Changeset fd5013 in git


Ignore:
Timestamp:
Aug 2, 2006, 5:40:53 PM (18 years ago)
Author:
Hans Schönemann <hannes@…>
Branches:
(u'spielwiese', '17f1d200f27c5bd38f5dfc6e8a0879242279d1d8')
Children:
f6f1dbfc1e8487ccbf1ca0d7a3d2600069315a9f
Parents:
d1932987874a4523d2a42efe3046b953ed52af76
Message:
*hannes/markwig: typos, format, doc


git-svn-id: file:///usr/local/Singular/svn/trunk@9385 2c84dea3-7e68-4137-9b89-c4e89433aadc
Location:
Singular/LIB
Files:
8 edited

Legend:

Unmodified
Added
Removed

  • Singular/LIB/finvar.lib

    rd193298 rfd5013  
    11///////////////////////////////////////////////////////////////////////////////
    2 version="$Id: finvar.lib,v 1.50 2006-07-18 15:48:13 Singular Exp $"
     2version="$Id: finvar.lib,v 1.51 2006-08-02 15:40:47 Singular Exp $"
    33category="Invariant theory";
    44info="
    … …  
    138138RETURN:  a <list>, the first list element will be a gxn <matrix> representing
    139139         the Reynolds operator if we are in the non-modular case; if the
    140          characteristic is >0, minpoly==0 and the finite group non-cyclic the
     140         characteristic is >0, minpoly==0 and the finite group is non-cyclic the
    141141         second list element is an <int> giving the lowest common multiple of
    142142         the matrix group elements' order (used in molien); in general all
    … …  
    148148         (or the generators themselves during the first run). All the ones that
    149149         have been generated before are thrown out and the program terminates
    150          when no new elements found in one run. Additionally each time a new
     150         when no new elements are found in one run. Additionally each time a new
    151151         group element is found the corresponding ring mapping of which the
    152152         Reynolds operator is made up is generated. They are stored in the rows
    … …  
    358358         elements generated by group_reynolds(), lcm is the second return value
    359359         of group_reynolds()
    360 RETURN:  in case of characteristic 0 a 1x2 <matrix> giving enumerator and
     360RETURN:  in case of characteristic 0 a 1x2 <matrix> giving numerator and
    361361         denominator of Molien series; in case of prime characteristic a ring
    362362         with the name `ringname` of characteristic 0 is created where the same
    … …  
    908908         new group element is found the corresponding ring mapping of which the
    909909         Reynolds operator is made up is generated. They are stored in the rows
    910          of the first return value. In characteristic 0 the terms 1/det(1-xE)
     910         of the first return value. In characteristic 0 the term 1/det(1-xE)
    911911         is computed whenever a new element E is found. In prime characteristic
    912912         a Brauer lift is involved and the terms are only computed after the
    913913         entire matrix group is generated (to avoid the modular case). The
    914          returned matrix gives enumerator and denominator of the expanded
     914         returned matrix gives numerator and denominator of the expanded
    915915         version where common factors have been canceled.
    916916EXAMPLE: example reynolds_molien; shows an example
    … …  
    20642064ASSUME:  REY is the first return value of group_reynolds or reynolds_molien and
    20652065         M the one of molien or the second one of reynolds_molien
    2066 DISPLAY: information about the various stages of the programme if v does not
     2066DISPLAY: information about the various stages of the program if v does not
    20672067         equal 0
    20682068RETURN:  primary invariants (type <matrix>) of the invariant ring
    … …  
    22072207         ringname gives the name of a ring of characteristic 0 that has been
    22082208         created by molien or reynolds_molien
    2209 DISPLAY: information about the various stages of the programme if v does not
     2209DISPLAY: information about the various stages of the program if v does not
    22102210         equal 0
    22112211RETURN:  primary invariants (type <matrix>) of the invariant ring
    … …  
    23602360         <int>
    23612361ASSUME:  REY is the first return value of group_reynolds or reynolds_molien
    2362 DISPLAY: information about the various stages of the programme if v does not
     2362DISPLAY: information about the various stages of the program if v does not
    23632363         equal 0
    23642364RETURN:  primary invariants (type <matrix>) of the invariant ring and an
    … …  
    25032503         <int>
    25042504ASSUME:  REY is the first return value of group_reynolds or reynolds_molien
    2505 DISPLAY: information about the various stages of the programme if v does not
     2505DISPLAY: information about the various stages of the program if v does not
    25062506         equal 0
    25072507RETURN:  primary invariants (type <matrix>) of the invariant ring  and an
    … …  
    26442644         G1,G2,...: <matrices> generating a finite matrix group, v: an optional
