source: git/Singular/LIB/presolve.lib @ b7b65ee

///////////////////////////////////////////////////////////////////////////////
//changed GMG 20.5.09: degreepart, linearpart, elimpart
///////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="Symbolic-numerical solving";
info="
LIBRARY:  presolve.lib     Pre-Solving of Polynomial Equations
AUTHOR:   Gert-Martin Greuel, email: greuel@mathematik.uni-kl.de,

PROCEDURES:
 degreepart(id,d1,d2);  elements of id of total degree >= d1 and <= d2, and rest
 elimlinearpart(id);    linear part eliminated from id
 elimpart(id[,n]);      partial elimination of vars [among first n vars]
 elimpartanyr(i,p);     factors of p partially eliminated from i in any ring
 fastelim(i,p[..]);     fast elimination of factors of p from i [options]
 findvars(id[..]);      ideal of variables occuring in id [more information]
 hilbvec(id[,c,o]);     intvec of Hilberseries of id [in char c and ord o]
 linearpart(id);        elements of id of total degree <=1
 tolessvars(id[,]);     maps id to new basering having only vars occuring in id
 solvelinearpart(id);   reduced std-basis of linear part of id
 sortandmap(id[..]);    map to new basering with vars sorted w.r.t. complexity
 sortvars(id[n1,p1..]); sort vars w.r.t. complexity in id [different blocks]
 valvars(id[..]);       valuation of vars w.r.t. to their complexity in id
 idealSplit(id,tF,fS);  a list of ideals such that their intersection
                        has the same radical as id
                       ( parameters in square brackets [] are optional)
";

LIB "inout.lib";
LIB "general.lib";
LIB "matrix.lib";
LIB "ring.lib";
LIB "elim.lib";
///////////////////////////////////////////////////////////////////////////////
proc shortid (id,int n,list #)
"USAGE:   shortid(id,n[,e]); id= ideal/module, n,e=integers
RETURN:  - if called with two arguments or e=0:
@*       same type as id, containing generators of id having <= n terms.
@*       - if called with three arguments and e!=0:
@*       a list L:
@*       L[1]: same type as id, containing generators of id having <= n terms.
@*       L[2]: number of corresponding generator of id
NOTE:    May be used to compute partial standard basis in case id is to hard
EXAMPLE: example shortid; shows an example
"
{
  intvec v;
  int ii;
  for(ii=1; ii<=ncols(id); ii++)
  {
   if (size(id[ii]) <=n and id[ii]!=0 )
   {
     v=v,ii;
   }
   if (size(id[ii]) > n )
   {
       id[ii]=0;
   }
  }
  if( size(v)>1 )
  {
    v = v[2..size(v)];
  }
  id = simplify(id,2);
  list L = id,v;
  if ( size(#)==0 )
  {
    return(id);
  }
  if ( size(#)!=0 )
  {
    if(#[1]==0)
    {
      return(id);
    }
    if(#[1]!=0)
    {
      return(L);
    }
  }
}
example
{ "EXAMPLE:"; echo = 2;
   ring s=0,(x,y,z,w),dp;
   ideal i = (x3+y2+yw2)^2,(xz+z2)^2,xyz-w2-xzw;
   shortid(i,3);
}
///////////////////////////////////////////////////////////////////////////////

proc degreepart (id,int d1,int d2,list #)
"USAGE:   degreepart(id,d1,d2[,v]);  id=ideal/module, d1,d1=integers, v=intvec
RETURN:  list of size 2,
         _[1]: generators of id of [v-weighted] total degree >= d1 and <= d2
          (default: v = 1,...,1)
         _[2]: remaining generators of id
NOTE:    if id is of type int/number/poly it is converted to ideal, if id is
         of type intmat/matrix/vector to module and then the corresponding
         generators are computed
EXAMPLE: example degreepart; shows an example
"
{
   if( typeof(id)=="int" or typeof(id)=="number"
       or typeof(id)=="ideal" or typeof(id)=="poly" )
   {
      ideal dpart = ideal(id);
   }
   if( typeof(id)=="intmat" or typeof(id)=="matrix"
       or typeof(id)=="module" or typeof(id)=="vector")
   {
      module dpart = module(id);
   }

   def epart = dpart;
   int s,ii = ncols(id),0;
   if ( size(#)==0 )
   {
      for ( ii=1; ii<=s; ii++ )
      {
         dpart[ii] = (jet(id[ii],d1-1)==0)*(id[ii]==jet(id[ii],d2))*id[ii];
         epart[ii] = (size(dpart[ii])==0) * id[ii];
      }
   }
   else
   {
      for ( ii=1; ii<=s; ii=ii+1 )
      {
      dpart[ii]=(jet(id[ii],d1-1,#[1])==0)*(id[ii]==jet(id[ii],d2,#[1]))*id[ii];
       epart[ii] = (size(dpart[ii])==0)*id[ii];
      }
   }
   list L = simplify(dpart,2),simplify(epart,2);
   return(L);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   ideal i=1+x+x2+x3+x4,3,xz+y3+z8;
   degreepart(i,0,4);

   module m=[x,y,z],x*[x3,y2,z],[1,x2,z3,0,1];
   intvec v=2,3,6;
   show(degreepart(m,8,8,v));
}
///////////////////////////////////////////////////////////////////////////////

proc linearpart (id)
"USAGE:   linearpart(id);  id=ideal/module
RETURN:  list of size 2,
         _[1]: generators of id of total degree <= 1
         _[2]: remaining generators of id
NOTE:    all variables have degree 1 (independent of ordering of basering)
EXAMPLE: example linearpart; shows an example
"
{
   return(degreepart(id,0,1));
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   ideal i=1+x+x2+x3,3,x+3y+5z;
   linearpart(i);

   module m=[x,y,z],x*[x3,y2,z],[1,x2,z3,0,1];
   show(linearpart(m));
}
///////////////////////////////////////////////////////////////////////////////

proc elimlinearpart (ideal i,list #)
"USAGE:   elimlinearpart(i[,n]);  i=ideal, n=integer,@*
          default: n=nvars(basering)
RETURN:   list L with 5 entries:
  @format
  L[1]: ideal obtained from i by substituting from the first n variables those
        which appear in a linear part of i, by putting this part into triangular
        form
  L[2]: ideal of variables which have been substituted
  L[3]: ideal, j-th element defines substitution of j-th var in [2]
  L[4]: ideal of variables of basering, eliminated ones are set to 0
  L[5]: ideal, describing the map from the basering to itself such that
        L[1] is the image of i
  @end format
NOTE:    the procedure always interreduces the ideal i internally w.r.t.
         ordering dp.
EXAMPLE: example elimlinearpart; shows an example
"
{
   int ii,n,k,ringchange;
   string o;
   intvec getoption = option(get);
   option(redSB);
   def BAS = basering;
   n = nvars(BAS);
   list gnirlist = ringlist(basering);
   list g3 = gnirlist[3];
   list g32 = g3[size(g3)];

