source: git/Singular/LIB/standard.lib @ 6e11a25

//////////////////////////////////////////////////////////////////////////////
//major revision Jan/Feb. 2007, GMG
//groebner mit Optionen versehen
//////////////////////////////////////////////////////////////////////////////
version="$Id: standard.lib,v 1.93 2007-06-01 13:17:23 Singular Exp $";
category="Miscellaneous";
info="
LIBRARY: standard.lib   Procedures which are always loaded at Start-up

PROCEDURES:
 stdfglm(ideal[,ord])   standard basis of ideal via fglm [and ordering ord]
 stdhilb(ideal[,h])     Hilbert driven Groebner basis of ideal
 groebner(ideal,...)    standard basis using a heuristically chosen method
 res(ideal/module,[i])  free resolution of ideal or module
 sprintf(fmt,...)       returns fomatted string
 fprintf(link,fmt,..)   writes formatted string to link
 printf(fmt,...)        displays formatted string
 weightKB(stc,dd,vl)    degree dd part of a kbase wrt. some weigths
";
// qslimgb(i)             computes a standard basis with slimgb in a qring
// hilbRing([i])          create a ring containing the homogenized i
// par2varRing([i])       create a ring with pars to vars together with i
// quotientList(L,...)    a list, say QL, s.t. ring(QL) creates a correct qring

//////////////////////////////////////////////////////////////////////////////

proc stdfglm (i, list #)
"SYNTAX: @code{stdfglm (} ideal_expression @code{)} @*
         @code{stdfglm (} ideal_expression@code{,} string_expression @code{)}
TYPE:    ideal
PURPOSE: computes the standard basis of the ideal in the basering
         via @code{fglm} from the ordering given as the second argument
         to the ordering of the basering. If no second argument is given,
         \"dp\" is used. The standard basis for the given ordering (resp. for
         \"dp\") is computed via the command groebner except if a further
         argument \"std\" or \"slimgb\" is given in which case std resp.
         slimgb is used.
SEE ALSO: fglm, groebner, std, slimgb, stdhilb
KEYWORDS: fglm
EXAMPLE: example stdfglm; shows an example"
{
  if (nrows(i) > 1)
  {
    ERROR("first argument of 'stdfglm' must be an ideal");
  }
  string os;
  int s = size(#);
  def P= basering;
  string algorithm;
  int ii;
  for( ii=1; ii<=s; ii++)
  {
    if ( typeof(#[ii])== "string" )
    {
      if ( #[ii]=="std" || #[ii]=="slimgb" )
      {
        algorithm =  #[ii];
        # = delete(#,ii);
        s--;
        ii--;
      }
    }
  }

  if( s > 0 && (typeof(#[1]) == "string") )
  {
    os = #[1];
    ideal Qideal = ideal(P);
    int sQ = size(Qideal);
    int sM = size(minpoly);
    if ( sM!=0 )
    {
      string mpoly = string(minpoly);
    }
    if (sQ!=0 )
    {
      execute("ring Rfglm=("+charstr(P)+"),("+varstr(P)+"),"+os+";");
      ideal Qideal = fetch(P,Qideal);
      qring Pfglm = groebner(Qideal,"std","slimgb");
    }
    else
    {
      execute("ring Pfglm=("+charstr(P)+"),("+varstr(P)+"),"+os+";");
    }
    if ( sM!=0 )
    {
      execute("minpoly="+mpoly+";");
    }
  }
  else
  {
    list BRlist = ringlist(P);
    int nvarP = nvars(P);
    intvec w;                       //for ringweights of basering P
    int k;
    for(k=1;  k <= nvarP; k++)
    {
      w[k]=deg(var(k));
    }

    BRlist[3] = list();
    if( s==0 or (typeof(#[1]) != "string") )
    {
      if( w==1 )
      {
        BRlist[3][1]=list("dp",w);
      }
      else
      {
        BRlist[3][1]=list("wp",w);
      }
      BRlist[3][2]=list("C",intvec(0));
      def Pfglm = ring(quotientList(BRlist));
      setring Pfglm;
    }
  }
  ideal i = fetch(P,i);

  intvec opt = option(get);            //save options
  option(redSB);
  if (size(algorithm) > 0)
  {
    i = groebner(i,algorithm);
  }
  else
  {
    i = groebner(i);
  }
  option(set,opt);
  setring P;
  return (fglm(Pfglm,i));
}
example
{ "EXAMPLE:"; echo = 2;
   ring r  = 0,(x,y,z),lp;
   ideal i = y3+x2,x2y+x2,x3-x2,z4-x2-y;
   stdfglm(i);                   //uses fglm from "dp" (with groebner) to "lp"
   stdfglm(i,"std");             //uses fglm from "dp" (with std) to "lp"

   ring s  = (0,x),(y,z,u,v),lp;
   minpoly = x2+1;
   ideal i = u5-v4,zv-u2,zu3-v3,z2u-v2,z3-uv,yv-zu,yu-z2,yz-v,y2-u,u-xy2;
   weight(i);
   stdfglm(i,"(a(2,3,4,5),dp)"); //uses fglm from "(a(2,3,4,5),dp)" to "lp"
}

/////////////////////////////////////////////////////////////////////////////

proc stdhilb(i,list #)
"SYNTAX: @code{stdhilb (} ideal_expression @code{)} @*
         @code{stdhilb (} module_expression @code{)} @*
         @code{stdhilb (} ideal_expression@code{,} intvec_expression @code{)}
         @code{stdhilb (} ideal_expression@code{,} list of string_expressions
               and intvec_expressin @code{)} @*
TYPE:    type of the first argument
PURPOSE: Compute a Groebner basis of the ideal/module in the basering by
         using the Hilbert driven Groebner basis algorithm.
         If an argument of type string @code{\"std\"} resp. @code{\"slimgb\"}
         is given, the standard basis computation uses @code{std} or
         @code{slimgb}, otherwise a heuristically chosen method (default)
THEORY:  If the ideal is not homogeneous compute first a Groebner basis
         of the homogenization of the ideal, then the Hilbert function and,
         finally, a Groebner basis in the original ring by using the
         computed Hilbert function.@*
         If the ideal is homogeneous and a second argument of type intvec
         is given it will be used as 1st Hilbert function in the Hilbert
         driven algorithm.
NOTE:    'homogeneous' means weighted homogeneous with respect to the weights
         w[i] of the variables var(i) of the basering. Parameters are not
         converted to variables.
ASSUME:  The argument of type intvec is the 1st Hilbert series, computed
         by @code{hilb} using an intvector w, w[i]=deg(var(i)), as third
         argument
SEE ALSO: stdfglm, std, slimgb, groebner
KEYWORDS: Hilbert function
EXAMPLE: example stdhilb;  shows an example"
{

//--------------------- save data from basering --------------------------
  def P=basering;
  int nr = nrows(i);            //nr=1 iff i is an ideal
  ideal Qideal = ideal(P);      //defining the quotient ideal if P is a qring
  int was_qring;                //remembers if basering was a qring
  int is_homog =homog(i);       //check for homogeneity of i and Qideal
  if (size(Qideal) > 0)
  {
     was_qring = 1;
  }

  // save ordering of basering P for later use
  list ord_P = ringlist(P)[3];     //ordering of basering in ringlist
  string ordstr_P = ordstr(P);     //ordering of basering as string
  int nvarP = nvars(P);
  intvec w;                        //for ringweights of basering P
  int k;
  for(k=1;  k<=nvarP; k++)
  {
    w[k]=deg(var(k));
  }
  int neg=1-attrib (P,"global");

  //save options:
  int p_opt;
  string s_opt = option();
  if (find(s_opt, "prot"))  { p_opt = 1; }

//--------------------- check the given method ---------------------------
  string method;
  for (k=1; k<=size(#); k++)
  {
    if (typeof(#[k]) == "intvec")
    {
      intvec hi = #[k];
    }
    if (typeof(#[k]) == "string")
    {
      method = method + "," + #[k];
    }
  }

  if (npars(P) > 0)             //clear denominators of parameters
  {
    for( k=ncols(i); k>0; k-- )
    {
      i[k]=cleardenom(i[k]);
    }
  }

//---------- exclude cases to which stdhilb should no be applied  ----------
//Note that quotient ideal of qring must be homogeneous too

  if( find(ordstr_P,"s") || find(ordstr_P,"M")
      || find(ordstr_P,"a") || (neg > 0) )
  {
    if( defined(hi) && is_homog )
    {
      if (p_opt){"std with given Hilbert function in basering";}
      return( std(i,hi,w) );
    }
    if (p_opt){"//--stdhilb not implemented, use std in basering";}
    //if ( neg )
    //{
    //  "//*** WARNING: non-positive ring weights, computation may not finish";
    //}
    return( std(i) );
  }