    26452645         <int>
    2646 DISPLAY: information about the various stages of the programme if v does not
     2646DISPLAY: information about the various stages of the program if v does not
    26472647         equal 0
    26482648RETURN:  primary invariants (type <matrix>) of the invariant ring
    … …  
    27842784         G1,G2,...: <matrices> generating a finite matrix group, flags: an
    27852785         optional <intvec> with three entries, if the first one equals 0 (also
    2786          the default), the programme attempts to compute the Molien series and
    2787          Reynolds operator, if it equals 1, the programme is told that the
     2786         the default), the program attempts to compute the Molien series and
     2787         Reynolds operator, if it equals 1, the program is told that the
    27882788         Molien series should not be computed, if it equals -1 characteristic 0
    27892789         is simulated, i.e. the Molien series is computed as if the base field
    27902790         were characteristic 0 (the user must choose a field of large prime
    2791          characteristic, e.g. 32003) and if the first one is anything else, it
     2791         characteristic, e.g. 32003), and if the first one is anything else, it
    27922792         means that the characteristic of the base field divides the group
    27932793         order, the second component should give the size of intervals between
    … …  
    27972797         common factors should always be canceled when the expansion is simple
    27982798         (the root of the extension field occurs not among the coefficients)
    2799 DISPLAY: information about the various stages of the programme if the third
     2799DISPLAY: information about the various stages of the program if the third
    28002800         flag does not equal 0
    28012801RETURN:  primary invariants (type <matrix>) of the invariant ring and if
    … …  
    28922892        else
    28932893        { if (v)
    2894           { "  Since it is impossible for this programme to calculate the Molien series for";
     2894          { "  Since it is impossible for this program to calculate the Molien series for";
    28952895            "  invariant rings over extension fields of prime characteristic, we have to";
    28962896            "  continue without it.";
    … …  
    31993199ASSUME:  REY is the first return value of group_reynolds or reynolds_molien and
    32003200         M the one of molien or the second one of reynolds_molien
    3201 DISPLAY: information about the various stages of the programme if v does not
     3201DISPLAY: information about the various stages of the program if v does not
    32023202         equal 0
    32033203RETURN:  primary invariants (type <matrix>) of the invariant ring
    … …  
    33463346         ringname gives the name of a ring of characteristic 0 that has been
    33473347         created by molien or reynolds_molien
    3348 DISPLAY: information about the various stages of the programme if v does not
     3348DISPLAY: information about the various stages of the program if v does not
    33493349         equal 0
    33503350RETURN:  primary invariants (type <matrix>) of the invariant ring
    … …  
    34993499         bases elements, v: an optional <int>
    35003500ASSUME:  REY is the first return value of group_reynolds or reynolds_molien
    3501 DISPLAY: information about the various stages of the programme if v does not
     3501DISPLAY: information about the various stages of the program if v does not
    35023502         equal 0
    35033503RETURN:  primary invariants (type <matrix>) of the invariant ring  and an
    … …  
    36463646         bases elements, v: an optional <int>
    36473647ASSUME:  REY is the first return value of group_reynolds or reynolds_molien
    3648 DISPLAY: information about the various stages of the programme if v does not
     3648DISPLAY: information about the various stages of the program if v does not
    36493649         equal 0
    36503650RETURN:  primary invariants (type <matrix>) of the invariant ring  and an
    … …  
    37923792         where -|r| to |r| is the range of coefficients of the random
    37933793         combinations of bases elements, v: an optional <int>
    3794 DISPLAY: information about the various stages of the programme if v does not
     3794DISPLAY: information about the various stages of the program if v does not
    37953795         equal 0
    37963796RETURN:  primary invariants (type <matrix>) of the invariant ring
    … …  
    39433943         where -|r| to |r| is the range of coefficients of the random
    39443944         combinations of bases elements, flags: an optional <intvec> with three
    3945          entries, if the first one equals 0 (also the default), the programme
     3945         entries, if the first one equals 0 (also the default), the program
    39463946         attempts to compute the Molien series and Reynolds operator, if it
    3947          equals 1, the programme is told that the Molien series should not be