//---------------------------------- start ------------------------------------
   if ( size(#)!=0 ) {  n=#[1]; }
   ideal maxi,rest = maxideal(1),0;
   if ( n < nvars(BAS) )
   {
      rest = maxi[n+1..nvars(BAS)];     //variables which are not substituted
   }
   attrib(rest,"isSB",1);

//-------------------- find linear part and reduce rest ----------------------
// Perhaps for big systems, check only those generators of id
// which do not contain elements not to be eliminated

   //ideal id = interred(i);
   //## gmg, geaendert 9/2008: interred sehr lange z.B. bei Leonard1 in normal,
   //daher interred ersetzt durch: std nur auf linearpart angewendet
   //Wechsel zu dp Ordnung (da Lin affin linear)

//--------------- replace ordering different from dp by dp -------------------
   o = "dp("+string(n)+")";
   if( ! find(ordstr(BAS),o) or find(ordstr(BAS),"a") )
   {
      ringchange = 1;                         //remember change of ring
      intvec V;
      V[n]=0; V=V+1;                          //weights for dp ordering
      gnirlist[3] = list("dp",V), g32;
      def newBAS = ring(gnirlist);            //change of ring to dp ordering
      setring newBAS;
      ideal rest = imap(BAS,rest);
      attrib(rest,"isSB",1);
      ideal i = imap(BAS,i);
   }

   list  Lin = linearpart(i);
   ideal lin = std(Lin[1]);          //SB of ideal generated by polys of i
                                     //having at most degree 1
   ideal id = Lin[2];                //remaining polys from i, of deg > 1
   id = simplify(NF(id,lin),2);      //instead of subst
   ideal id1 = linearpart(id)[1];
   while( size(id1) != 0 )           //repeat to find linear parts
   {
      lin = lin,id1;
      lin = std(lin);
      id = simplify(NF(id,lin),2);   //instead of subst, (### is faster)
      id1 = linearpart(id)[1];
   }
//------------- check for special case of unit ideal and return ---------------
   int check;
   if( lin[1] == 1 )
   {
     check = 1;
   }
   else
   {
     for (ii=1; ii<=size(id); ii++ )
     {
       if ( id[ii] == 1 )
       {
         check = 1; break;
        }
      }
    }

   if (check == 1)        //case of a unit ideal
   {
     setring BAS;
     list L = ideal(1), ideal(0), ideal(0), maxideal(1), maxideal(1);
     option(set,getoption);
     return(L);
   }
//----- remove generators from lin containing vars not to be eliminated  ------
   if ( n < nvars(BAS) )
   {
      for ( ii=1; ii<=size(lin); ii++ )
      {
         if ( reduce(lead(lin[ii]),rest) == 0 )
         {
            id=lin[ii],id;
            lin[ii] = 0;
         }
      }
   }
   lin = simplify(lin,1);
   ideal eva = lead(lin);               //vars to be eliminated
   attrib(eva,"isSB",1);
   ideal neva = NF(maxideal(1),eva);    //vars not to be eliminated
//------------------ go back to original ring end return  ---------------------

   if ( ringchange  )                   //i.e there was a ring change
   {
      setring BAS;
      ideal id = imap(newBAS,id);
      ideal eva = imap(newBAS,eva);
      ideal lin = imap(newBAS,lin);
      ideal neva = imap(newBAS,neva);
   }

   eva = eva[ncols(eva)..1];  // sorting according to variables in basering
   lin = lin[ncols(lin)..1];
   ideal phi = neva;
   k = 1;
   for( ii=1; ii<=n; ii++ )
   {
      if( neva[ii] == 0 )
      {
         phi[ii] = eva[k]-lin[k];
         k=k+1;
      }
   }

   list L = id, eva, lin, neva, phi;
   option(set,getoption);
   return(L);
}
example
{ "EXAMPLE:"; echo = 2;
   ring s=0,(u,x,y,z),dp;
   ideal i = u3+y3+z-x,x2y2+z3,y+z+1,y+u;
   elimlinearpart(i);
}
///////////////////////////////////////////////////////////////////////////////
proc elimpart (ideal i,list #)
"USAGE:   elimpart(i [,n,e] );  i=ideal, n,e=integers
         n   : only the first n vars are considered for substitution,@*
         e =0: substitute from linear part of i (same as elimlinearpart)@*
         e!=0: eliminate also by direct substitution@*
         (default: n = nvars(basering), e = 1)
RETURN:  list of 5 objects:
  @format
  [1]: ideal obtained by substituting from the first n variables those
       from i, which appear in the linear part of i (or, if e!=0, which
       can be expressed directly in the remaining vars)
  [2]: ideal, variables which have been substituted
  [3]: ideal, i-th element defines substitution of i-th var in [2]
  [4]: ideal of variables of basering, substituted ones are set to 0
  [5]: ideal, describing the map from the basering, say k[x(1..m)], to
       itself onto k[..variables from [4]..] and [1] is the image of i
  @end format
  The ideal i is generated by [1] and [3] in k[x(1..m)], the map [5]
  maps [3] to 0, hence induces an isomorphism
  @format
            k[x(1..m)]/i -> k[..variables from [4]..]/[1]
  @end format
NOTE:    Applying elimpart to interred(i) may result in more substitutions.
         However, interred may be more expansive than elimpart for big ideals
EXAMPLE: example elimpart; shows an example
"
{
   def BAS = basering;
   int n,e = nvars(BAS),1;
   if ( size(#)==1 ) {  n=#[1]; }
   if ( size(#)==2 ) {  n=#[1]; e=#[2];}
//----------- interreduce linear part with proc elimlinearpart ----------------
// lin = ideal i after interreduction with linear part
// eva = eliminated (substituted) variables
// sub = polynomials defining substitution
// neva= not eliminated variables
// phi = map describing substitution

   list L = elimlinearpart(i,n);
   ideal lin, eva, sub, neva, phi = L[1], L[2], L[3], L[4], L[5];
   if ( e == 0 )
   {
       return(L);
   }