//------------------------ change to hilbRing ----------------------------

  list hiRi = hilbRing(i);  //The ground field of P and Philb coincide
  intvec W = hiRi[2];       //Philb has an extra variable @ or @(k)
  def Philb = hiRi[1];      //Philb is no qring and the predefined
  setring Philb;            //ideal/module Id(1) in Philb is homogeneous
                            //Parameters of P are not converted in Philb
//-------- compute Hilbert function of homogenized ideal in Philb ---------
//Philb has only 1 block. There are three cases

  string algorithm;       //possibilities: std, slimgb, stdorslimgb
  //define algorithm:
  if( find(method,"std") && !find(method,"slimgb") )
  {
    algorithm = "std";
  }
  if( find(method,"slimgb") && !find(method,"std") )
  {
    algorithm = "slimgb";
  }
  if( find(method,"std") && find(method,"slimgb") ||
    (!find(method,"std") && !find(method,"slimgb")) )
  {
    algorithm = "stdorslimgb";
  }

  if ( algorithm=="std" || ( algorithm=="stdorslimgb" && char(P)>0 ) )
  {
    if (p_opt) {"std in ring " + string(Philb);}
    intvec hi = hilb( std(Id(1)),1,W );
  }
  if ( algorithm=="slimgb" || ( algorithm=="stdorslimgb" && char(P)==0 ) )
  {
    intvec hi = hilb(qslimgb(Id(1)),1,W);
  }

  //-------------- we need another intermediate ring Phelp ----------------
  //In Phelp we change the ordering from Philb, otherwise it coincides with
  //Philb, that is, it has in addition to P an extra homogenizing variable
  //with name @, resp. @(i) if @ and @(1), ..., @(i-1) are defined.
  //Phelp has the same ordering as P on common variables. In Phelp
  //a quotient ideal from P is added to the input

  list BRlist = ringlist(Philb);
  BRlist[3] = list();
  int so = size(ord_P);
  if( ord_P[so][1] =="c" || ord_P[so][1] =="C" )
  {
    list moduleord = ord_P[so];
    so = so-1;
  }
  for (k=1; k<=so; k++)
  {
    BRlist[3][k] = ord_P[k];
  }

  BRlist[3][so+1] = list("dp",1);
  w = w,1;

  if( defined(moduleord) )
  {
    BRlist[3][so+2] = moduleord;
  }

//------ change to extended ring and compute std with hilbert series ------
  def Phelp = ring(quotientList(BRlist));
  setring Phelp;
  def i = imap(Philb, Id(1));
  kill Philb;

  // compute std with Hilbert series
  if (w ==1)
  {
    if (p_opt){ "std with hilb in " + string(Phelp);}
    i = std(i, hi);
  }
  else
  {
    if(p_opt){"std with weighted hilb in "+string(Phelp);}
    i = std(i, hi, w);
  }

//-------------------- go back to original ring ---------------------------
//The main computation is done. Do not forget to simplfy before maping.

  // subst 1 for homogenizing var
  if ( p_opt ) { "dehomogenization"; }
  i = subst(i, var(nvars(basering)), 1);

  if (p_opt) { "simplification"; }
  i= simplify(i,34);

  setring P;
  if (p_opt) { "imap to ring "+string(P); }
  i = imap(Phelp,i);
  kill Phelp;
  if( was_qring)
  {
    i = NF(i,std(0));
  }
  i = simplify(i,34);

  // compute reduced SB
  if (find(s_opt, "redSB") > 0)
  {
    if (p_opt) { "//interreduction"; }
    i=interred(i);
  }
  attrib(i, "isSB", 1);
  return (i);
}
example
{ "EXAMPLE:"; echo = 2;
   ring  r = 0,(x,y,z),lp;
   ideal i = y3+x2,x2y+x2z2,x3-z9,z4-y2-xz;
   ideal j = stdhilb(i); j;

   ring  r1 = 0,(x,y,z),wp(3,2,1);
   ideal  i = y3+x2,x2y+x2z2,x3-z9,z4-y2-xz;  //ideal is homogeneous
   ideal j = stdhilb(i,"std"); j;
   //this is equivalent to:
   intvec v = hilb(std(i),1);
   ideal j1 = std(i,v,intvec(3,2,1)); j1;
   size(NF(j,j1))+size(NF(j1,j));            //j and j1 define the same ideal
}

///////////////////////////////////////////////////////////////////////////////
proc quotientList (list RL, list #)
"SYNTAX: @code{quotientList (} list_expression @code{)} @*
         @code{quotientList (} list_expression @code{,} string_expression@code{)}
TYPE:    list
PURPOSE: define a ringlist, say QL, of the first argument, say RL, which is
         assumed to be the ringlist of a qring, but where the quotient ideal
         RL[4] is not a standard basis with respect to the given monomial
         order in RL[3]. Then QL will be obtained from RL just by replacing
         RL[4] by a standard of it with respect to this order. RL itself
         will be returnd if size(RL[4]) <= 1 (in which case it is known to be
         a standard basis w.r.t. any ordering) or if a second argument
         \"isSB\" of type string is given.
NOTE:    the command ring(quotientList(RL)) defines a quotient ring correctly
         and should be used instead of ring(RL) if the quotient ideal RL[4]
         is not (or not known to be) a standard basis with respect to the
         monomial ordering specified in RL[3].
SEE ALSO: ringlist, ring
EXAMPLE: example quotientList; shows an example"
{
  def P = basering;
  if( size(#) > 0 )
  {
    if ( #[1] == "isSB")
    {
      return (RL);
    }
  }
  ideal Qideal  = RL[4];  //##Achtung: falls basering Nullteiler hat, kann
                           //die SB eines Elements mehrere Elemente enthalten
  if( size(Qideal) <= 0)
  {
    return (RL);
  }

  RL[4] = ideal(0);
  def Phelp = ring(RL);
  setring Phelp;
  ideal Qideal = groebner(fetch(P,Qideal));
  setring P;
  RL[4]=fetch(Phelp,Qideal);
  return (RL);
}
example
{ "EXAMPLE:"; echo = 2;
   ring P = 0,(y,z,u,v),lp;
   ideal i = y+u2+uv3, z+uv3;            //i is an lp-SB but not a dp_SB
   qring Q = std(i);
   list LQ = ringlist(Q);
   LQ[3][1][1]="dp";
   def Q1 = ring(quotientList(LQ));
   setring Q1;
   Q1;

   setring Q;
   ideal q1 = uv3+z, u2+y-z, yv3-zv3-zu; //q1 is a dp-standard basis
   LQ[4] = q1;
   def Q2 = ring(quotientList(LQ,"isSB"));
   setring Q2;
   Q2;
}

///////////////////////////////////////////////////////////////////////////////
proc par2varRing (list #)
"USAGE:   par2varRing([l]); l list of ideals/modules [default:l=empty list]
RETURN:  list, say L, with L[1] a ring where the parameters of the
         basering have been converted to an additional last block of
         variables, all of weight 1, and ordering dp.
         If a list l with l[i] an ideal/module is given, then
         l[i] + minpoly*freemodule(nrows(l[i])) is mapped to an ideal/module
         in L[1] with name Id(i).
         If the basering has no parameters then L[1] is the basering.
EXAMPLE: example par2varRing; shows an example"
{
  def P = basering;
  int npar = npars(P);  //number of parameters
  int s = size(#);
  int ii;
  if ( npar == 0)
  {
    dbprint(printlevel-voice+3,"// ** no parameters, ring was not changed");
    for( ii = 1; ii <= s; ii++)
    {
      def Id(ii) = #[ii];
      export (Id(ii));
    }
    return(list(P));
  }

  list rlist = ringlist(P);
  list parlist = rlist[1];
  rlist[1] = parlist[1];
  poly Minpoly = minpoly;     //check for minpoly:
  int sm = size(Minpoly);

  //now create new ring
  for( ii = 1; ii <= s; ii++)
  {
    def Id(ii) = #[ii];
  }
  int nvar = size(rlist[2]);
  int nblock = size(rlist[3]);
  int k;
  for (k=1; k<=npar; k++)
  {
    rlist[2][nvar+k] = parlist[2][k];   //change variable list
  }

  //converted parameters get one block dp. If module ordering was in front
  //it stays in front, otherwise it will be moved to the end
  intvec OW = 1:npar;
  if( rlist[3][nblock][1] =="c" || rlist[3][nblock][1] =="C" )
  {
    rlist[3][nblock+1] = rlist[3][nblock];
    rlist[3][nblock] = list("dp",OW);
  }
  else
  {
    rlist[3][nblock+1] = list("dp",OW);
  }

  def Ppar2var = ring(quotientList(rlist));
  setring Ppar2var;
  if ( sm == 0 )
  {
    for( ii = 1; ii <= s; ii++)
    {
      def Id(ii) = imap(P,Id(ii));
      export (Id(ii));
    }
  }
  else
  {
    if( find(option(),"prot") ){"//add minpoly to input";}
    poly Minpoly = imap(P,Minpoly);
    for( ii = 1; ii <= s; ii++)
    {
      def Id(ii) = imap(P,Id(ii));
      Id(ii) = Id(ii),Minpoly*freemodule(nrows(Id(ii)));
      export (Id(ii));
    }
  }
  list Lpar2var = Ppar2var;
  return(Lpar2var);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = (0,x),(y,z,u,v),lp;
   minpoly = x2+1;
   ideal i = x3,x2+y+z+u+v,xyzuv-1; i;
   def P = par2varRing(i)[1]; P;
   setring(P);
   Id(1);

   setring R;
   module m = x3*[1,1,1], (xyzuv-1)*[1,0,1];
   def Q = par2varRing(m)[1]; Q;
   setring(Q);
   print(Id(1));
}