     3947         equals 1, the program is told that the Molien series should not be
    39483948         computed, if it equals -1 characteristic 0 is simulated, i.e. the
    39493949         Molien series is computed as if the base field were characteristic 0
    39503950         (the user must choose a field of large prime characteristic, e.g.
    3951          32003) and if the first one is anything else, it means that the
     3951         32003), and if the first one is anything else, it means that the
    39523952         characteristic of the base field divides the group order, the second
    39533953         component should give the size of intervals between canceling common
    … …  
    39573957         always be canceled when the expansion is simple (the root of the
    39583958         extension field does not occur among the coefficients)
    3959 DISPLAY: information about the various stages of the programme if the third
     3959DISPLAY: information about the various stages of the program if the third
    39603960         flag does not equal 0
    39613961RETURN:  primary invariants (type <matrix>) of the invariant ring and if
    … …  
    40604060        else
    40614061        { if (v)
    4062           { "  Since it is impossible for this programme to calculate the Molien series for";
     4062          { "  Since it is impossible for this program to calculate the Molien series for";
    40634063            "  invariant rings over extension fields of prime characteristic, we have to";
    40644064            "  continue without it.";
    … …  
    59545954         Molien series is computed as if the base field were characteristic 0
    59555955         (the user must choose a field of large prime characteristic, e.g.
    5956          32003) and if the first one is anything else, it means that the
     5956         32003), and if the first one is anything else, it means that the
    59575957         characteristic of the base field divides the group order (i.e. it will
    59585958         not even be attempted to compute the Reynolds operator or Molien
    … …  
    60516051        else
    60526052        { if (v)
    6053           { "  Since it is impossible for this programme to calculate the Molien
     6053          { "  Since it is impossible for this program to calculate the Molien
    60546054 series for";
    60556055            "  invariant rings over extension fields of prime characteristic, we
    … …  
    61716171         i.e. the Molien series is computed as if the base field were
    61726172         characteristic 0 (the user must choose a field of large prime
    6173          characteristic, e.g.  32003) and if the first one is anything else,
     6173         characteristic, e.g.  32003), and if the first one is anything else,
    61746174         then the characteristic of the base field divides the group order
    61756175         (i.e. we will not even attempt to compute the Reynolds operator or
    … …  
    62866286        else
    62876287        { if (v)
    6288           { "  Since it is impossible for this programme to calculate the Molien
     6288          { "  Since it is impossible for this program to calculate the Molien
    62896289 series for";
    62906290            "  invariant rings over extension fields of prime characteristic, we
    … …  
    64036403THEORY:  The ideal of algebraic relations of the invariant ring generators is
    64046404         calculated, then the variables of the original ring are eliminated and
    6405          the polynomials that are left over define the orbit variety
     6405         the polynomials that are left over define the orbit variety.
    64066406EXAMPLE: example orbit_variety; shows an example
    64076407"
    … …  
    66206620@*       s: a <string> giving a name for a new ring
    66216621RETURN:  The procedure ends with a new ring named s.
    6622          It contains a Groebner basis
    6623          (type <ideal>, named G) for the ideal defining the
    6624          relative orbit variety with respect to I in the new ring.
     6622         It contains a Groebner basis (type <ideal>, named G) for the ideal
     6623         defining the relative orbit variety with respect to I in the new ring.
    66256624THEORY:  A Groebner basis of the ideal of algebraic relations of the invariant
    66266625         ring generators is calculated, then one of the basis elements plus the
    … …  
    67076706@*       F: a 1xm <matrix> defining an invariant ring of some matrix group
    67086707RETURN:  The <ideal> defining the image under that group of the variety defined
    6709          by I
     6708         by I.
    67106709THEORY:  rel_orbit_variety(I,F) is called and the newly introduced
    67116710@*       variables in the output are replaced by the generators of the
    67126711@*       invariant ring. This ideal in the original variables defines the image
    6713 @*       of the variety defined by I
     6712@*       of the variety defined by I.
    67146713EXAMPLE: example image_of_variety; shows an example
    67156714"