//-------- direct substitution of variables if possible and if e!=0 -----------
// first find terms lin1 in lin of pure degree 1 in each polynomial of lin
// k1 = pure degree 1 part, i.e. nonzero elts of lin1, renumbered
// k2 = lin2 (=matrix(lin) - matrix(lin2)), renumbered
// kin = matrix(k1)+matrix(k2) = those polys of lin which contained a pure
// degree 1 part.
/*
Alte Version mit interred:
// Then go to ring newBAS with ordering c,dp(n) and create a matrix with
// size(k1) colums and 2 rows, such that if [f1,f2] is a column of M then f1+f2
// is one of the polys of lin containing a pure degree 1 part and f1 is this
// part interreduce this matrix (i.e. Gauss elimination on linear part, with
// rest transformed accordingly).
//Ist jetzt durch direkte Substitution gemacht (schneller!)
         //Variante falls wieder interred angewendet werden soll:
         //ideal k12 = k1,k2;
         //matrix M = matrix(k12,2,kk);     //degree 1 part is now in row 1
         //M = interred(M);
         //### interred zu teuer, muss nicht sein. Wenn interred angewendet
         //werden soll, vorher in Ring mit Ordnung (c,dp) wechseln!
         //Abfrage:  if( ordstr(BAS) != "c,dp("+string(n)+")" )
         //auf KEINEN Fall std (wird zu gross)
         //l = ncols(M);
         //k1 = M[1,1..l];
         //k2 = M[2,1..l];
Interred ist jetzt ganz weggelassen. Aber es gibt Beispiele wo interred polys
mit Grad 1 Teilen produziert, die vorher nicht da waren (aus polys, die einen konstanten Term haben).
z.B. i=xy2-xu4-x+y2,x2y2+z3+zy,y+z2+1,y+u2;, interred(i)=z2+y+1,y2-x,u2+y,x3-z
-z ergibt ich auch i[2]-z*i[3] mit option(redThrough)
statt interred kann man hier auch NF(i,i[3])+i[3] verwenden
hier lifert elimpart(i) 2 Substitutionen (x,y) elimpart(interred(i))
aber 3 (x,y,z)
Da interred oder NF aber die Laenge der polys vergroessern kann, nicht gemacht
*/
   int ii, kk;
   ideal k1, k2, lin2;
   int l = ncols(lin);                  // lin=i after applying elimlinearpart
   ideal lin1 = ideal(matrix(jet(lin,1))-matrix(jet(lin,0)));  // part of pure degree 1
   //Note: If i,i1,i2 are ideals, then i = i1 - i2 is equivalent to
   //i = ideal(matrix(i1) - matrix(i2))

   if (size(lin1) == 0 )
   {
       return(L);
   }

   //-------- check candidates for direct substitution of variables ----------
   //since lin1 != 0 there are candidates for substituting variables

   lin2 = matrix(lin) - matrix(lin1);      //difference as matrix
   // rest of lin, part of pure degree 1 substracted from each generator of lin

   for( ii=1; ii<=l; ii++ )
   {
      if( lin1[ii] != 0 )
      {
         kk = kk+1;
         k1[kk] = lin1[ii];  // part of pure degree 1, renumbered
         k2[kk] = lin2[ii];  // rest of those polys which had a degree 1 part
         lin2[ii] = 0;
      }
   }
   //Now each !=0 generator of lin2 contains only constant terms or terms of
   //degree >= 2, hence lin 2 can never be used for further substitutions
   //We have: lin = ideal(matrix(k1)+matrix(k2)), lin2

   ideal kin = matrix(k1)+matrix(k2);
   //kin = polys of lin which contained a pure degree 1 part.
   kin = simplify(kin,2);
   l = size(kin);                      //l != 0 since lin1 != 0
   poly p,kip,vip, cand;
   int count=1;
   while ( count != 0 )
   {
         count = 0;
         for ( ii=1; ii<=n; ii++  )    //start direct substitution of var(ii)
         {
            for (kk=1; kk<=l; kk++ )
            {
               p = kin[kk]/var(ii);
               //if ( deg(p) == 0 )
               //old test, does not work if some var has deg 0
               //geaendert Mai 09 gmg

               if( p!=0 & p == jet(p,0) )
                   //this means that kin[kk]= p*var(ii) + h,
                   //with p=const !=0 and h not depending on var(ii)
               {
                  //we look for the shortest candidate to substitute var(ii)
                  if ( cand == 0 )
                  {
                     cand = kin[kk];  //candidate for substituting var(ii)
                  }
                  else
                  {
                     if ( size(kin[kk]) < size(cand) )
                     {
                        cand = kin[kk];
                     }
                  }
                }
            }
            if ( cand != 0 )
            {
                  p = cand/var(ii);
                  kip = cand/p;     //normalized polynomial of kin w.r.t var(ii)
                  eva = eva+var(ii); //var(ii) added to list of elimin. vars
                  neva[ii] = 0;
                  sub = sub+kip;     //polynomial defining substituion
                  //## gmg: geaendert 08/2008, map durch subst ersetzt
                  //(viel schneller)
                  vip = var(ii) - kip;  //polynomial to be substituted
                  lin = subst(lin, var(ii), vip);  //subst in rest
                  lin = simplify(lin,2);
                  kin = subst(kin, var(ii), vip);  //subst in pure dgree 1 part
                  kin = simplify(kin,2);
                  l = size(kin);
                  count = 1;
            }
            cand=0;
         }
   }

   lin = kin+lin;

   for( ii=1; ii<=size(lin); ii++ )
   {
      lin[ii] = cleardenom(lin[ii]);
   }

   for( ii=1; ii<=n; ii++ )
   {
      for( kk=1; kk<=size(eva); kk++ )
      {
         if (phi[ii] == eva[kk] )
         {  phi[ii] = eva[kk]-sub[kk]; break; }
      }
   }
   map psi = BAS,phi;
   ideal phi1 = maxideal(1);
   for(ii=1; ii<=size(eva); ii++)
   {
      phi1=psi(phi1);
   }
   L = lin, eva, sub, neva, phi1;
   return(L);
}
example
{ "EXAMPLE:"; echo = 2;
   ring s=0,(u,x,y,z),dp;
   ideal i = xy2-xu4-x+y2,x2y2+z3+zy,y+z2+1,y+u2;
   elimpart(i);

   i = interred(i); i;
   elimpart(i);

