Changeset b1645e in git


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Timestamp:
Dec 19, 2013, 5:08:20 PM (11 years ago)
Author:
Oleksandr Motsak <motsak@…>
Branches:
(u'fieker-DuVal', '117eb8c30fc9e991c4decca4832b1d19036c4c65')(u'spielwiese', 'b52fc4b2495505785981d640dcf7eb3e456778ef')
Children:
f4f4033f0b5880920c8e4bfa1a3ab7c2f9e749b8
Parents:
789d6fd59d9c7d6f129c8226b0d7ad732069ada8
git-author:
Oleksandr Motsak <motsak@mathematik.uni-kl.de>2013-12-19 17:08:20+01:00
git-committer:
Oleksandr Motsak <motsak@mathematik.uni-kl.de>2013-12-19 17:09:18+01:00
Message:
Moved general descriptiomn to the header + references
File:
1 edited

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  • Singular/LIB/schreyer.lib

    r789d6f rb1645e  
    66AUTHOR:  Oleksandr Motsak <U@D>, where U={motsak}, D={mathematik.uni-kl.de}
    77KEYWORDS: Schreyer ordering; Schreyer resolution; syzygy
     8OVERVIEW:
     9@* The library contains helper procedures for computing a Schreyer resultion (cf. [SFO])
     10   for @code{derham.lib} also in non-commutative setting (cf. [MO]).
     11   We call a free resulution a Schreyer resolution if any higher syzygy is a Goebner base of a previous one
     12   with respect to the corresponding Schreyer ordering.
     13   Schreyer resolution can be much bigger than a minimal resolution of the same module one but can be easier to construct.
     14@* Input for resolution computations is a set of vectors @code{M} in form of a module over some basering @code{R}.
     15   The ring @code{R} may be non-commutative, in which case the ring ordering should be global.
     16@* These procedures produce/work with partial Schreyer resolutions of @code{(R^rank(M))/M} in form of
     17   a ring (endowed with a special ring ordering that will be extended in the course of a resolution computation)
     18   containing a list of modules @code{RES} and a module @code{MRES}:
     19@* The list of modules @code{RES} contains the images of maps (also called syzygies) substituting the
     20   computed beginning of a Schreyer resolution, that is, each syzygy module is a Groebner Basis
     21   of the previous one with respect to corresponding Schreyer induced ordering.
     22@* The list @code{RES} starts with a zero map given by @code{rank(M)} zero generators indicating that the image of
     23   the first differential map is zero. The second map @code{RES[2]} is given by @code{M}, which indicates that
     24   the resolution is of @code{(R^rank(M))/M} is being computed.
     25@* The module @code{MRES} is a direct sum of modules from @code{RES} and thus comprises all computed differentials.
     26@* Syzygies are shifted so that @code{gen(i)} is mapped to @code{MRES[i]} under the differential map.
     27@* Schreyer ordering extends the starting module ordering on @code{M} (defined in Singular by the basering @code{R})
     28   and is extended to higher syzygies using the following definition:
     29@*        a < b if and only if (d(a) < d(b)) OR ( (d(a) = d(b) AND (comp(a) < comp(b)) ),
     30@* where @code{d(a)} is the image of a under the differential (given by @code{MRES}),
     31   and @code{comp(a)} is the mod. component, for any module terms @code{a} and @code{b} from the same free module.
     32REFERENCES:
     33[SFO] Schreyer, F.O.: Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrassschen Divisionssatz,
     34      Master's thesis, Univ. Hamburg, 1980.
     35[MO]  Motsak, O.: Non-commutative Computer Algebra with applications: Graded commutative algebra and related
     36      structures in Singular with applications, Ph.D. thesis, TU Kaiserslautern, 2010
     37
     38NOTE:  requires the dynamic or built-in module @code{syzextra}
    839
    940PROCEDURES:
    10 Sres(M,len)     compute Schreyer resolution of module M of maximal length len
    11 Ssyz(M)         compute Schreyer resolution of module M of length 1
    12 Scontinue(len)  extend currently active resolution by (at most) len syszygies
    13 
    14 NOTE:  requires the dynamic or built-in module: syzextra
     41  Sres(M,len)     compute Schreyer resolution of module M of maximal length len
     42  Ssyz(M)         compute Schreyer resolution of module M of length 1
     43  Scontinue(len)  extend currently active resolution by (at most) len syszygies
    1544";
    1645
     
    332361RETURN:  nothing, instead it changes the currently active resolution
    333362PURPOSE: extends the currently active resolution by at most len syzygies
    334 NOTE:    must be used within a ring returned by Sres or Ssyz
     363ASSUME:  must be used within a ring returned by Sres or Ssyz
    335364EXAMPLE: example Scontinue; shows an example
    336365"
     
    362391"USAGE:  Ssyz(module M)
    363392RETURN:  ring, containing a Schreyer resolution
    364 PURPOSE: computes a Schreyer resolution of M of length 1
    365 NOTE:    the output is explained in Sres
     393PURPOSE: computes a Schreyer resolution of M of length 1 (see the library overview)
     394SEE ALSO: Sres
    366395EXAMPLE: example Ssyz; shows an example
    367396"
     
    394423"USAGE:  Sres(module M, int len)
    395424RETURN:  ring, containing a Schreyer resolution
    396 PURPOSE: computes a Schreyer resolution of (basering^rank(M))/M with at most len syzygy modules
    397 NOTE:    input is a set of vectors M over a basering. The ring basering may be non-commutative.
    398 @*       If given len is zero then nvars(basering) + 1 is used instead.
    399 @*       Schreyer resolution is represented by a list of modules RES and a module MRES
    400          belonging to a specially constructed ring, which is endowed with a Schreyer ordering.
    401 @*       The list of modules RES contains the images of maps (also called syzygies) subsituting the
    402          computed beginning of a Schreyer free resolution of (baseRing^rank(M))/M.
    403 @*       The leading zero map RES[1] with rank(M) zero generators indicates that the image of
    404          the first differential map is zero. The second map RES[2] is given by M, which indicates that
    405          the resolution is of (baseRing^rank(M))/M is being computed.
    406 @*       The module MRES is a direct sum of modules from RES and comprises all computed differential maps.
    407 @*       Syzygies are shifted so that gen(i) is mapped to MRES[i] under the differential.
    408 @*       Schreyer ordering extends an arbitrary starting module ordeing (defined by basering)
    409          and is extended to higher syzygt modules using the following definition:
    410 @*       a < b if and only if (d(a) < d(b)) OR ( (d(a) = d(b) AND (comp(a) < comp(b)) ),
    411 @*       where d(a) is the image of a under the differential (given by MRES),
    412          and comp(a) is the mod. component, for any module terms a and b.
    413 @*       Syzygies are given by Groebner bases with respect to corresponding Schreyer orderings.
     425PURPOSE: computes a Schreyer resolution of M of length at most len (see the library overview)
     426NOTE:    If given len is zero then nvars(basering) + 1 is used instead.
     427SEE ALSO: Ssyz
    414428EXAMPLE: example Sres; shows an example
    415429"
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