  • Singular/LIB/gmspoly.lib

    rd193298 rfd5013  
    11///////////////////////////////////////////////////////////////////////////////
    2 version="$Id: gmspoly.lib,v 1.13 2005-05-11 13:02:32 Singular Exp $";
     2version="$Id: gmspoly.lib,v 1.14 2006-08-02 15:40:50 Singular Exp $";
    33category="Singularities";
    44
    … …  
    146146  attrib(V0,"isSB",1);
    147147  module V1=B;
    148   option("redSB");
     148  option(redSB);
    149149  while(size(reduce(V1,V0))>0)
    150150  {

  • Singular/LIB/gmssing.lib

    rd193298 rfd5013  
    11///////////////////////////////////////////////////////////////////////////////
    2 version="$Id: gmssing.lib,v 1.9 2005-05-18 18:16:32 Singular Exp $";
     2version="$Id: gmssing.lib,v 1.10 2006-08-02 15:40:50 Singular Exp $";
    33category="Singularities";
    44
    … …  
    220220RETURN:
    221221  list nf;
    222   ideal nf[1];  projection of p to <gmsbasis>C{{s}} mod s^(K+1)
     222  ideal nf[1];  projection of p to <gmsbasis>C{{s}} mod s^(K+1) @*
    223223  ideal nf[2];  p==nf[1]+nf[2]
    224 NOTE:     computation can be continued by setting p=nf[2]
     224NOTE:     computation can be continued by setting p to nf[2][1]
    225225EXAMPLE:  example gmsnf; shows examples
    226226"
    … …  
    301301  ideal l[2];  p==matrix(gmsbasis)*l[1]+l[2]
    302302@end format
    303 NOTE:     computation can be continued by setting p=l[2]
     303NOTE:     computation can be continued by setting p to l[2]
    304304EXAMPLE:  example gmscoeffs; shows examples
    305305"
    … …  
    580580ASSUME:   characteristic 0; local degree ordering;
    581581          isolated critical point 0 of t
    582 RETURN:   ideal r;  roots of the Bernstein polynomial b excluding the root -1
     582RETURN:   list: 
     583          roots of the Bernstein polynomial b (ideal) and its multiplicies
    583584NOTE:     the roots of b are negative rational numbers and -1 is a root of b
    584585KEYWORDS: Bernstein polynomial
    … …  
    12321233proc sppnf(list sp)
    12331234"USAGE:   sppnf(list(a,w[,m])); ideal a, intvec w, intvec m
    1234 ASSUME:  ncols(e)==size(w)==size(m)
     1235ASSUME:  ncols(a)==size(w)==size(m)
    12351236RETURN:  order (a[i][,w[i]]) with multiplicity m[i] lexicographically
    12361237EXAMPLE: example sppnf; shows examples