   elimpart(i,2);
}

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

proc elimpartanyr (ideal i, list #)
"USAGE:   elimpartanyr(i [,p,e] );  i=ideal, p=polynomial, e=integer@*
         p: product of vars to be eliminated,@*
         e =0: substitute from linear part of i (same as elimlinearpart)@*
         e!=0: eliminate also by direct substitution@*
         (default: p=product of all vars, e=1)
RETURN:  list of 6 objects:
  @format
  [1]: (interreduced) ideal obtained by substituting from i those vars
       appearing in p, which occur in the linear part of i (or which can
       be expressed directly in the remaining variables, if e!=0)
  [2]: ideal, variables which have been substituted
  [3]: ideal, i-th element defines substitution of i-th var in [2]
  [4]: ideal of variables of basering, substituted ones are set to 0
  [5]: ideal, describing the map from the basering, say k[x(1..m)], to
       itself onto k[..variables fom [4]..] and [1] is the image of i
  [6]: int, # of vars considered for substitution (= # of factors of p)
  @end format
  The ideal i is generated by [1] and [3] in k[x(1..m)], the map [5]
  maps [3] to 0, hence induces an isomorphism
  @format
            k[x(1..m)]/i -> k[..variables fom [4]..]/[1]
  @end format
NOTE:    the procedure uses @code{execute} to create a ring with ordering dp
         and vars placed correctly and then applies @code{elimpart}.
EXAMPLE: example elimpartanyr; shows an example
"
{
   def P = basering;
   int j,n,e = 0,0,1;
   poly p = product(maxideal(1));
   if ( size(#)==1 ) { p=#[1]; }
   if ( size(#)==2 ) { p=#[1]; e=#[2]; }
   string a,b;
   for ( j=1; j<=nvars(P); j++ )
   {
      if (deg(p/var(j))>=0) { a = a+varstr(j)+","; n = n+1; }
      else { b = b+varstr(j)+","; }
   }
   if ( size(b) != 0 ) { b = b[1,size(b)-1]; }
   else { a = a[1,size(a)-1]; }
   execute("ring gnir ="+charstr(P)+",("+a+b+"),dp;");
   ideal i = imap(P,i);
   list L = elimpart(i,n,e)+list(n);
   setring P;
   list L = imap(gnir,L);
   return(L);
}
example
{ "EXAMPLE:"; echo = 2;
   ring s=0,(x,y,z),dp;
   ideal i = x3+y2+z,x2y2+z3,y+z+1;
   elimpartanyr(i,z);
}
///////////////////////////////////////////////////////////////////////////////

proc fastelim (ideal i, poly p, list #)
"USAGE:   fastelim(i,p[h,o,a,b,e,m]); i=ideal, p=polynomial; h,o,a,b,e=integers@*
          p: product of variables to be eliminated;@*
  Optional parameters:
  @format
  - h !=0: use Hilbert-series driven std-basis computation
  - o !=0: use proc @code{valvars} for a - hopefully - optimal ordering of vars
  - a !=0: order vars to be eliminated w.r.t. increasing complexity
  - b !=0: order vars not to be eliminated w.r.t. increasing complexity
  - e !=0: use @code{elimpart} first to eliminate easy part
  - m !=0: compute a minimal system of generators
  @end format
  (default: h,o,a,b,e,m = 0,1,0,0,0,0)
RETURN:  ideal obtained from i by eliminating those variables, which occur in p
EXAMPLE: example fastelim; shows an example.
"
{
   def P = basering;
   int h,o,a,b,e,m = 0,1,0,0,0,0;
   if ( size(#) == 1 ) { h=#[1]; }
   if ( size(#) == 2 ) { h=#[1]; o=#[2]; }
   if ( size(#) == 3 ) { h=#[1]; o=#[2]; a=#[3]; }
   if ( size(#) == 4 ) { h=#[1]; o=#[2]; a=#[3]; b=#[4];}
   if ( size(#) == 5 ) { h=#[1]; o=#[2]; a=#[3]; b=#[4]; e=#[5]; }
   if ( size(#) == 6 ) { h=#[1]; o=#[2]; a=#[3]; b=#[4]; e=#[5]; m=#[6]; }
   list L = elimpartanyr(i,p,e);
   poly q = product(L[2]);     //product of vars which are already eliminated
   if ( q==0 ) { q=1; }
   p = p/q;                    //product of vars which must still be eliminated
   int nu = size(L[5])-size(L[2]);   //number of vars which must still be eliminated
   if ( p==1 )                 //ready if no vars are left
   {                           //compute minbase if 3-rd argument !=0
      if ( m != 0 ) { L[1]=minbase(L[1]); }
      return(L);
   }
//---------------- create new ring with remaining variables -------------------
   string newvar = string(L[4]);
   L = L[1],p;
   execute("ring r1=("+charstr(P)+"),("+newvar+"),"+"dp;");
   list L = imap(P,L);
//------------------- find "best" ordering of variables  ----------------------
   newvar = string(maxideal(1));
   if ( o != 0 )
   {
      list ordevar = valvars(L[1],a,L[2],b);
      intvec v = ordevar[1];
      newvar=string(sort(maxideal(1),v)[1]);
//------------ create new ring with "best" ordering of variables --------------
      def r0=changevar(newvar);
      setring r0;
      list L = imap(r1,L);
      kill r1;
      def r1 = r0;
      kill r0;
   }
//----------------- h==0: eliminate remaining vars directly -------------------
   if ( h == 0 )
   {
      L[1] = eliminate(L[1],L[2]);
      def r2 = r1;
   }
   else
//------- h!=0: homogenize and compute Hilbert series using hilbvec ----------
   {
      intvec hi = hilbvec(L[1]);         // Hilbert series of i
      execute("ring r2=("+charstr(P)+"),("+varstr(basering)+",@homo),dp;");
      list L = imap(r1,L);
      L[1] = homog(L[1],@homo);          // @homo = homogenizing var
//---- use Hilbert-series to eliminate variables with Hilbert-driven std -----
      L[1] = eliminate(L[1],L[2],hi);
      L[1]=subst(L[1],@homo,1);          // dehomogenize by setting @homo=1
   }
   if ( m != 0 )                         // compute minbase
   {
      if ( #[1] != 0 ) { L[1] = minbase(L[1]); }
   }
   def id = L[1];
   setring P;
   return(imap(r2,id));
}
example
{ "EXAMPLE:"; echo = 2;
   ring s=31991,(e,f,x,y,z,t,u,v,w,a,b,c,d),dp;
   ideal i = w2+f2-1, x2+t2+a2-1,  y2+u2+b2-1, z2+v2+c2-1,
            d2+e2-1, f4+2u, wa+tf, xy+tu+ab;
   fastelim(i,xytua,1,1);       //with hilb,valvars
   fastelim(i,xytua,1,0,1);     //with hilb,minbase
}
///////////////////////////////////////////////////////////////////////////////

