Changeset 7abf40 in git


Ignore:
Timestamp:
Feb 18, 2011, 10:09:44 AM (13 years ago)
Author:
Frank Seelisch <seelisch@…>
Branches:
(u'spielwiese', '8e0ad00ce244dfd0756200662572aef8402f13d5')
Children:
607fc78e355f385cd271cf9b760410b83c6a6a75
Parents:
673ee3c38462387be8b4ece27179dfbf664273ba
Message:
new lib version from Mathias Schulze

git-svn-id: file:///usr/local/Singular/svn/trunk@13865 2c84dea3-7e68-4137-9b89-c4e89433aadc
Location:
Singular/LIB
Files:
2 edited

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  • Singular/LIB/gmspoly.lib

    r673ee3 r7abf40  
    11///////////////////////////////////////////////////////////////////////////////
    2 version="$Id$";
     2version="$Id: gmspoly.lib 12529 2010-02-08 15:57:48Z seelisch $";
    33category="Singularities";
    44
    … …  
    66LIBRARY:  gmspoly.lib  Gauss-Manin System of Tame Polynomials
    77
    8 AUTHOR:   Mathias Schulze, email: mschulze@mathematik.uni-kl.de
    9 
    10 OVERVIEW: A library to compute invariants related to the Gauss-Manin system
    11           of a cohomologically tame polynomial
     8AUTHOR:   Mathias Schulze, mschulze at mathematik.uni-kl.de
     9
     10OVERVIEW:
     11A library for computing the Gauss-Manin system of a cohomologically tame
     12polynomial f. Schulze's algorithm [Sch05], based on Sabbah's theory [Sab98],
     13is used to compute a good basis of (the Brieskorn lattice of) the Gauss-Manin system and the differential operation of f in terms of this basis.
     14In addition, there is a test for tameness in the sense of Broughton.
     15Tame polynomials can be considered as an affine algebraic analogue of local
     16analytic isolated hypersurface singularities. They have only finitely many
     17citical points, and those at infinity do not give rise to atypical values
     18in a sense depending on the precise notion of tameness considered. Well-known
     19notions of tameness like tameness, M-tameness, Malgrange-tameness, and
     20cohomological tameness, and their relations, are reviewed in [Sab98,§8].
     21For ordinary tameness, see Broughton [Bro88,§3].
     22Sabbah [Sab98] showed that the Gauss-Manin system, the D-module direct image
     23of the structure sheaf, of a cohomologically tame polynomial carries a
     24similar structure as in the isolated singularity case, coming from a Mixed
     25Hodge structure on the cohomology of the Milnor (typical) fibre (see
     26gmssing.lib). The data computed by this library encodes the differential structure of the Gauss-Manin system, and the Mixed Hodge structure of the Milnor fibre over the complex numbers. As a consequence, it yields the Hodge numbers, spectral pairs, and monodromy at infinity.
     27
     28REFERENCES:
     29[Bro88] S. Broughton: Milnor numbers and the topology of polynomial
     30        hypersurfaces. Inv. Math. 92 (1988) 217-241.
     31[Sab98] C. Sabbah: Hypergeometric periods for a tame polynomial.
     32        arXiv.org math.AG/9805077.
     33[Sch05] M. Schulze: Good bases for tame polynomials.
     34        J. Symb. Comp. 39,1 (2005), 103-126.
    1235
    1336PROCEDURES:
    14   isTame(f);  test if the polynomial f is tame
    15   goodBasis(f);  a good basis of the Brieskorn lattice of a cohomologically tame f
     37  isTame(f);     test whether the polynomial f is tame
     38  goodBasis(f);  good basis of Brieskorn lattice of cohom. tame polynomial f
    1639
    1740SEE ALSO: gmssing_lib
    … …  
    6083@format
    6184int k=
     85  1;  if f is tame in the sense of Broughton [Bro88,§3]
    6286  0;  if f is not tame
    63   1;  if f is tame
    6487@end format
     88REMARKS:  procedure implements Proposition 3.1 in [Bro88]
    6589KEYWORDS: tame polynomial
    6690EXAMPLE:  example isTame; shows examples
    … …  
    138162///////////////////////////////////////////////////////////////////////////////
    139163
    140 static proc vfiltTuple(matrix B,int d)
     164static proc vfiltmat(matrix B,int d)
    141165{
    142166  int mu=ncols(B);
    … …  
    426450proc goodBasis(poly f)
    427451"USAGE:    goodBasis(f); poly f
    428 ASSUME:   f is cohomologically tame
     452ASSUME:   f is cohomologically tame in the sense of Sabbah [Sab98,§8]
    429453RETURN:
    430454@format
    … …  
    433457  matrix A;  A(s)=A0+s*A1 and t[matrix(b)]=[matrix(b)](A(s)+s^2*(d/ds))
    434458@end format
     459REMARKS:  procedure implements Algorithm 6 in [Sch05]
    435460KEYWORDS: tame polynomial; Gauss-Manin system; Brieskorn lattice;
    436461          mixed Hodge structure; V-filtration; weight filtration;
    437462          monodromy; spectrum; spectral pairs; good basis
     463SEE ALSO: gmssing_lib
    438464EXAMPLE:  example goodBasis; shows examples
    439465"
    … …  
    517543        B=pidbasis(K2,L2);
    518544      }
     545
    519546      B0=std(coeffs(reduce(matrix(K,nrows(K),nrows(B))*B,JS),b0));
    520547      b=matrix(K,nrows(K),nrows(B))*B;
    … …  
    527554    setring(@S);
    528555
    529     e0,s0,V,B=vfiltTuple(imap(@XS,A),d);
     556    e0,s0,V,B=vfiltmat(imap(@XS,A),d);
    530557    a,e0,e,s,V,B,G=spec(e0,s0,V,B);
    531558