//////////////////////////////////////////////////////////////////////////////
proc hilbRing ( list # )
"USAGE:   hilbRing([l]); l list of ideals/modules [default:l=empty list]
RETURN:  list, say L: L[1] is a ring and L[2] an intvec
         L[1] is a ring whith an extra homogenizing variable with name @,
         resp. @(i) if @ and @(1), ..., @(i-1) are defined.
         The monomial ordering of L[1] is 1 block dp if the
         weights of the variables of the basering, say R, are all 1, resp.
         wp(w,1) wehre w is the intvec of weights of the variables of R.
         If R is a quotient ring P/Q, then L[1] is not a quotient ring but
         contains the ideal @Qidealhilb@, the homogenized ideal Q of P.
         (Parameters of R are not touched).
         If a list l is given with l[i] an ideal, then l[i] is
         mapped to the homogenized ideal Id(i) in L[1].
         L[2] is the intvec (w,1)
PURPOSE: Prepare a ring for computing the (weighted) hilbert series of
         an ideal with an easy monomial ordering.
EXAMPLE: example hilbRing; shows an example
"
{
  def P = basering;
  ideal Qideal = ideal(P);    //defining the quotient ideal if P is a qring
  if( size(Qideal) != 0 )
  {
    int is_qring =1;
  }
  list BRlist = ringlist(P);
  BRlist[4] = ideal(0);

  int nvarP = nvars(P);
  int s = size(#);
  intvec w;                   //for ringweights of basering P
  int k;
  for(k=1;  k<=nvarP; k++)
  {
    w[k]=deg(var(k));
  }

  for(k = 1; k <= s; k++)
  {
    def Id(k) = #[k];
    int nr(k) = nrows(Id(k));
  }

  // a homogenizing variable is added:
  // call it @, resp. @(k) if @(1),...,@(k-1) are defined
  string homvar;
  if ( defined(@)==0 )
  {
    homvar = "@";
  }
  else
  {
    k=1;
    while( defined(@(k)) != 0 )
    {
      k++;
    }
    homvar = "@("+string(k)+")";
  }
  BRlist[2][nvarP+1] = homvar;
  w[nvarP +1]=1;

  //ordering is set to (dp,C) if weights of all variables are 1
  //resp. to (wp(w,1),C) where w are the ringweights of basering P
  //homogenizing var gets weight 1:

  BRlist[3] = list();
  BRlist[3][2]=list("C",intvec(0));
  if(w==1)
  {
    BRlist[3][1]=list("dp",w);
  }
  else
  {
    BRlist[3][1]=list("wp",w);
  }

  //change ring and get ideal from previous ring
  def Philb = ring(quotientList(BRlist));
  kill BRlist;
  setring Philb;
  if( defined(is_qring) )
  {
    ideal @Qidealhilb@ =  homog( imap(P,Qideal), `homvar` );
    export(@Qidealhilb@);

    if( find(option(),"prot") ){"add quotient ideal to input";}
    for(k = 1; k <= s; k++)
    {  //homogenize
      def Id(k) =  homog( imap(P,Id(k)), `homvar` );
      Id(k) =  Id(k),@Qidealhilb@*freemodule(nr(k)) ;
      export(Id(k));
    }
  }
  else
  {
    for(k = 1; k <= s; k++)
    { //homogenize
      def Id(k) =  homog( imap(P,Id(k)), `homvar` );
      export(Id(k));
    }
  }
  list Lhilb = Philb,w;
  return(Lhilb);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,(x,y,z,u,v),lp;
   ideal i = x+y2+z3,xy+xv+yz+zu+uv,xyzuv-1;
   def P = hilbRing(i)[1];  P;
   setring P;
   Id(1);
   hilb(std(Id(1)),1);

   ring S =  0,(x,y,z,u,v),lp;
   qring T = std(x+y2+z3);
   ideal i = xy+xv+yz+zu+uv,xyzuv-v5;
   module m = i*[0,1,1] + (xyzuv-v5)*[1,1,0];
   def Q = hilbRing(m)[1];  Q;
   setring Q;
   print(Id(1));
}

//////////////////////////////////////////////////////////////////////////////
proc qslimgb (i)
"USAGE:   qslimgb(i); i ideal or module
RETURN:  same type as input, a standard basis of i computed with slimgb
NOTE:    As long as slimgb does not know qrings qslimgb should be used in case
         the basering is (possibly) a quotient ring. The quotient ideal is
         added to the input and slimgb is applied.
EXAMPLE: example qslimgb; shows an example"
{
  def P = basering;
  ideal Qideal = ideal(P);      //defining the quotient ideal if P is a qring
  int p_opt;
  if( find(option(),"prot") )
  {
    p_opt=1;
  }
  if (size(Qideal) == 0)
  {
    if (p_opt) { "slimgb in ring " + string(P); }
    return(slimgb(i));
  }

  //case of a qring; since slimgb does not know qrings we
  //delete the quotient ideal and add it to i

  list BRlist = ringlist(P);
  BRlist[4] = ideal(0);
  def Phelp = ring(BRlist);
  kill BRlist;
  setring Phelp;
  // module case:
  def iq = imap(P,i);
  iq = iq, imap(P,Qideal)*freemodule(nrows(iq));
  if (p_opt)
  {
    "slimgb in ring " + string(Phelp);
    "(with quotient ideal added to input)";
  }
  iq = slimgb(iq);

  setring P;
  if (p_opt) { "//imap to original ring"; }
  i = imap(Phelp,iq);
  kill Phelp;

  if (find(option(),"redSB") > 0)
  {
    if (p_opt) { "//interreduction"; }
    i=interred(i);
  }
  attrib(i, "isSB", 1);
  return (i);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R  = (0,v),(x,y,z,u),dp;
   qring Q = std(x2-y3);
   ideal i = x+y2,xy+yz+zu+u*v,xyzu*v-1;
   ideal j = qslimgb(i); j;

   module m = [x+y2,1,0], [1,1,x2+y2+xyz];
   print(qslimgb(m));
}

//////////////////////////////////////////////////////////////////////////////
proc groebner(def i_par, list #)
"SYNTAX: @code{groebner (} ideal_expression @code{)} @*
         @code{groebner (} module_expression @code{)} @*
         @code{groebner (} ideal_expression@code{,} int_expression @code{)} @*
         @code{groebner (} module_expression@code{,} int_expression @code{)}
         @code{groebner (} ideal_expression@code{,} list of string_expressions
               @code{)} @*
         @code{groebner (} ideal_expression@code{,} list of string_expressions
               and int_expression @code{)} @*
         @code{groebner (} ideal_expression@code{,} int_expression @code{)} @*
TYPE:    type of the first argument
PURPOSE: computes a standard basis of the first argument @code{I}
         (ideal or module), by a heuristically chosen method (default)
         or by a method specified by further arguments of type string.
         Possible methods are:  @*
         - the direct methods @code{\"std\"} or @code{\"slimgb\"} without
           conversion @*
         - conversion methods @code{\"hilb\"} or @code{\"fglm\"} where
           a Groebner basis is first computed with an \"easy\" ordering
           and then converted to the ordering of the basering by the
           Hilbert driven Groebner basis computation or by linear algebra.
           The actual computation of the Groebner basis can be
           specified by @code{\"std\"} or by @code{\"slimgb\"}
           (not implemented for all orderings) @*
         A further string @code{\"par2var\"} converts parameters to an extra
         block of variables before a Groebner basis computation (and
         afterwards back).
         @code{option(prot)} tells about the chosen method.
NOTE:    If a further argument, say @code{wait}, of type int is given,
         then the computation proceeds at most @code{wait} seconds.
         That is, if no result could be computed in @code{wait} seconds,
         then the computation is interrupted, 0 is returned, a warning
         message is displayed, and the global variable
         @code{Standard::groebner_error} is defined.
         This feature uses MP and is hence only available on UNIX platforms.
HINT:    Since there exists no uniform best method for computing standard
         bases, and since the difference in performance of a method on
         different examples can be huge, it is recommended to test, for hard
         examples, first various methods on a simplified example (e.g. use
         characteristic 32003 instead of 0 or substitute a subset of
         parameters/variables by integers, etc.). @*
SEE ALSO: stdhilb, stdfglm, std, slimgb
KEYWORDS: time limit on computations; MP, groebner basis computations
EXAMPLE: example groebner;  shows an example"

{
//Vorgabe einer Teilmenge aus {hilb,fglm,par2var,std,slimgb}
//Aktuelle Einstellungen (Jan 2007):
//---------------------------------
//0. Immer Aufruf von std unabhaengig von der Vorgabe:
//   gemischte Ordnungen, extra Gewichtsvektor, Matrix Ordnungen

//1. Keine Vorgabe: es wirkt die aktuelle Heuristk:
//   - Char p: std
//   - Char = 0: slimgb (im qring wird Quotientenideal zum Input addiert)
//   - 1-Block-Ordnungen/non-commutative: direkt Aufruf von std oder slimgb
//   - Komplizierte Ordnungen (lp oder > 1 Block): hilb
//   - Parameter werden grundsaetzlich nicht in Variable umgewandelt
//   ? alternativ: more than 1 parameter will be converted to ring variable ?
//   - fglm is keine Heuristik, da sonst vorher dim==0 peprueft werden muss

//2. Vorgabe aus {std,slimgb}: es wird wo immer moeglich das angegebene
//   gewaehlt (da slimgb keine Hilbertfunktion kennt, wird std verwendet).
//   Bei slimgb im qring, wird das Quotientenideal zum Ideal addiert.
//   Bei Angabe von std zusammen mit slimgb (aequivalent zur Angabe von
//   keinem von beidem) wirkt obige Heuristik.