  • Singular/LIB/hnoether.lib

    rd193298 rfd5013  
    1 version="$Id: hnoether.lib,v 1.52 2006-07-18 15:48:19 Singular Exp $";
     1version="$Id: hnoether.lib,v 1.53 2006-08-02 15:40:51 Singular Exp $";
    22category="Singularities";
    33info="
    … …  
    16281628   contains the multiplicities of the branches at their infinitely near point
    16291629   of 0 in its (i-1) order neighbourhood (i.e., i=1: multiplicity of the
    1630    branches themselves, i=2: multiplicity of their 1st quadratic transformed,
     1630   branches themselves, i=2: multiplicity of their 1st quadratic transform,
    16311631   etc., @*
    16321632   Hence, @code{multsequence(INPUT)[1][*,j]} is the multiplicity sequence
    … …  
    17881788     with:
    17891789       @code{a_1 , ... , a_n} the sequence of multiplicities of the 1st branch,
    1790        @code{[...]} the multiplicities of the j-th transformed of all branches,
     1790       @code{[...]} the multiplicities of the j-th transform of all branches,
    17911791       @code{(...)} indicating branches meeting in an infinitely near point.
    17921792@end format
    … …  
    22422242         mu (optional) is Milnor number of f.@*
    22432243         NP (optional) is output of @code{newtonpoly(f)}.
    2244 RETURN:  int: 1 if f in Newton non-degenerate, 0 otherwise.
     2244RETURN:  int: 1 if f is Newton non-degenerate, 0 otherwise.
    22452245SEE ALSO: newtonpoly
    22462246KEYWORDS: Newton non-degenerate; Newton polygon
    … …  
    42364236RETURN:  int, the delta invariant of the singularity at 0, that is, the vector
    42374237         space dimension of R~/R, (R~ the normalization of the local ring of
    4238          the singularity.
     4238         the singularity).
    42394239NOTE:    In case the Hamburger-Noether expansion of the curve f is needed
    42404240         for other purposes as well it is better to calculate this first

  • Singular/LIB/qhmoduli.lib

    rd193298 rfd5013  
    3939           moduli space (note : Spec(R) is the moduli space)
    4040OPTIONS: 1 compute equations of the mod. space,
    41          2 use a primary decomposition
    42          4 compute E_f0, i.e., the image of G_f0
    43          To combine options, add their value, default: opt =7
     41         2 use a primary decomposition,
     42         4 compute E_f0, i.e., the image of G_f0,
     43         to combine options, add their value, default: opt =7
    4444EXAMPLE: example ModEqn; shows an example
    4545"
    … …  
    202202            if opt = 0, 2, it is the ideal defining the equivariant embedding
    203203OPTIONS: 1 compute equations of the quotient,
    204          2 use a primary decomposition when computing the Reynolds operator @*
    205          To combine options, add their value, default: opt =3.
     204         2 use a primary decomposition when computing the Reynolds operator,@*
     205         to combine options, add their value, default: opt =3.
    206206EXAMPLE: example QuotientEquations; shows an example
    207207"
    … …  
    410410
    411411proc StabOrder(list #)
    412 "USAGE:   StabOrder(f); poly f;
     412"USAGE:   StabOrder(f); poly f
    413413PURPOSE: compute the order of the stabilizer group of f.
    414414ASSUME:  f quasihomogeneous polynomial with an isolated singularity at 0
    … …  
    433433PURPOSE: compute the equations of the isometry group of f.
    434434ASSUME:  f semiquasihomogeneous polynomial with an isolated singularity at 0
    435 RETURN:  list of two ring 'S1', 'S2'
     435RETURN:  list of two rings 'S1', 'S2'
    436436         - 'S1' contians the equations of the stabilizer (ideal 'stabid') @*
    437437         - 'S2' contains the action of the stabilizer (ideal 'actionid')
    … …  
    469469proc StabEqnId(ideal data, intvec wt)
    470470"USAGE:   StabEqn(I, w); I ideal, w intvec
    471 PURPOSE: compute the equations of the isometry group of the ideal I
     471PURPOSE: compute the equations of the isometry group of the ideal I,
    472472         each generator of I is fixed by the stabilizer.
    473473ASSUME:  I semiquasihomogeneous ideal wrt 'w' with an isolated singularity at 0
    474 RETURN:  list of two ring 'S1', 'S2'
     474RETURN:  list of two rings 'S1', 'S2'
    475475         - 'S1' contians the equations of the stabilizer (ideal 'stabid') @*
    476476         - 'S2' contains the action of the stabilizer (ideal 'actionid')
    … …  
    14171417
    14181418proc Max(data)
    1419 "USAGE:   Max(data); intvec/list of integers data
     1419"USAGE:   Max(data); intvec/list of integers
    14201420PURPOSE: find the maximal integer contained in 'data'
    14211421RETURN:  list
    … …  
    14311431  return(max);
    14321432}
     1433example
     1434{"EXAMPLE:";  echo = 2;
     1435  Max(list(1,2,3));
     1436}
    14331437
    14341438///////////////////////////////////////////////////////////////////////////////
    14351439
    14361440proc Min(data)
    1437 "USAGE:   Min(data); intvec/list of integers data
     1441"USAGE:   Min(data); intvec/list of integers
    14381442PURPOSE: find the minimal integer contained in 'data'
    14391443RETURN:  list
    … …  
    14491453  return(min);
    14501454}
    1451 
    1452 ///////////////////////////////////////////////////////////////////////////////
     1455example
     1456{"EXAMPLE:";  echo = 2;
     1457  Min(intvec(1,2,3));
     1458}
     1459
     1460///////////////////////////////////////////////////////////////////////////////