proc faststd (@id, list #)
"USAGE:   faststd(id [,\"hilb\",\"sort\",\"dec\",o,\"blocks\"]);
         id=ideal/module, o=string (allowed:\"lp\",\"dp\",\"Dp\",\"ls\",
         \"ds\",\"Ds\"),  \"hilb\",\"sort\",\"dec\",\"block\" options for
         Hilbert-driven std, and the procedure sortandmap
RETURN:  a ring R, in which an ideal STD_id is stored: @*
         - the ring R differs from the active basering only in the choice
         of monomial ordering and in the sorting of the variables.
         - STD_id is a standard basis for the image (under imap) of the input
         ideal/module id with respect to the new monomial ordering. @*
NOTE:    Using the optional input parameters, we may modify the computations
         performed: @*
         - \"hilb\"  : use Hilbert-driven standard basis computation@*
         - \"sort\"  : use 'sortandmap' for a best sorting of the variables@*
         - \"dec\"   : order vars w.r.t. decreasing complexity (with \"sort\")@*
         - \"block\" : create block ordering, each block having ordstr=o, s.t.
                     vars of same complexity are in one block (with \"sort\")@*
         - o       : defines the basic ordering of the resulting ring@*
         [default: o=ordering of 1st block of basering (if allowed, else o=\"dp\"],
                  \"sort\", if none of the optional parameters is given @*
         This procedure is only useful for hard problems where other methods fail.@*
         \"hilb\" is useful for hard orderings (as \"lp\") or for characteristic 0,@*
         it is correct for \"lp\",\"dp\",\"Dp\" (and for block orderings combining
         these) but not for s-orderings or if the vars have different weights.@*
         There seem to be only few cases in which \"dec\" is fast.
SEE ALSO: groebner
EXAMPLE: example faststd; shows an example.
"
{
   def @P = basering;
   int @h,@s,@n,@m,@ii = 0,0,0,0,0;
   string @o,@va,@c = ordstr(basering),"","";
//-------------------- prepare ordering and set options -----------------------
   if ( @o[1]=="c" or @o[1]=="C")
      {  @o = @o[3,2]; }
   else
      { @o = @o[1,2]; }
   if( @o[1]!="d" and @o[1]!="D" and @o[1]!="l")
      { @o="dp"; }

   if (size(#) == 0 )
      { @s = 1; }
   for ( @ii=1; @ii<=size(#); @ii++ )
   {
      if ( typeof(#[@ii]) != "string" )
      {
         "// wrong syntax! type: help faststd";
         return();
      }
      else
      {
         if ( #[@ii] == "hilb"  ) { @h = 1; }
         if ( #[@ii] == "dec"   ) { @n = 1; }
         if ( #[@ii] == "block" ) { @m = 1; }
         if ( #[@ii] == "sort"  ) { @s = 1; }
         if ( #[@ii]=="lp" or #[@ii]=="dp" or #[@ii]=="Dp" or #[@ii]=="ls"
              or #[@ii]=="ds" or #[@ii]=="Ds" ) { @o = #[@ii]; }
      }
   }
   if( voice==2 ) { "// chosen options, hilb sort dec block:",@h,@s,@n,@m; }

//-------------------- nosort: create ring with new name ----------------------
   if ( @s==0 )
   {
      execute("ring @S1 =("+charstr(@P)+"),("+varstr(@P)+"),("+@o+");");
      def STD_id = imap(@P,@id);
      if ( @h==0 ) { STD_id = std(STD_id); }
   }

//---------------------- no hilb: compute SB directly -------------------------
   if ( @s != 0 and @h == 0 )
   {
      intvec getoption = option(get);
      option(redSB);
      @id = interred(sort(@id)[1]);
      poly @p = product(maxideal(1),1..nvars(@P));
      def @S1=sortandmap(@id,@n,@p,0,@o,@m);
      setring @S1;
      option(set,getoption);
      def STD_id=imap(@S1,IMAG);
      STD_id = std(STD_id);
   }
//------- hilb: homogenize and compute Hilbert-series using hilbvec -----------
// this uses another standardbasis computation
   if ( @h != 0 )
   {
      execute("ring @Q=("+charstr(@P)+"),("+varstr(@P)+",@homo),("+@o+");");
      def @id = imap(@P,@id);
      @id = homog(@id,@homo);               // @homo = homogenizing var
      if ( @s != 0 )
      {
        intvec getoption = option(get);
        option(redSB);
        @id = interred(sort(@id)[1]);
        poly @p = product(maxideal(1),1..(nvars(@Q)-1));
        def @S1=sortandmap(@id,@n,@p,0,@o,@m);
        setring @S1;
        option(set,getoption);
        kill @Q;
        def @Q= basering;
        def @id = IMAG;
      }
      intvec @hi;                     // encoding of Hilbert-series of i
      @hi = hilbvec(@id);
      //if ( @s!=0 ) { @hi = hilbvec(@id,"32003",ordstr(@Q)); }
      //else { @hi = hilbvec(@id); }
//-------------------------- use Hilbert-driven std --------------------------
      @id = std(@id,@hi);
      @id = subst(@id,@homo,1);             // dehomogenize by setting @homo=1
      @va = varstr(@Q)[1,size(varstr(@Q))-6];
      if ( @s!=0 )
      {
         @o = ordstr(@Q);
         if ( @o[1]=="c" or @o[1]=="C") { @o = @o[1,size(@o)-6]; }
         else { @o = @o[1,size(@o)-8] + @o[size(@o)-1,2]; }
      }
      kill @S1;
      execute("ring @S1=("+charstr(@Q)+"),("+@va+"),("+@o+");");
      def STD_id = imap(@Q,@id);
   }
   attrib(STD_id,"isSB",1);
   export STD_id;
   if (defined(IMAG)) { kill IMAG; }
   setring @P;
   dbprint(printlevel-voice+3,"
// 'faststd' created a ring, in which an object STD_id is stored.
// To access the object, type (if the name R was assigned to the return value):
        setring R; STD_id; ");
   return(@S1);
}
example
{ "EXAMPLE:"; echo = 2;
   system("--ticks-per-sec",100); // show time in 1/100 sec
   ring s = 0,(e,f,x,y,z,t,u,v,w,a,b,c,d),(c,lp);
   ideal i = w2+f2-1, x2+t2+a2-1,  y2+u2+b2-1, z2+v2+c2-1,
            d2+e2-1, f4+2u, wa+tf, xy+tu+ab;
   option(prot); timer=1;
   int time = timer;
   ideal j=std(i);
   timer-time;
   dim(j),mult(j);

   time = timer;
   def R=faststd(i);                      // use "best" ordering of vars
   timer-time;
   show(R);setring R;dim(STD_id),mult(STD_id);

   setring s;kill R;time = timer;
   def R=faststd(i,"hilb");                // hilb-std only
   timer-time;
   show(R);setring R;dim(STD_id),mult(STD_id);