//3. Nichtleere Vorgabe aus {hilb,fglm,std,slimgb}:
//   es wird nur das angegebene und moegliche sowie das notwendige verwendet
//   und bei Wahlmoeglickeit je nach Heuristik.
//   Z.B. Vorgabe von {hilb} ist aequivalent zu {hilb,std,slimgb} und es wird
//   hilb und nach Heuristik std oder slimgb verwendet, aber nicht par2var;
//   bei Vorgabe von {hilb,slimgb} wird hilb und wo moeglich slimgb verwendet.

//4. Bei Vorgabe von {par2var} wird par2var immer mit hilb und nach Heuristik
//   std oder slimgb verwendet. Zu Variablen konvertierte Parameter haben
//   extra letzten Block und Gewichte 1.


  def P=basering;
  if ((typeof(i_par)=="vector")||(typeof(i_par)=="module")) {module i=i_par;}
  else {ideal i=i_par; }
  kill i_par;

//----------------------- save the given method ---------------------------
  string method;
  list Method;
  int k;
  for (k=1; k<=size(#); k++)
  {
     if (typeof(#[k]) == "int")
     {
       int wait = #[k];
     }
     if (typeof(#[k]) == "string")
     {
       method = method + "," + #[k];
       Method = Method + list(#[k]);
     }
  }

 //======= we have an argument of type int -- try to use MPfork links =======
  if ( defined(wait) == voice )
  {
    if ( system("with", "MP") )
    {
        int j = 10;
        string bs = nameof(basering);
        link l_fork = "MPtcp:fork";
        open(l_fork);
        write(l_fork, quote(system("pid")));
        int pid = read(l_fork);
//        write(l_fork, quote(groebner(eval(i))));
        write(l_fork, quote(groebner(eval(i),eval(Method))));
//###Fehlermeldung:
// ***dError: undef. ringorder used
// occured at:

        // sleep in small intervalls for appr. one second
        if (wait > 0)
        {
          while(j < 1000000)
          {
            if (status(l_fork, "read", "ready", j)) {break;}
            j = j + j;
          }
        }

        // sleep in intervalls of one second from now on
        j = 1;
        while (j < wait)
        {
          if (status(l_fork, "read", "ready", 1000000)) {break;}
          j = j + 1;
        }

        if (status(l_fork, "read", "ready"))
        {
          def result = read(l_fork);
          if (bs != nameof(basering))
          {
            def PP = basering;
            setring P;
            def result = imap(PP, result);
            kill PP;
          }
          if (defined(groebner_error))
          {
            kill groebner_error;
          }
          kill l_fork;
        }
        else
        {
          ideal result;
          if (! defined(groebner_error))
          {
            int groebner_error = 1;
            export groebner_error;
          }
          "** groebner did not finish";
          j = system("sh", "kill " + string(pid));
        }
        return (result);
    }
    else
    {
      "** groebner with a time limit on computation is not supported
          in this configuration";
    }
  }

 //=========== we are still here -- do the actual computation =============

//--------------------- save data from basering ---------------------------
  poly Minpoly = minpoly;      //minimal polynomial
  int was_minpoly;             //remembers if there was a minpoly in P
  if (size(Minpoly) > 0)
  {
     was_minpoly = 1;
  }

  ideal Qideal = ideal(P);      //defining the quotient ideal if P is a qring
  int was_qring;                //remembers if basering was a qring
  //int is_homog = 1;
  if (size(Qideal) > 0)
  {
     was_qring = 1;
     //is_homog = homog(Qideal); //remembers if Qideal was homog (homog(0)=1)
  }
  list BRlist = ringlist(P);

  // save ordering of basering P for later use
  list ord_P = BRlist[3];       //should be available in all rings
  string ordstr_P = ordstr(P);
  int nvars_P = nvars(P);
  int npars_P = npars(P);
  intvec w;                     //for ringweights of basering P
  for(k=1;  k<=nvars_P; k++)
  {
     w[k]=deg(var(k));
  }
  int neg=1-attrib (P,"global");

  //save options:
  intvec opt=option(get);
  string s_opt = option();
  int p_opt;
  if (find(s_opt, "prot"))  { p_opt = 1; }

//------------------ cases where std is always used ------------------------
//If other methods are not implemented or do not make sense, i.e. for
//local or mixed orderings, matrix orderings, extra weight vector and modules

  if(  //( find(ordstr_P,"s") > 0 ) || // covered by neg
       ( find(ordstr_P,"M") > 0 )
    || ( find(ordstr_P,"a") > 0 )
    || ( neg>0 ) )
  {
    if (p_opt) { "std in basering"; }
    return(std(i));
  }

//now we have:
//ideal or module, global ordering, no matrix ordering, no extra weight vector
//The interesting cases start now.

 //------------------ classify the possible settings ---------------------
  string algorithm;       //possibilities: std, slimgb, stdorslimgb
  string conversion;      //possibilities: hilb, fglm, hilborfglm, no
  string partovar;        //possibilities: yes, no
  string order;           //possibilities: simple, !simple
  string direct;          //possibilities: yes, no

  //define algorithm:
  if( find(method,"std") && !find(method,"slimgb") )
  {
    algorithm = "std";
  }
  if( find(method,"slimgb") && !find(method,"std") )
  {
    algorithm = "slimgb";
  }
  if( find(method,"std") && find(method,"slimgb") ||
      (!find(method,"std") && !find(method,"slimgb")) )
  {
    algorithm = "stdorslimgb";
  }

  //define conversion:
  if( find(method,"hilb") && !find(method,"fglm") )
  {
     conversion = "hilb";
  }
  if( find(method,"fglm") && !find(method,"hilb") )
  {
    conversion = "fglm";
  }
  if( find(method,"fglm") && find(method,"hilb") )
  {
    conversion = "hilborfglm";
  }
  if( !find(method,"fglm") && !find(method,"hilb") )
  {
    conversion = "no";
  }

  //define partovar:
  if( find(method,"par2var") && npars_P > 0 )
  {
     partovar = "yes";
  }
  else
  {
     partovar = "no";
  }

  //define order:
  if (system("nblocks") <= 2)
  {
    if ( find(ordstr_P,"M")+find(ordstr_P,"lp")+find(ordstr_P,"rp") <= 0 )
    {
      order = "simple";
    }
  }

  //define direct:
  if ( (order=="simple" && (size(method)==0)) ||
       (size(BRlist)>4) ||
        (order=="simple" && (method==",par2var" && npars_P==0 )) ||
         (conversion=="no" && partovar=="no" &&
           (algorithm=="std" || algorithm=="slimgb" ||
            (find(method,"std") && find(method,"slimgb")) ) ) )
  {
    direct = "yes";
  }
  else
  {
    direct = "no";
  }

  //order=="simple" means that the ordering of the variables consists of one
  //block which is not a matrix ordering and not a lexicographical ordering.
  //(Note:Singular counts always least 2 blocks, one is for module component):
  //Call a method "direct" if conversion=="no" && partovar="no" which means
  //that we apply std or slimgb dircet in the basering (exception
  //as long as slimgb does not know qrings: in a qring of a ring P
  //the ideal Qideal is added to the ideal and slimgb is applied in P).
  //We apply a direct method if we have a simple monomial ordering, if no
  //conversion (fglm or hilb) is specified and if the parameters shall
  //not be made to variables

//---------------------------- direct methods -----------------------------
  if ( direct == "yes" )
  {
    if ( algorithm=="std" || (algorithm=="stdorslimgb" && char(P)>0) )
    {
      if (p_opt) { "std in " + string(P); }
      return(std(i));
    }
    if ( algorithm=="slimgb" || (algorithm=="stdorslimgb" && char(P)==0) )
    {
      return(qslimgb(i));
    }
  }

//--------------------------- indirect methods -----------------------------
//indirect methods are methods where a conversion is used with a ring change
//We are in the following situation:
//direct=="no" (i.e. "hilb" or "fglm" or "par2var" is given)
//or no method is given and we have a complicated monomial ordering
//Note thar "par2var" is not a default strategy, it must be explicitely
//given in order to be performed.