  • Singular/LIB/sing.lib

    rd193298 rfd5013  
    1 // $Id: sing.lib,v 1.29 2005-05-06 14:39:12 hannes Exp $
     1// $Id: sing.lib,v 1.30 2006-08-02 15:40:52 Singular Exp $
    22//(GMG/BM, last modified 26.06.96)
    33///////////////////////////////////////////////////////////////////////////////
    4 version="$Id: sing.lib,v 1.29 2005-05-06 14:39:12 hannes Exp $";
     4version="$Id: sing.lib,v 1.30 2006-08-02 15:40:52 Singular Exp $";
    55category="Singularities";
    66info="
    … …  
    182182"USAGE:   is_reg(f,id); f poly, id ideal or module
    183183RETURN:  1 if multiplication with f is injective modulo id, 0 otherwise
    184 NOTE:    let R be the basering and id a submodule of R^n. The procedure checks
     184NOTE:    Let R be the basering and id a submodule of R^n. The procedure checks
    185185         injectivity of multiplication with f on R^n/id. The basering may be a
    186          quotient ring
     186         quotient ring.
    187187EXAMPLE: example is_reg; shows an example
    188188"
    … …  
    214214"USAGE:   is_regs(i[,id]); i poly, id ideal or module (default: id=0)
    215215RETURN:  1 if generators of i are a regular sequence modulo id, 0 otherwise
    216 NOTE:    let R be the basering and id a submodule of R^n. The procedure checks
     216NOTE:    Let R be the basering and id a submodule of R^n. The procedure checks
    217217         injectivity of multiplication with i[k] on R^n/id+i[1..k-1].
    218          The basering may be a quotient ring
     218         The basering may be a quotient ring.
    219219         printlevel >=0: display comments (default)
    220220         printlevel >=1: display comments during computation
    … …  
    431431
    432432proc qhspectrum (poly f, intvec w)
    433 "USAGE:   qhspectrum(f,w);  f=poly, w=intvec;
     433"USAGE:   qhspectrum(f,w);  f=poly, w=intvec
    434434ASSUME:  f is a weighted homogeneous isolated singularity w.r.t. the weights
    435435         given by w; w must consist of as many positive integers as there
    … …  
    440440         d = w-degree(f)  and  si/d = i-th spectral-number(f)
    441441         No return value if basering has parameters or if f is no isolated
    442          singularity, displays a warning in this case
     442         singularity, displays a warning in this case.
    443443EXAMPLE: example qhspectrum; shows an example
    444444"
    … …  
    568568         all relevant information can be obtained. The most important are
    569569         probably vdim(T_1(id)); (which computes the Tjurina number),
    570          hilb(T_1(id)); and kbase(T_1(id));
    571          If T_1 is called with two argument, then matrix([2])*(kbase([1]))
     570         hilb(T_1(id)); and kbase(T_1(id)).
     571         If T_1 is called with two arguments, then matrix([2])*(kbase([1]))
    572572         represents a basis of 1st order semiuniversal deformation of id
    573573         (use proc 'deform', to get this in a direct way).
    574          For a complete intersection the proc Tjurina is faster
     574         For a complete intersection the proc Tjurina is faster.
    575575EXAMPLE: example T_1; shows an example
    576576"
    … …  
    653653DISPLAY: k-dimension of T_2(id) if printlevel >= 0 (default)
    654654NOTE:    The most important information is probably vdim(T_2(id)).
    655          Use proc miniversal to get equations of miniversal deformation.
     655         Use proc miniversal to get equations of the miniversal deformation.
    656656EXAMPLE: example T_2; shows an example
    657657"
    … …  
    727727DISPLAY: k-dimension of T_1 and T_2 if printlevel >= 0 (default)
    728728NOTE:    Use proc miniversal from deform.lib to get miniversal deformation of i,
    729          the list contains all objects used by proc miniversal
     729         the list contains all objects used by proc miniversal.
    730730EXAMPLE: example T_12; shows an example
    731731"
    … …  
    800800RETURN:  int, which is:
    801801         1. the codimension of id2 in id1, i.e. the vectorspace dimension of
    802             id1/id2 if id2 is contained in id1 and if this number is finite
    803          2. -1 if the dimension of id1/id2 is infinite
     802            id1/id2 if id2 is contained in id1 and if this number is finite@*
     803         2. -1 if the dimension of id1/id2 is infinite@*
    804804         3. -2 if id2 is not contained in id1,
    805 COMPUTE: consider the two hilberseries iv1(t) and iv2(t), then, in case 1.,
     805COMPUTE: consider the two Hilbert series iv1(t) and iv2(t), then, in case 1.,
    806806         q(t)=(iv2(t)-iv1(t))/(1-t)^n must be rational, and the result is the
    807807         sum of the coefficients of q(t) (n dimension of basering)
    … …  
    933933"USAGE:   tangentcone(id [,n]); id = ideal, n = int
    934934RETURN:  the tangent cone of id
    935 NOTE:    the procedure works for any monomial ordering.
    936          If n=0 use std w.r.t. local ordering ds, if n=1 use locstd
     935NOTE:    The procedure works for any monomial ordering.
     936         If n=0 use std w.r.t. local ordering ds, if n=1 use locstd.
    937937EXAMPLE: example tangentcone; shows an example
    938938"