  • Singular/LIB/gmssing.lib

    r673ee3 r7abf40  
    11///////////////////////////////////////////////////////////////////////////////
    2 version="$Id$";
     2version="$Id: gmssing.lib 13548 2010-10-19 07:45:40Z hannes $";
    33category="Singularities";
    44
    … …  
    66LIBRARY: gmssing.lib  Gauss-Manin System of Isolated Singularities
    77
    8 AUTHOR:   Mathias Schulze, email: mschulze@mathematik.uni-kl.de
    9 
    10 OVERVIEW: A library to compute invariants related to the the Gauss-Manin system
    11           of an isolated hypersurface singularity
     8AUTHOR:  Mathias Schulze, mschulze at mathematik.uni-kl.de
     9
     10OVERVIEW:
     11A library for computing invariants related to the Gauss-Manin system of an
     12isolated hypersurface singularity.
     13
     14REFERENCES:
     15[Sch01] M. Schulze: Algorithms for the Gauss-Manin connection. J. Symb. Comp.
     16        32,5 (2001), 549-564.
     17[Sch02] M. Schulze: The differential structure of the Brieskorn lattice.
     18        In: A.M. Cohen et al.: Mathematical Software - ICMS 2002.
     19        World Scientific (2002).
     20[Sch03] M. Schulze: Monodromy of Hypersurface Singularities.
     21        Acta Appl. Math. 75 (2003), 3-13.
     22[Sch04] M. Schulze: A normal form algorithm for the Brieskorn lattice.
     23        J. Symb. Comp. 38,4 (2004), 1207-1225.
    1224
    1325PROCEDURES:
    … …  
    3850KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice;
    3951          mixed Hodge structure; V-filtration; weight filtration;
    40           Bernstein-Sato polynomial; local Bernstein-Sato polynomial;
    41           monodromy; spectrum; spectral pairs; good basis
     52          Bernstein-Sato polynomial; monodromy; spectrum; spectral pairs;
     53          good basis
    4254";
    4355
    … …  
    590602@end format
    591603KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice;
    592           Bernstein-Sato polynomial; local Bernstein-Sato polynomial
     604          Bernstein-Sato polynomial
    593605EXAMPLE:  example bernstein; shows examples
    594606"
    … …  
    12401252proc sppnf(list sp)
    12411253"USAGE:   sppnf(list(a,w[,m])); ideal a, intvec w, intvec m
    1242 ASSUME:  ncols(e)==size(w)==size(m)
     1254ASSUME:  ncols(a)==size(w)==size(m)
    12431255RETURN:  order (a[i][,w[i]]) with multiplicity m[i] lexicographically
    12441256EXAMPLE: example sppnf; shows examples
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