//------------ case where no parameters are made to variables  -------------
  if (  partovar == "no" && conversion == "hilb"
    || (partovar == "no" && conversion == "fglm" )
    || (partovar == "no" && conversion == "hilborfglm" )
    || (partovar == "no" && conversion == "no" && direct == "no") )
  //last case: heuristic
  {
    if ( conversion=="fglm" )
    {
      if ( algorithm=="std" || (algorithm=="stdorslimgb" && char(P)>0) )
      {
        return (stdfglm(i,"std"));
      }
      if ( algorithm=="slimgb" || (algorithm=="stdorslimgb" && char(P)==0) )
      {
        return (stdfglm(i,"slimgb"));
      }
    }
    else
    {
      if ( algorithm=="std" || (algorithm=="stdorslimgb" && char(P)>0) )
      {
        return (stdhilb(i,"std"));
      }
      if ( algorithm=="slimgb" || (algorithm=="stdorslimgb" && char(P)==0) )
      {
        return (stdhilb(i,"slimgb"));
      }
    }
  }

//------------ case where parameters are made to variables  ----------------
//define a ring Phelp via par2varRing in which the parameters are variables

  else
  {
    // reset options
    option(none);
    // turn on options prot, mem, redSB, intStrategy if previously set
    if ( find(s_opt, "prot") )
      { option(prot); }
    if ( find(s_opt, "mem") )
      { option(mem); }
    if ( find(s_opt, "redSB") )
      { option(redSB); }
    if ( find(s_opt, "intStrategy") )
      { option(intStrategy); }

    //first clear denominators of parameters
    if (npars_P > 0)
    {
      for( k=ncols(i); k>0; k-- )
      { i[k]=cleardenom(i[k]); }
    }

    def Phelp = par2varRing(i)[1];   //minpoly is mapped with i
    setring Phelp;
    def i = Id(1);
    //is_homog = homog(i);

    //If parameters are converted to ring variables, they appear in an extra
    //block. Therefore we use always hilb for this block ordering:
    if ( conversion=="fglm" )
    {
      i = (stdfglm(i));       //only uesful for 1 parameter with minpoly
    }
    else
    {
      if ( algorithm=="std" || (algorithm=="stdorslimgb" && char(P)>0) )
      {
        i = stdhilb(i,"std");
      }
      if ( algorithm=="slimgb" || (algorithm=="stdorslimgb" && char(P)==0) )
      {
        i = stdhilb(i,"slimgb");
      }
    }
  }

//-------------------- go back to original ring ---------------------------
//The main computation is done. However, the SB coming from a ring with
//extra variables is in general too big. We simplify it befor mapping it
//to the basering.

  if (p_opt) { "//simplification"; }

  if (was_minpoly)
  {
    ideal Minpoly = imap(P,Minpoly);
    attrib(Minpoly,"isSB",1);
    i = simplify(NF(i,Minpoly),2);
  }

  def Li = lead(i);
  setring P;
  def Li = imap(Phelp,Li);
  Li = simplify(Li,32);
  intvec vi;
  for (k=1; k<=ncols(Li); k++)
  {
    vi[k] = Li[k]==0;
  }

  setring Phelp;
  for (k=1;  k<=size(i) ;k++)
  {
    if(vi[k]==1)
    {
      i[k]=0;
    }
  }
  i = simplify(i,2);

  setring P;
  if (p_opt) { "//imap to original ring"; }
  i = imap(Phelp,i);
  kill Phelp;
  i = simplify(i,34);

  // clean-up time
  option(set, opt);
  if (find(s_opt, "redSB") > 0)
  {
    if (p_opt) { "//interreduction"; }
    i=interred(i);
  }
  attrib(i, "isSB", 1);
  return (i);
}
example
{ "EXAMPLE: "; echo=2;
  intvec opt = option(get);
  option(prot);
  ring r  = 0,(a,b,c,d),dp;
  ideal i = a+b+c+d,ab+ad+bc+cd,abc+abd+acd+bcd,abcd-1;
  groebner(i);

  ring s  = 0,(a,b,c,d),lp;
  ideal i = imap(r,i);
  groebner(i,"hilb");

  ring R  = (0,a),(b,c,d),lp;
  minpoly = a2+1;
  ideal i = a+b+c+d,ab+ad+bc+cd,abc+abd+acd+bcd,d2-c2b2;
  groebner(i,"par2var","slimgb");

  groebner(i,"fglm");          //computes a reduced standard basis

  if (system("with","MP")) {groebner(i,10,"std");}
  defined(Standard::groebner_error);
  option(set,opt);
}

//////////////////////////////////////////////////////////////////////////

proc res(list #)
"@c we do texinfo here:
@cindex resolution, computation of
@table @code
@item @strong{Syntax:}
@code{res (} ideal_expression@code{,} int_expression @code{[,} any_expression @code{])}
@*@code{res (} module_expression@code{,} int_expression @code{[,} any_expression @code{])}
@item @strong{Type:}
resolution
@item @strong{Purpose:}
computes a (possibly minimal) free resolution of an ideal or module using
a heuristically chosen method.
@* The second (int) argument (say, @code{k}) specifies the length of
the resolution. If it is not positive then @code{k} is assumed to be the
number of variables of the basering.
@* If a third argument is given, the returned resolution is minimized.

Depending on the input, the returned resolution is computed using the
following methods:
@table @asis
@item @strong{quotient rings:}
@code{nres} (classical method using syzygies) , see @ref{nres}.

@item @strong{homogeneous ideals and k=0:}
@code{lres} (La'Scala's method), see @ref{lres}.

@item @strong{not minimized resolution and (homogeneous input with k not 0, or local rings):}
@code{sres} (Schreyer's method), see @ref{sres}.

@item @strong{all other inputs:}
@code{mres} (classical method), see @ref{mres}.
@end table
@item @strong{Note:}
Accessing single elements of a resolution may require that some partial
computations have to be finished and may therefore take some time.
@end table
@c ref
See also
@ref{betti};
@ref{ideal};
@ref{minres};
@ref{module};
@ref{mres};
@ref{nres};
@ref{lres};
@ref{hres};
@ref{sres}.
@ref{resolution}
@c ref
"
{
   def P=basering;
   if (size(#) < 2)
   {
     ERROR("res: need at least two arguments: ideal/module, int");
   }

   def m=#[1]; //the ideal or module
   int i=#[2]; //the length of the resolution
   if (i< 0) { i=0;}

   string varstr_P = varstr(P);

   int p_opt;
   string s_opt = option();
   // set p_opt, if option(prot) is set
   if (find(s_opt, "prot"))
   {
     p_opt = 1;
   }

   if(size(ideal(basering)) > 0)
   {
     // the quick hack for qrings - seems to fit most needs
     // (lres is not implemented for qrings, sres is not so efficient)
     if (p_opt) { "using nres";}
     return(nres(m,i));
   }

   if(homog(m)==1)
   {
      resolution re;
      if (((i==0) or (i>=nvars(basering))) && typeof(m) != "module")
      {
        //LaScala for the homogeneous case and i == 0
        if (p_opt) { "using lres";}
        re=lres(m,i);
        if(size(#)>2)
        {
           re=minres(re);
        }
      }
      else
      {
        if(size(#)>2)
        {
          if (p_opt) { "using mres";}
          re=mres(m,i);
        }
        else
        {
          if (p_opt) { "using sres";}
          re=sres(std(m),i);
        }
      }
      return(re);
   }

   //mres for the global non homogeneous case
   if(find(ordstr(P),"s")==0)
   {
      string ri= "ring Phelp ="
                  +string(char(P))+",("+varstr_P+"),(dp,C);";
      execute(ri);
      def m=imap(P,m);
      if (p_opt) { "using mres in another ring";}
      list re=mres(m,i);
      setring P;
      resolution result=imap(Phelp,re);
      if (size(#) > 2) {result = minres(result);}
      return(result);
   }

   //sres for the local case and not minimal resolution
   if(size(#)<=2)
   {
      string ri= "ring Phelp ="
                  +string(char(P))+",("+varstr_P+"),(ls,c);";
      execute(ri);
      def m=imap(P,m);
      m=std(m);
      if (p_opt) { "using sres in another ring";}
      list re=sres(m,i);
      setring P;
      resolution result=imap(Phelp,re);
      return(result);
   }

   //mres for the local case and minimal resolution
   string ri= "ring Phelp ="
                  +string(char(P))+",("+varstr_P+"),(ls,C);";
   execute(ri);
   def m=imap(P,m);
    if (p_opt) { "using mres in another ring";}
   list re=mres(m,i);
   setring P;
   resolution result=imap(Phelp,re);
   result = minres(result);
   return(result);
}
example
{"EXAMPLE:"; echo = 2;
  ring r=0,(x,y,z),dp;
  ideal i=xz,yz,x3-y3;
  def l=res(i,0); // homogeneous ideal: uses lres
  l;
  print(betti(l), "betti"); // input to betti may be of type resolution
  l[2];         // element access may take some time
  i=i,x+1;
  l=res(i,0);   // inhomogeneous ideal: uses mres
  l;
  ring rs=0,(x,y,z),ds;
  ideal i=imap(r,i);
  def l=res(i,0); // local ring not minimized: uses sres
  l;
  res(i,0,0);     // local ring and minimized: uses mres
}
/////////////////////////////////////////////////////////////////////////

proc quot (m1,m2,list #)
"SYNTAX: @code{quot (} module_expression@code{,} module_expression @code{)}
         @*@code{quot (} module_expression@code{,} module_expression@code{,}
            int_expression @code{)}
         @*@code{quot (} ideal_expression@code{,} ideal_expression @code{)}
         @*@code{quot (} ideal_expression@code{,} ideal_expression@code{,}
            int_expression @code{)}
TYPE:    ideal
SYNTAX:  @code{quot (} module_expression@code{,} ideal_expression @code{)}
TYPE:    module
PURPOSE: computes the quotient of the 1st and the 2nd argument.
         If a 3rd argument @code{n} is given the @code{n}-th method is used
         (@code{n}=1...5).
SEE ALSO: quotient
EXAMPLE: example quot; shows an example"
{
  if (((typeof(m1)!="ideal") and (typeof(m1)!="module"))
     or ((typeof(m2)!="ideal") and (typeof(m2)!="module")))
  {
    "USAGE:   quot(m1, m2[, n]); m1, m2 two submodules of k^s,";
    "         n (optional) integer (1<= n <=5)";
    "RETURN:  the quotient of m1 and m2";
    "EXAMPLE: example quot; shows an example";
    return();
  }
  if (typeof(m1)!=typeof(m2))
  {
    return(quotient(m1,m2));
  }
  if (size(#)>0)
  {
    if (typeof(#[1])=="int" )
    {
      return(quot1(m1,m2,#[1]));
    }
  }
  else
  {
    return(quot1(m1,m2,2));
  }
}
example
{ "EXAMPLE:"; echo = 2;
  ring r=181,(x,y,z),(c,ls);
  ideal id1=maxideal(4);
  ideal id2=x2+xyz,y2-z3y,z3+y5xz;
  option(prot);
  ideal id3=quotient(id1,id2);
  id3;
  ideal id4=quot(id1,id2,1);
  id4;
  ideal id5=quot(id1,id2,2);
  id5;
}

