyy <levandov@…>

Branches:

(u'spielwiese', '0d6b7fcd9813a1ca1ed4220cfa2b104b97a0a003')

Children:

40e810dce492e702bf0e751809ae911eaf670a12

Parents:

237b3e4eb8747703701bf0ed61e0bbd289339b4a

Message:

*levandov: keywords added, minor cleanup

git-svn-id: file:///usr/local/Singular/svn/trunk@13056 2c84dea3-7e68-4137-9b89-c4e89433aadc

File:

• Singular/LIB/dmodvar.lib (modified) (3 diffs)

Singular/LIB/dmodvar.lib

@@ -9 +9 @@
        Jorge Martin-Morales,   jorge@unizar.es
 
-THEORY: Let K be a field of characteristic 0. Given a polynomial ring
-      R = K[x_1,...,x_n] and 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 for symbolic discrete
-      (that is shiftable) variables s_1,..., s_r.
-      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
-      - 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>
-      - S is the universal enveloping algebra of gl_r, generated by s_{ij}, where s_{ii}=s_i.
- One is interested in the following data:
- - the left ideal Ann F^s in D<S>, usually denoted by LD in the output
- - global Bernstein polynomial in one variable s = s_1 + ...+ s_r, denoted by bs,
- - its minimal integer root s0, the list of all roots of bs, which are known
-   to be rational, with their multiplicities, which is denoted by BS
- - an r-tuple of operators in D<S>, denoted by PS, such that the functional equality
-     sum(k=1 to k=r) P_k*f_k*F^s = bs*F^s holds in R[1/F]*F^s.
+THEORY: Let K be a field of characteristic 0. Given a polynomial ring R = K[x_1,...,x_n] and 
+@* 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 
+@* for symbolic discrete (that is shiftable) variables s_1,..., s_r.
+@* 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
+@*   - 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>
+@*   - S is the universal enveloping algebra of gl_r, generated by s_{ij}, where s_{ii}=s_i.
+@* One is interested in the following data:
+@*   - the left ideal Ann F^s in D<S>, usually denoted by LD in the output
+@*   - global Bernstein polynomial in one variable s = s_1 + ...+ s_r, denoted by bs,
+@*   - its minimal integer root s0, the list of all roots of bs, which are known
+@*     to be negative rational numbers, with their multiplicities, which is denoted by BS
+@*   - an r-tuple of operators in D<S>, denoted by PS, such that the functional equality
+@*     sum(k=1 to k=r) P_k*f_k*F^s = bs*F^s holds in R[1/F]*F^s.
 
 REFERENCES:
@@ -36 +34 @@
 AUXILIARY PROCEDURES:
 makeIF(F[,ORD]);    create the Malgrange ideal, associated with F = F[1],..,F[P]
+
+SEE ALSO: bfun_lib, dmod_lib, dmodapp_lib, gmssing_lib
+
+KEYWORDS: D-module; D-module structure; Bernstein-Sato polynomial for variety; global Bernstein-Sato polynomial for variety;
+Weyl algebra; parametric annihilator for variety; Budur-Mustata-Saito approach; initial ideal approach;
 ";
 
@@ -362 +365 @@
 
 proc bfctVarAnn (ideal F, list #)
-"USAGE:  bfctVarAnn(F[,gid,eng]);  F an ideal, gid,eng optional ints
-RETURN:  list of ideal and intvec
-PURPOSE: computes the roots of the Bernstein-Sato polynomial and their
-@*       multiplicities for an affine algebraic variety defined by
-@*        F = F[1],..,F[r].
+"USAGE:  bfctVarAnn(F[,gid,eng]); F an ideal, gid,eng optional ints
+RETURN:  list of an ideal and an intvec
+PURPOSE: computes the roots of the Bernstein-Sato polynomial and their multiplicities 
+@*       for an affine algebraic variety defined by F = F[1],..,F[r].
 ASSUME:  The basering is a commutative polynomial ring in char 0.
 BACKGROUND: In this proc, the annihilator of f^s in D[s] is computed and then a
 @*       system of linear equations is solved by linear reductions in order to
-@*       find the minimal polynomial of S = s(1)(1) + ... + s(P)(P), following
-@*       the ideas by Noro.
-NOTE:    In the output list, the ideal contains all the roots and the invec their
-@*        multiplicities.
+@*       find the minimal polynomial of S = s(1)(1) + ... + s(P)(P)
+NOTE:    In the output list, the ideal contains all the roots and the intvec their multiplicities.
 @*       If gid<>0, the ideal is used as given. Otherwise, and by default, a
 @*       heuristically better suited generating set is used.