   setring s;kill R;time = timer;
   def R=faststd(i,"hilb","sort");         // hilb-std,"best" ordering
   timer-time;
   show(R);setring R;dim(STD_id),mult(STD_id);

   setring s;kill R;time = timer;
   def R=faststd(i,"hilb","sort","block","dec"); // hilb-std,"best",blocks
   timer-time;
   show(R);setring R;dim(STD_id),mult(STD_id);

   setring s;kill R;time = timer;
   timer-time;time = timer;
   def R=faststd(i,"sort","block","Dp"); //"best",decreasing,Dp-blocks
   timer-time;
   show(R);setring R;dim(STD_id),mult(STD_id);
}
///////////////////////////////////////////////////////////////////////////////

proc findvars(id, list #)
"USAGE:   findvars(id [,any] ); id=poly/ideal/vector/module/matrix, any=any type
RETURN:  if no second argument is present: ideal of variables occuring in id,@*
         if a second argument is given (of any type): list L with 4 entries:
  @format
  L[1]: ideal of variables occuring in id
  L[2]: intvec of variables occuring in id
  L[3]: ideal of variables not occuring in id
  L[4]: intvec of variables not occuring in id
  @end format
EXAMPLE: example findvars; shows an example
"
{
   int ii,n;
   ideal found, notfound;
   intvec f,nf;
   n = nvars(basering);
   ideal i = simplify(ideal(matrix(id)),10);
   matrix M[ncols(i)][1] = i;
   vector v = module(M)[1];
   ideal max = maxideal(1);

   for (ii=1; ii<=n; ii++)
   {
      if ( v != subst(v,var(ii),0) )
      {
         found = found+var(ii);
         f = f,ii;
      }
      else
      {
         notfound = notfound+var(ii);
         nf = nf,ii;
      }
   }
   if ( size(f)>1 ) { f = f[2..size(f)]; }      //intvec of found vars
   if ( size(nf)>1 ) { nf = nf[2..size(nf)]; }  //intvec of vars not found
   if( size(#)==0 )  { return(found); }
   if( size(#)!=0 )  { list L = found,f,notfound,nf; return(L); }
}
example
{ "EXAMPLE:"; echo = 2;
   ring s  = 0,(e,f,x,y,t,u,v,w,a,d),dp;
   ideal i = w2+f2-1, x2+t2+a2-1;
   findvars(i);
   findvars(i,1);
}
///////////////////////////////////////////////////////////////////////////////

proc hilbvec (@id, list #)
"USAGE:   hilbvec(id[,c,o]); id=poly/ideal/vector/module/matrix, c,o=strings,@*
          c=char, o=ordering used by @code{hilb} (default: c=\"32003\", o=\"dp\")
RETURN:  intvec of 1st Hilbert-series of id, computed in char c and ordering o
NOTE:    id must be homogeneous (i.e. all vars have weight 1)
EXAMPLE: example hilbvec; shows an example
"
{
   def @P = basering;
   string @c,@o = "32003", "dp";
   if ( size(#) == 1 ) {  @c = #[1]; }
   if ( size(#) == 2 ) {  @c = #[1]; @o = #[2]; }
   string @si = typeof(@id)+" @i = "+string(@id)+";";  //** weg
   execute("ring @r=("+@c+"),("+varstr(basering)+"),("+@o+");");
   //**def i = imap(P,@id);
   execute(@si);                   //** weg
   //show(basering);
   @i = std(@i);
   intvec @hi = hilb(@i,1);         // intvec of 1-st Hilbert-series of id
   return(@hi);
}
example
{ "EXAMPLE:"; echo = 2;
   ring s   = 0,(e,f,x,y,z,t,u,v,w,a,b,c,d,H),dp;
   ideal id = w2+f2-1, x2+t2+a2-1,  y2+u2+b2-1, z2+v2+c2-1,
              d2+e2-1, f4+2u, wa+tf, xy+tu+ab;
   id = homog(id,H);
   hilbvec(id);
}
///////////////////////////////////////////////////////////////////////////////

proc tolessvars (id ,list #)
"USAGE:   tolessvars(id [,s1,s2] ); id poly/ideal/vector/module/matrix,
          s1=string (new ordering)@*
          [default: s1=\"dp\" or \"ds\" depending on whether the first block
          of the old ordering is a p- or an s-ordering, respectively]
RETURN:  If id contains all vars of the basering: empty list. @*
         Else: ring R with the same char as the basering, but possibly less
         variables (only those variables which actually occur in id). In R
         an object IMAG (image of id under imap) is stored.
DISPLAY: If printlevel >=0, display ideal of vars, which have been omitted
         from the old ring.
EXAMPLE: example tolessvars; shows an example
"
{
//---------------- initialisation and check occurence of vars -----------------
   int s,ii,n,fp,fs;
   string s2,newvar;
   int pr = printlevel-voice+3;  // p = printlevel+1 (default: p=1)
   def P = basering;
   s2 = ordstr(P);

   list L = findvars(id,1);
   newvar = string(L[1]);    // string of new variables
   n = size(L[1]);           // number of new variables
   if( n == 0 )
   {
      dbprint( pr,"","// no variable occured in "+typeof(id)+", no change of ring!");
      return(id);
   }
   if( n == nvars(P) )
   {
     dbprint(printlevel-voice+3,"
// All variables appear in input object.
// empty list returned. ");
     return(list());
   }
//----------------- prepare new ring, map to it and return --------------------
   if ( size(#) == 0 )
   {
       fp = find(s2,"p");
       fs = find(s2,"s");
       if( fs==0 or (fs>=fp && fp!=0) ) { s2="dp"; }
       else {  s2="ds"; }
   }
   if ( size(#) ==1 ) { s2=#[1]; }
   dbprint( pr,"","// variables which did not occur:",L[3] );
   execute("ring S1=("+charstr(P)+"),("+newvar+"),("+s2+");");
   def IMAG = imap(P,id);
   export IMAG;
   dbprint(printlevel-voice+3,"
// 'tolessvars' created a ring, in which an object IMAG is stored.
// To access the object, type (if the name R was assigned to the return value):
        setring R; IMAG; ");
   return(S1);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r  = 0,(x,y,z),dp;
   ideal i = y2-x3,x-3,y-2x;
   def R_r = tolessvars(i,"lp");
   setring R_r;
   show(basering);
   IMAG;
   kill R_r;
}
///////////////////////////////////////////////////////////////////////////////