static proc quot1 (module m1, module m2,int n)
"USAGE:   quot1(m1, m2, n); m1, m2 two submodules of k^s,
         n integer (1<= n <=5)
RETURN:  the quotient of m1 and m2
EXAMPLE: example quot1; shows an example"
{
  if (n==1)
  {
    return(quotient1(m1,m2));
  }
  else
  {
    if (n==2)
    {
      return(quotient2(m1,m2));
    }
    else
    {
      if (n==3)
      {
        return(quotient3(m1,m2));
      }
      else
      {
        if (n==4)
        {
          return(quotient4(m1,m2));
        }
        else
        {
          if (n==5)
          {
            return(quotient5(m1,m2));
          }
          else
          {
            return(quotient(m1,m2));
          }
        }
      }
    }
  }
}
example
{ "EXAMPLE:"; echo = 2;
  ring r=181,(x,y,z),(c,ls);
  ideal id1=maxideal(4);
  ideal id2=x2+xyz,y2-z3y,z3+y5xz;
  option(prot);
  ideal id6=quotient(id1,id2);
  id6;
  ideal id7=quot1(id1,id2,1);
  id7;
  ideal id8=quot1(id1,id2,2);
  id8;
}

static proc quotient0(module a,module b)
{
  module mm=b+a;
  resolution rs=lres(mm,0);
  list I=list(rs);
  matrix M=I[2];
  matrix A[1][nrows(M)]=M[1..nrows(M),1];
  ideal i=A;
  return (i);
}
proc quotient1(module a,module b)  //17sec
"USAGE:   quotient1(m1, m2); m1, m2 two submodules of k^s,
RETURN:  the quotient of m1 and m2"
{
  int i;
  a=std(a);
  module dummy;
  module B=NF(b,a)+dummy;
  ideal re=quotient(a,module(B[1]));
  for(i=2;i<=ncols(B);i++)
  {
     re=intersect1(re,quotient(a,module(B[i])));
  }
  return(re);
}
proc quotient2(module a,module b)    //13sec
"USAGE:   quotient2(m1, m2); m1, m2 two submodules of k^s,
RETURN:  the quotient of m1 and m2"
{
  a=std(a);
  module dummy;
  module bb=NF(b,a)+dummy;
  int i=ncols(bb);
  ideal re=quotient(a,module(bb[i]));
  bb[i]=0;
  module temp;
  module temp1;
  module bbb;
  int mx;
  i=i-1;
  while (1)
  {
    if (i==0) break;
    temp = a+bb*re;
    temp1 = lead(interred(temp));
    mx=ncols(a);
    if (ncols(temp1)>ncols(a))
    {
      mx=ncols(temp1);
    }
    temp1 = matrix(temp1,1,mx)-matrix(lead(a),1,mx);
    temp1 = dummy+temp1;
    if (deg(temp1[1])<0) break;
    re=intersect1(re,quotient(a,module(bb[i])));
    bb[i]=0;
    i = i-1;
  }
  return(re);
}
proc quotient3(module a,module b)   //89sec
"USAGE:   quotient3(m1, m2); m1, m2 two submodules of k^s,
         only for global rings
RETURN:  the quotient of m1 and m2"
{
  string s="ring @newr=("+charstr(basering)+
           "),("+varstr(basering)+",@t,@w),dp;";
  def @newP=basering;
  execute(s);
  module b=imap(@newP,b);
  module a=imap(@newP,a);
  int i;
  int j=ncols(b);
  vector @b;
  for(i=1;i<=j;i++)
  {
     @b=@b+@t^(i-1)*@w^(j-i+1)*b[i];
  }
  ideal re=quotient(a,module(@b));
  setring @newP;
  ideal re=imap(@newr,re);
  return(re);
}
proc quotient5(module a,module b)   //89sec
"USAGE:   quotient5(m1, m2); m1, m2 two submodules of k^s,
         only for global rings
RETURN:  the quotient of m1 and m2"
{
  string s="ring @newr=("+charstr(basering)+
           "),("+varstr(basering)+",@t),dp;";
  def @newP=basering;
  execute(s);
  module b=imap(@newP,b);
  module a=imap(@newP,a);
  int i;
  int j=ncols(b);
  vector @b;
  for(i=1;i<=j;i++)
  {
     @b=@b+@t^(i-1)*b[i];
  }
  @b=homog(@b,@w);
  ideal re=quotient(a,module(@b));
  setring @newP;
  ideal re=imap(@newr,re);
  return(re);
}
proc quotient4(module a,module b)   //95sec
"USAGE:   quotient4(m1, m2); m1, m2 two submodules of k^s,
         only for global rings
RETURN:  the quotient of m1 and m2"
{
  string s="ring @newr=("+charstr(basering)+
           "),("+varstr(basering)+",@t),dp;";
  def @newP=basering;
  execute(s);
  module b=imap(@newP,b);
  module a=imap(@newP,a);
  int i;
  vector @b=b[1];
  for(i=2;i<=ncols(b);i++)
  {
     @b=@b+@t^(i-1)*b[i];
  }
  matrix sy=modulo(@b,a);
  ideal re=sy;
  setring @newP;
  ideal re=imap(@newr,re);
  return(re);
}
static proc intersect1(ideal i,ideal j)
{
  def R=basering;
  execute("ring gnir = ("+charstr(basering)+"),
                       ("+varstr(basering)+",@t),(C,dp);");
  ideal i=var(nvars(basering))*imap(R,i)+(var(nvars(basering))-1)*imap(R,j);
  ideal j=eliminate(i,var(nvars(basering)));
  setring R;
  map phi=gnir,maxideal(1);
  return(phi(j));
}

//////////////////////////////////////////////////////////////////
///
/// sprintf, fprintf printf
///
proc sprintf(string fmt, list #)
"SYNTAX:  @code{sprintf (} string_expression @code{[,} any_expressions
               @code{] )}
RETURN:   string
PURPOSE:  @code{sprintf(fmt,...);} performs output formatting. The first
          argument is a format control string. Additional arguments may be
          required, depending on the content of the control string. A series
          of output characters is generated as directed by the control string;
          these characters are returned as a string. @*
          The control string @code{fmt} is simply text to be copied,
          except that the string may contain conversion specifications.@*
          Do @code{help print;} for a listing of valid conversion
          specifications. As an addition to the conversions of @code{print},
          the @code{%n} and @code{%2} conversion specification does not
          consume an additional argument, but simply generates a newline
          character.
NOTE:     If one of the additional arguments is a list, then it should be
          enclosed once more into a @code{list()} command, since passing a list
          as an argument flattens the list by one level.
SEE ALSO: fprintf, printf, print, string
EXAMPLE : example sprintf; shows an example
"
{
  int sfmt = size(fmt);
  if (sfmt  <= 1)
  {
    return (fmt);
  }
  int next, l, nnext;
  string ret;
  list formats = "%l", "%s", "%2l", "%2s", "%t", "%;", "%p", "%b", "%n", "%2";
  while (1)
  {
    if (size(#) <= 0)
    {
      return (ret + fmt);
    }
    nnext = 0;
    while (nnext < sfmt)
    {
      nnext = find(fmt, "%", nnext + 1);
      if (nnext == 0)
      {
        next = 0;
        break;
      }
      l = 1;
      while (l <= size(formats))
      {
        next = find(fmt, formats[l], nnext);
        if (next == nnext) break;
        l++;
      }
      if (next == nnext) break;
    }
    if (next == 0)
    {
      return (ret + fmt);
    }
    if (formats[l] != "%2" && formats[l] != "%n")
    {
      ret = ret + fmt[1, next - 1] + print(#[1], formats[l]);
      # = delete(#, 1);
    }
    else
    {
      ret = ret + fmt[1, next - 1] + print("", "%2s");
    }
    if (size(fmt) <= (next + size(formats[l]) - 1))
    {
      return (ret);
    }
    fmt = fmt[next + size(formats[l]), size(fmt)-next-size(formats[l]) + 1];
  }
}
example
{ "EXAMPLE:"; echo=2;
  ring r=0,(x,y,z),dp;
  module m=[1,y],[0,x+z];
  intmat M=betti(mres(m,0));
  list l = r, m, M;
  string s = sprintf("s:%s,%n l:%l", 1, 2); s;
  s = sprintf("s:%n%s", l); s;
  s = sprintf("s:%2%s", list(l)); s;
  s = sprintf("2l:%n%2l", list(l)); s;
  s = sprintf("%p", list(l)); s;
  s = sprintf("%;", list(l)); s;
  s = sprintf("%b", M); s;
}