  • Singular/LIB/spcurve.lib

    rd193298 rfd5013  
    11// (anne, last modified 31.5.99)
    22/////////////////////////////////////////////////////////////////////////////
    3 version="$Id: spcurve.lib,v 1.20 2005-05-06 14:39:18 hannes Exp $";
     3version="$Id: spcurve.lib,v 1.21 2006-08-02 15:40:53 Singular Exp $";
    44category="Singularities";
    55info="
    … …  
    274274"USAGE:   discr(sem,n);  sem ideal, n integer
    275275ASSUME:  sem is the versal deformation of an ideal of codimension 2. @*
    276          the first n variables of the ring are treated as variables
    277          all the others as parameters
     276         The first n variables of the ring are treated as variables
     277         all the others as parameters.
    278278RETURN:  ideal describing the discriminant
    279279NOTE:    This is not a powerful algorithm!

  • Singular/LIB/spectrum.lib

    rd193298 rfd5013  
    11///////////////////////////////////////////////////////////////////////////////
    2 version="$Id: spectrum.lib,v 1.18 2003-05-30 09:49:37 mschulze Exp $";
     2version="$Id: spectrum.lib,v 1.19 2006-08-02 15:40:53 Singular Exp $";
    33category="Singularities";
    44info="
    … …  
    5454  ring R=0,(x,y),ds;
    5555  poly f=x^31+x^6*y^7+x^2*y^12+x^13*y^2+y^29;
    56   spectrumnd(f);
     56  list s=spectrumnd(f);
     57  size(s[1]);
     58  s[1][22];
     59  s[2][22];
    5760}
    5861///////////////////////////////////////////////////////////////////////////////
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