proc solvelinearpart (id,list #)
"USAGE:   solvelinearpart(id [,n] );  id=ideal/module, n=integer (default: n=0)
RETURN:  (interreduced) generators of id of degree <=1 in reduced triangular
         form if n=0 [non-reduced triangular form if n!=0]
ASSUME:  monomial ordering is a global ordering (p-ordering)
NOTE:    may be used to solve a system of linear equations,
         see @code{gauss_row} from 'matrix.lib' for a different method
WARNING: the result is very likely to be false for 'real' coefficients, use
         char 0 instead!
EXAMPLE: example solvelinearpart; shows an example
"
{
   intvec getoption = option(get);
   option(redSB);
   if ( size(#)!=0 )
   {
      if(#[1]!=0) { option(noredSB); }
   }
   def lin = interred(degreepart(id,0,1)[1]);
   if ( size(#)!=0 )
   {
      if(#[1]!=0)
      {
         return(lin);
      }
   }
   option(set,getoption);
   return(simplify(lin,1));
}
example
{ "EXAMPLE:"; echo = 2;
   // Solve the system of linear equations:
   //         3x +   y +  z -  u = 2
   //         3x +  8y + 6z - 7u = 1
   //        14x + 10y + 6z - 7u = 0
   //         7x +  4y + 3z - 3u = 3
   ring r = 0,(x,y,z,u),lp;
   ideal i= 3x +   y +  z -  u,
           13x +  8y + 6z - 7u,
           14x + 10y + 6z - 7u,
            7x +  4y + 3z - 3u;
   ideal j= 2,1,0,3;
   j = matrix(i)-matrix(j);        // difference of 1x4 matrices
                                   // compute reduced triangular form, setting
   solvelinearpart(j);             // the RHS equal 0 gives the solutions!
   solvelinearpart(j,1); "";       // triangular form, not reduced
}
///////////////////////////////////////////////////////////////////////////////

proc sortandmap (@id, list #)
"USAGE:   sortandmap(id [,n1,p1,n2,p2...,o1,m1,o2,m2...]);@*
         id=poly/ideal/vector/module,@*
         p1,p2,...= polynomials (product of variables),@*
         n1,n2,...= integers,@*
         o1,o2,...= strings,@*
         m1,m2,...= integers@*
         (default: p1=product of all vars, n1=0, o1=\"dp\",m1=0)
         the last pi (containing the remaining vars) may be omitted
RETURN:  a ring R, in which a poly/ideal/vector/module IMAG is stored: @*
         - the ring R differs from the active basering only in the choice
         of monomial ordering and in the sorting of the variables.@*
         - IMAG is the image (under imap) of the input ideal/module id @*
         The new monomial ordering and sorting of vars is as follows:
  @format
  - each block of vars occuring in pi is sorted w.r.t. its complexity in id,
  - ni controls the sorting in i-th block (= vars occuring in pi):
    ni=0 (resp. ni!=0) means that least complex (resp. most complex) vars come
    first
  - oi and mi define the monomial ordering of the i-th block:
    if mi =0, oi=ordstr(i-th block)
    if mi!=0, the ordering of the i-th block itself is a blockordering,
      each subblock having ordstr=oi, such that vars of same complexity are
      in one block
  @end format
         Note that only simple ordstrings oi are allowed: \"lp\",\"dp\",\"Dp\",
         \"ls\",\"ds\",\"Ds\". @*
NOTE:    We define a variable x to be more complex than y (with respect to id)
         if val(x) > val(y) lexicographically, where val(x) denotes the
         valuation vector of x:@*
         consider id as list of polynomials in x with coefficients in the
         remaining variables. Then:@*
         val(x) = (maximal occuring power of x,  # of all monomials in leading
         coefficient, # of all monomials in coefficient of next smaller power
         of x,...).
EXAMPLE: example sortandmap; shows an example
"
{
   def @P = basering;
   int @ii,@jj;
   intvec @v;
   string @o;
//----------------- find o in # and split # into 2 lists ---------------------
   # = # +list("dp",0);
   for ( @ii=1; @ii<=size(#); @ii++)
   {
      if ( typeof(#[@ii])=="string" )  break;
   }
   if ( @ii==1 ) { list @L1 = list(); }
   else { list @L1 = #[1..@ii-1]; }
   list @L2 = #[@ii..size(#)];
   list @L = sortvars(@id,@L1);
   string @va = string(@L[1]);
   list @l = @L[2];   //e.g. @l[4]=intvec describing permutation of 1-st block
//----------------- construct correct ordering with oi and mi ----------------
   for ( @ii=4; @ii<=size(@l); @ii=@ii+4 )
   {
      @L2=@L2+list("dp",0);
      if ( @L2[@ii/2] != 0)
      {
         @v = @l[@ii];
         for ( @jj=1; @jj<=size(@v); @jj++ )
         {
           @o = @o+@L2[@ii/2 -1]+"("+string(@v[@jj])+"),";
         }
      }
      else
      {
         @o = @o+@L2[@ii/2 -1]+"("+string(size(@l[@ii/2]))+"),";
      }
   }
   @o=@o[1..size(@o)-1];
   execute("ring @S1 =("+charstr(@P)+"),("+@va+"),("+@o+");");
   def IMAG = imap(@P,@id);
   export IMAG;
   dbprint(printlevel-voice+3,"
// 'sortandmap' created a ring, in which an object IMAG is stored.
// To access the object, type (if the name R was assigned to the return value):
        setring R; IMAG; ");
   return(@S1);
}
example
{ "EXAMPLE:"; echo = 2;
   ring s = 32003,(x,y,z),dp;
   ideal i=x3+y2,xz+z2;
   def R_r=sortandmap(i);
   show(R_r);
   setring R_r; IMAG;
   kill R_r; setring s;
   def R_r=sortandmap(i,1,xy,0,z,0,"ds",0,"lp",0);
   show(R_r);
   setring R_r; IMAG;
   kill R_r;
}
///////////////////////////////////////////////////////////////////////////////