proc printf(string fmt, list #)
"SYNTAX:  @code{printf (} string_expression @code{[,} any_expressions@code{] )}
RETURN:   none
PURPOSE:  @code{printf(fmt,...);} performs output formatting. The first
          argument is a format control string. Additional arguments may be
          required, depending on the content of the control string. A series
          of output characters is generated as directed by the control string;
          these characters are displayed (i.e., printed to standard out). @*
          The control string @code{fmt} is simply text to be copied, except
          that the string may contain conversion specifications. @*
          Do @code{help print;} for a listing of valid conversion
          specifications. As an addition to the conversions of @code{print},
          the @code{%n} and @code{%2} conversion specification does not
          consume an additional argument, but simply generates a newline
          character.
NOTE:     If one of the additional arguments is a list, then it should be
          enclosed once more into a @code{list()} command, since passing a
          list as an argument flattens the list by one level.
SEE ALSO: sprintf, fprintf, print, string
EXAMPLE : example printf; shows an example
"
{
  write("", sprintf(fmt, #));
}
example
{ "EXAMPLE:"; echo=2;
  ring r=0,(x,y,z),dp;
  module m=[1,y],[0,x+z];
  intmat M=betti(mres(m,0));
  list l=r,m,matrix(M);
  printf("s:%s,l:%l",1,2);
  printf("s:%s",l);
  printf("s:%s",list(l));
  printf("2l:%2l",list(l));
  printf("%p",matrix(M));
  printf("%;",matrix(M));
  printf("%b",M);
}


proc fprintf(link l, string fmt, list #)
"SYNTAX:  @code{fprintf (} link_expression@code{,} string_expression @code{[,}
                any_expressions@code{] )}
RETURN:   none
PURPOSE:  @code{fprintf(l,fmt,...);} performs output formatting.
          The second argument is a format control string. Additional
          arguments may be required, depending on the content of the
          control string. A series of output characters is generated as
          directed by the control string; these characters are
          written to the link l.
          The control string @code{fmt} is simply text to be copied, except
          that the string may contain conversion specifications.@*
          Do @code{help print;} for a listing of valid conversion
          specifications. As an addition to the conversions of @code{print},
          the @code{%n} and @code{%2} conversion specification does not
          consume an additional argument, but simply generates a newline
          character.
NOTE:     If one of the additional arguments is a list, then it should be
          enclosed once more into a @code{list()} command, since passing
          a list as an argument flattens the list by one level.
SEE ALSO: sprintf, printf, print, string
EXAMPLE : example fprintf; shows an example
"
{
  write(l, sprintf(fmt, #));
}
example
{ "EXAMPLE:"; echo=2;
  ring r=0,(x,y,z),dp;
  module m=[1,y],[0,x+z];
  intmat M=betti(mres(m,0));
  list l=r,m,M;
  link li="";   // link to stdout
  fprintf(li,"s:%s,l:%l",1,2);
  fprintf(li,"s:%s",l);
  fprintf(li,"s:%s",list(l));
  fprintf(li,"2l:%2l",list(l));
  fprintf(li,"%p",list(l));
  fprintf(li,"%;",list(l));
  fprintf(li,"%b",M);
}

//////////////////////////////////////////////////////////////////////////

/*
proc minres(list #)
{
  if (size(#) == 2)
  {
    if (typeof(#[1]) == "ideal" || typeof(#[1]) == "module")
    {
      if (typeof(#[2] == "int"))
      {
        return (res(#[1],#[2],1));
      }
    }
  }

  if (typeof(#[1]) == "resolution")
  {
    return minimizeres(#[1]);
  }
  else
  {
    return minimizeres(#);
  }

}
*/
///////////////////////////////////////////////////////////////////////////////

proc weightKB(def stc, int dd, list wim)
"SYNTAX: @code{weightKB (} module_expression@code{,} int_expression @code{,}
            list_expression @code{)}@*
         @code{weightKB (} ideal_expression@code{,} int_expression@code{,}
            list_expression @code{)}
RETURN:  the same as the input type of the first argument
PURPOSE: If @code{I,d,wim} denotes the three arguments then weightKB
         computes the weighted degree- @code{d} part of a vector space basis
         (consisting of monomials) of the quotient ring, resp. of the
         quotient module, modulo @code{I} w.r.t. weights given by @code{wim}
         The information about the weights is given as a list of two intvec:
            @code{wim[1]} weights for all variables (positive),
            @code{wim[2]} weights for the module generators.
NOTE:    This is a generalisation for the command @code{kbase} with the same
         first two arguments.
SEE ALSO: kbase
EXAMPLE: example weightKB; shows an example
"
{
  if(checkww(wim)){ERROR("wrong weights";);}
  kbclass();
  wwtop=wim[1];
  stc=interred(lead(stc));
  if(typeof(stc)=="ideal")
  {
    stdtop=stc;
    ideal out=widkbase(dd);
    delkbclass();
    return(out);
  }
  list mbase=kbprepare(stc);
  module mout;
  int im,ii;
  if(size(wim)>1){mmtop=wim[2];}
  else{mmtop=0;}
  for(im=size(mbase);im>0;im--)
  {
    stdtop=mbase[im];
    if(im>size(mmtop)){ii=dd;}
    else{ii=dd-mmtop[im];}
    mout=mout+widkbase(ii)*gen(im);
  }
  delkbclass();
  return(mout);
}
example
{ "EXAMPLE:"; echo=2;
  ring R=0, (x,y), wp(1,2);
  weightKB(ideal(0),3,intvec(1,2));
}

///////////////////////////////////////////////////////////////////////////////
// construct global values
static proc kbclass()
{
  intvec wwtop,mmtop;
  export (wwtop,mmtop);
  ideal stdtop,kbtop;
  export (stdtop,kbtop);
}
// delete global values
static proc delkbclass()
{
  kill wwtop,mmtop;
  kill stdtop,kbtop;
}
//  select parts of the modul
static proc kbprepare(module mstc)
{
  list rr;
  ideal kk;
  int i1,i2;
  mstc=transpose(mstc);
  for(i1=ncols(mstc);i1>0;i1--)
  {
    kk=0;
    for(i2=nrows(mstc[i1]);i2>0;i2--)
    {
      kk=kk+mstc[i1][i2];
    }
    rr[i1]=kk;
  }
  return(rr);
}
//  check for weights
static proc checkww(list vv)
{
  if(typeof(vv[1])!="intvec"){return(1);}
  intvec ww=vv[1];
  int mv=nvars(basering);
  if(size(ww)<mv){return(1);}
  while(mv>0)
  {
    if(ww[mv]<=0){return(1);}
    mv--;
  }
  if(size(vv)>1)
  {
    if(typeof(vv[2])!="intvec"){return(1);}
  }
  return(0);
}
// The "Caller" for ideals
//    dd   - the degree of the result
static proc widkbase(int dd)
{
  if((size(stdtop)==1)&&(deg(stdtop[1])==0)){return(0);}
  if(dd<=0)
  {
    if(dd<0){return(0);}
    else{return(1);}
  }
  int m1,m2;
  m1=nvars(basering);
  while(wwtop[m1]>dd)
  {
    m1--;
    if(m1==0){return(0);}
  }
  attrib(stdtop,"isSB",1);
  poly mo=1;
  if(m1==1)
  {
    m2=dd/wwtop[1];
    if((m2*wwtop[1])==dd)
    {
      mo=var(1)^m2;
      if(reduce(mo,stdtop)==mo){return(mo);}
      else{return(0);}
    }
  }
  kbtop=0;
  m2=dd;
  weightmon(m1-1,m2,mo);
  while(m2>=wwtop[m1])
  {
    m2=m2-wwtop[m1];
    mo=mo*var(m1);
    if(m2==0)
    {
      if(reduce(mo,stdtop)==mo)
      {
        kbtop=kbtop+mo;
        return(kbtop);
      }
    }
    weightmon(m1-1,m2,mo);
  }
  return(kbtop);
}
// the recursive prozedur
//    va     - number of the variable
//    drest  - rest of the degree
//    mm     - the candidate
static proc weightmon(int va, int drest, poly mm)
{
  while(wwtop[va]>drest)
  {
    va--;
    if(va==0){return();}
  }
  int m2;
  if(va==1)
  {
    m2=drest/wwtop[1];
    if((m2*wwtop[1])==drest)
    {
      mm=mm*var(1)^m2;
      if(reduce(mm,stdtop)==mm){kbtop=kbtop+mm;}
    }
    return();
  }
  m2=drest;
  weightmon(va-1,m2,mm);
  while(m2>=wwtop[va])
  {
    m2=m2-wwtop[va];
    mm=mm*var(va);
    if(m2==0)
    {
      if(reduce(mm,stdtop)==mm)
      {
        kbtop=kbtop+mm;
        return();
      }
    }
    weightmon(va-1,m2,mm);
  }
  return();
}
example
{ "EXAMPLE:"; echo=2;
  ring r=0,(x,y,z),dp;
  ideal i = x6,y4,xyz;
  intvec w = 2,3,6;
  weightKB(i, 12, list(w));
}
//////////////////////////////////////////////////////////////////////////////

/*
Versuche:
///////////////////////////////////////////////////////////////////////////////
proc downsizeSB (I, list #)
"USAGE:   downsizeSB(I [,l]); I ideal, l list of integers [default: l=0]
RETURN:  intvec, say v, with v[j] either 1 or 0. We have v[j]=1 if
         leadmonom(I[j]) is divisible by some leadmonom(I[k]) or if
         leadmonom(i[j]) == leadmonom(i[k]) and l[j] >= l[k], with k!=j.
PURPOSE: The procedure is applied in a situation where the standard basis
         computation in the basering R is done via a conversion through an
         overring Phelp with additional variables and where a direct
         imap from Phelp to R is too expensive.
         Assume Phelp is created by the procedure @code{par2varRing} or
         @code{hilbRing} and IPhelp is a SB in Phelp [ with l[j]=
         length(IPhelp(j)) or any other integer reflecting the complexity
         of a IPhelp[j] ]. Let I = lead(IPhelp) mapped to R and compute
         v = downsizeSB(imap(Phelp,I),l) in R. Then, if Ihelp[j] is deleted
         for all j with v[j]=1, we can apply imap to the remaining generators
         of Ihelp and still get SB in R  (in general not minimal).
EXAMPLE: example downsizeSB; shows an example"
{
   int k,j;
   intvec v,l;
   poly M,N,W;
   int c=size(I);
   if( size(#) != 0 )
   {
     if ( typeof(#[1]) == "intvec" )
     {
       l = #[1];
     }
     else
     {
        ERROR("// 2nd argument must be an intvec");
     }
   }

   l[c+1]=0;
   v[c]=0;

   j=0;
   while(j<c-1)
   {
     j++;
     M = leadmonom(I[j]);
     if( M != 0 )
     {
        for( k=j+1; k<=c; k++ )
        {
          N = leadmonom(I[k]);
          if( N != 0 )
          {
             if( (M==N) && (l[j]>l[k]) )
             {
               I[j]=0;
               v[j]=1;
               break;
             }
             if( (M==N) && (l[j]<=l[k]) || N/M != 0 )
             {
               I[k]=0;
               v[k]=1;
             }
          }
        }
      }
   }
   return(v);
}
example
{ "EXAMPLE:"; echo = 2;
   ring  r = 0,(x,y,z,t),(dp(3),dp);
   ideal i = x+y+z+t,xy+yz+xt+zt,xyz+xyt+xzt+yzt,xyzt-t4;
   ideal Id = std(i);
   ideal I = lead(Id);  I;
   ring S = (0,t),(x,y,z),dp;
   downsizeSB(imap(r,I));
   //Id[5] can be deleted, we still have a SB of i in the ring S

   ring R = (0,x),(y,z,u),lp;
   ideal i = x+y+z+u,xy+xu+yz+zu,xyz+xyu+xzu+yzu,xyzu-1;
   def Phelp = par2varRing()[1];
   setring Phelp;
   ideal IPhelp = std(imap(R,i));
   ideal I = lead(IPhelp);
   setring R;
   ideal I = imap(Phelp,I); I;
   intvec v = downsizeSB(I); v;
}
///////////////////////////////////////////////////////////////////////////
// PROBLEM: Die Prozedur funktioniert nur fuer Ringe die global bekannt
//          sind, also interaktiv, aber nicht aus einer Prozedur.
//          Z.B. funktioniert example imapDownsize; nicht

proc imapDownsize (string R, string I)
"SYNTAX: @code{imapDownsize (} string @code{,} string @code{)} *@
         First string must be the string of the name of a ring, second
         string must be the string of the name of an object in the ring.
TYPE:    same type as the object with name the second string
PURPOSE: maps the object given by the second string to the basering.
         If R resp. I are the first resp. second string, then
         imapDownsize(R,I) is equivalent to simplify(imap(`R`,`I`),34).
NOTE:    imapDownsize is usually faster than imap if `I` is large and if
         simplify has a great effect, since the procedure maps only those
         generators from `I` which are not killed by simplify( - ,34).
         This is useful if `I` is a standard bases for a block ordering of
         `R` and if some variables from the last block in `R` are mapped
         to parameters. Then the returned result is a standard basis in
         the basering.
SEE ALSO: imap, fetch, map
EXAMPLE: example imapDownsize; shows an example"
{
       def BR = basering;
       int k;

       setring `R`;
       def @leadI@ = lead(`I`);
       int s = ncols(@leadI@);
       setring BR;
       ideal @leadI@ = simplify(imap(`R`,@leadI@),32);
       intvec vi;
       for (k=1; k<=s; k++)
       {
         vi[k] = @leadI@[k]==0;
       }
       kill @leadI@;

       setring `R`;
       kill @leadI@;
       for (k=1;  k<=s; k++)
       {
           if( vi[k]==1 )
           {
              `I`[k]=0;
           }
       }
       `I` = simplify(`I`,2);

       setring BR;
       return(imap(`R`,`I`));
}
example
{ "EXAMPLE:"; echo = 2;
   ring  r = 0,(x,y,z,t),(dp(3),dp);
   ideal i = x+y+z+t,xy+yz+xt+zt,xyz+xyt+xzt+yzt,xyzt-1;
   i = std(i); i;

   ring s = (0,t),(x,y,z),dp;
   imapDownsize("r","i");     //i[5] is omitted since lead(i[2]) | lead(i[5])
}
///////////////////////////////////////////////////////////////////////////////
//die folgende proc war fuer groebner mit fglm vorgesehen
//um die projektive Dimension korrekt zu berechnen, muss man aber
//voerher ein SB bzgl. einer Gradordnung berechnen und dann homogenisieren.
//Sonst koennen hoeherdimensionale Komponenten in Unendlich entstehen

proc projInvariants(ideal i,list #)
"SYNTAX: @code{projInvariants (} ideal_expression @code{)} @*
         @code{projInvariants (} ideal_expression@code{,} list of string_expres          sions@code{)}
TYPE:    list, say L, with L[1] and L[2] of type int and L[3] of type intvec
PURPOSE: Computes the (projective) dimension (L[1]), degree (L[2]) and the
         first Hilbert series (L[3], as intvec) of the homogenized ideal
         in the ring given by the procedure @code{hilbRing} with global
         ordering dp (resp. wp if the variables have weights >1)
         If an argument of type string @code{\"std\"} resp. @code{\"slimgb\"}
         is given, the standard basis computatuion uses @code{std} or
         @code{slimgb}, otherwise a heuristically chosen method (default)
NOTE:    Homogenized means weighted homogenized with respect to the weights
         w[i] of the variables var(i) of the basering. The returned dimension,
         degree and Hilbertseries are the respective invariants of the
         projective variety defined by the homogenized ideal. The dimension
         is equal to the (affine) dimension of the ideal in the basering
         (degree and Hilbert series make only sense for homogeneous ideals).
SEE ALSO: dim, dmult, hilb
KEYWORDS: dimension, degree, Hilbert function
EXAMPLE: example projInvariants;  shows an example"
{
  def P = basering;
  int p_opt;
  string s_opt = option();
  if (find(option(), "prot"))  { p_opt = 1; }

//---------------- check method and clear denomintors --------------------
  int k;
  string method;
  for (k=1; k<=size(#); k++)
  {
     if (typeof(#[k]) == "string")
     {
       method = method + "," + #[k];
     }
  }

  if (npars(P) > 0)             //clear denominators of parameters
  {
    for( k=ncols(i); k>0; k-- )
    {
       i[k]=cleardenom(i[k]);
    }
  }

//------------------------ change to hilbRing ----------------------------
     list hiRi = hilbRing(i);
     intvec W = hiRi[2];
     def Philb = hiRi[1];      //note: Philb is no qring and the predefined
     setring Philb;            //ideal Id(1) in Philb is homogeneous
     int di, de;               //for dimension, degree
     intvec hi;                //for hilbert series

//-------- compute Hilbert function of homogenized ideal in Philb ---------
//Philb has only 1 block. There are three cases

     string algorithm;       //possibilities: std, slimgb, stdorslimgb
     //define algorithm:
     if( find(method,"std") && !find(method,"slimgb") )
     {
        algorithm = "std";
     }
     if( find(method,"slimgb") && !find(method,"std") )
     {
         algorithm = "slimgb";
     }
     if( find(method,"std") && find(method,"slimgb") ||
         (!find(method,"std") && !find(method,"slimgb")) )
     {
         algorithm = "stdorslimgb";
     }

     if ( algorithm=="std" || ( algorithm=="stdorslimgb" && char(P)>0 ) )
     {
        if (p_opt) {"std in ring " + string(Philb);}
        Id(1) = std(Id(1));
        di = dim(Id(1))-1;
        de = mult(Id(1));
        hi = hilb( Id(1),1,W );
     }
     if ( algorithm=="slimgb" || ( algorithm=="stdorslimgb" && char(P)==0 ) )
     {
        if (p_opt) {"slimgb in ring " + string(Philb);}
        Id(1) = slimgb(Id(1));
        di = dim( Id(1) );
        if (di > -1)
        {
           di = di-1;
        }
        de = mult( Id(1) );
        hi = hilb( Id(1),1,W );
     }
     kill Philb;
     list L = di,de,hi;
     return(L);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r = 32003,(x,y,z),lp;
   ideal i = y2-xz,x2-z;
   projInvariants(i);

   ring R = (0),(x,y,z,u,v),lp;
   //minpoly = x2+1;
   ideal i = x2+1,x2+y+z+u+v,xyzuv-1;
   projInvariants(i);
   qring S =std(x2+1);
   ideal i = imap(R,i);
   projInvariants(i);
}

*/