proc sortvars (id, list #)
"USAGE:   sortvars(id[,n1,p1,n2,p2,...]);@*
         id=poly/ideal/vector/module,@*
         p1,p2,...= polynomials (product of vars),@*
         n1,n2,...= integers@*
         (default: p1=product of all vars, n1=0)
         the last pi (containing the remaining vars) may be omitted
COMPUTE: sort variables with respect to their complexity in id
RETURN:  list of two elements, an ideal and a list:
  @format
  [1]: ideal, variables of basering sorted w.r.t their complexity in id
       ni controls the ordering in i-th block (= vars occuring in pi):
       ni=0 (resp. ni!=0) means that less (resp. more) complex vars come first
  [2]: a list with 4 entries for each pi:
       _[1]: ideal ai : vars of pi in correct order,
       _[2]: intvec vi: permutation vector describing the ordering in ai,
       _[3]: intmat Mi: valuation matrix of ai, the columns of Mi being the
                  valuation vectors of the vars in ai
       _[4]: intvec wi: size of 1-st, 2-nd,... block of identical columns of Mi
                  (vars with same valuation)
  @end format
NOTE:    We define a variable x to be more complex than y (with respect to id)
         if val(x) > val(y) lexicographically, where val(x) denotes the
         valuation vector of x:@*
         consider id as list of polynomials in x with coefficients in the
         remaining variables. Then:@*
         val(x) = (maximal occuring power of x,  # of all monomials in leading
         coefficient, # of all monomials in coefficient of next smaller power
         of x,...).
EXAMPLE: example sortvars; shows an example
"
{
   int ii,jj,n,s;
   list L = valvars(id,#);
   list L2, L3 = L[2], L[3];
   list K; intmat M; intvec v1,v2,w;
   ideal i = sort(maxideal(1),L[1])[1];
   for ( ii=1; ii<=size(L2); ii++ )
   {
      M = transpose(L3[2*ii]);
      M = M[L2[ii],1..nrows(L3[2*ii])];
      w = 0; s = 0;
      for ( jj=1; jj<=nrows(M)-1; jj++ )
      {
         v1 = M[jj,1..ncols(M)];
         v2 = M[jj+1,1..ncols(M)];
         if ( v1 != v2 ) { n=jj-s; s=s+n; w = w,n; }
      }
      w=w,nrows(M)-s; w=w[2..size(w)];
      K = K+sort(L3[2*ii-1],L2[ii])+list(transpose(M))+list(w);
   }
   L = i,K;
   return(L);
}
example
{ "EXAMPLE:"; echo = 2;
   ring s=0,(x,y,z,w),dp;
   ideal i = x3+y2+yw2,xz+z2,xyz-w2;
   sortvars(i,0,xy,1,zw);
}
///////////////////////////////////////////////////////////////////////////////

proc valvars (id, list #)
"USAGE:   valvars(id[,n1,p1,n2,p2,...]);@*
         id=poly/ideal/vector/module,@*
         p1,p2,...= polynomials (product of vars),@*
         n1,n2,...= integers,

         ni controls the ordering of vars occuring in pi: ni=0 (resp. ni!=0)
         means that less (resp. more) complex vars come first (default: p1=product of all vars, n1=0),@*
         the last pi (containing the remaining vars) may be omitted
COMPUTE: valuation (complexity) of variables with respect to id.@*
         ni controls the ordering of vars occuring in pi:@*
         ni=0 (resp. ni!=0) means that less (resp. more) complex vars come first.
RETURN:  list with 3 entries:
  @format
  [1]: intvec, say v, describing the permutation such that the permuted
       ring variables are ordered with respect to their complexity in id
  [2]: list of intvecs, i-th intvec, say v(i) describing permutation
       of vars in a(i) such that v=v(1),v(2),...
  [3]: list of ideals and intmat's, say a(i) and M(i), where
       a(i): factors of pi,
       M(i): valuation matrix of a(i), such that the j-th column of M(i)
             is the valuation vector of j-th generator of a(i)
         @end format
NOTE:    Use @code{sortvars} in order to actually sort the variables!
         We define a variable x to be more complex than y (with respect to id)
         if val(x) > val(y) lexicographically, where val(x) denotes the
         valuation vector of x:@*
         consider id as list of polynomials in x with coefficients in the
         remaining variables. Then:@*
         val(x) = (maximal occuring power of x,  # of all monomials in leading
         coefficient, # of all monomials in coefficient of next smaller power
         of x,...).
EXAMPLE: example valvars; shows an example
"
{
//---------------------------- initialization ---------------------------------
   int ii,jj,kk,n;
   list L;                    // list of valuation vectors in one block
   intvec vec;                // describes permutation of vars (in one block)
   list blockvec;             // i-th element = vec of i-th block
   intvec varvec;             // result intvector
   list Li;                   // result list of ideals
   list LM;                   // result list of intmat's
   intvec v,w,s;              // w valuation vector for one variable
   matrix C;                  // coefficient matrix for different variables
   ideal i = simplify(ideal(matrix(id)),10);

//---- for each pii in # create ideal a(ii) intvec v(ii) and list L(ii) -------
// a(ii) = ideal of vars in product, v(ii)[j]=k <=> a(ii)[j]=var(k)

   v = 1..nvars(basering);
   int l = size(#);
   if ( l >= 2 )
   {
      ideal m=maxideal(1);
      for ( ii=2; ii<=l; ii=ii+2 )
      {
         int n(ii) = #[ii-1];
         ideal a(ii);
         intvec v(ii);
         for ( jj=1; jj<=nvars(basering); jj++ )
         {
            if ( #[ii]/var(jj) != 0)
            {
               a(ii) = a(ii) + var(jj);
               v(ii)=v(ii),jj;
               m[jj]=0;
               v[jj]=0;
            }
         }
         v(ii)=v(ii)[2..size(v(ii))];
      }
      if ( size(m)!=0 )
      {
         l = 2*(l/2)+2;
         ideal a(l) = simplify(m,2);
         intvec v(l) = compress(v);
         int n(l);
         if ( size(#)==l-1 ) { n(l) = #[l-1]; }
      }
   }
   else
   {
      l = 2;
      ideal a(2) = maxideal(1);
      intvec v(2) = v;
      int n(2);
      if ( size(#)==1 ) { n(2) = #[1]; }
   }
//------------- start loop to order variables in each a(ii) -------------------

   for ( kk=2; kk<=l; kk=kk+2 )
   {
      L = list();
      n = 0;
//---------------- get valuation of all variables in a(kk) --------------------
      for ( ii=1; ii<=size(a(kk)); ii++ )
      {
         C = coeffs(i,a(kk)[ii]);
         w = nrows(C); // =(maximal occuring power of a(kk)[ii])+1
         for ( jj=w[1]; jj>1; jj-- )
         {
            s = size(C[jj,1..ncols(C)]);
            w[w[1]-jj+2] = sum(s);
         }
         // w[1] should represent the maximal occuring power of a(kk)[ii] so it
         // has to be decreased by 1 since otherwise the constant term is also
         // counted
         w[1]=w[1]-1;

         L[ii]=w;
         n = size(w)*(size(w) > n) + n*(size(w) <= n);
      }
      intmat M(kk)[size(a(kk))][n];
      for ( ii=1; ii<=size(a(kk)); ii++ )
      {
         if ( n==1 ) { w = L[ii]; M(kk)[ii,1] = w[1]; }
         else  { M(kk)[ii,1..n] = L[ii]; }
      }
      LM[kk-1] = a(kk);
      LM[kk] = transpose(compress(M(kk)));
//------------------- compare valuation and insert in vec ---------------------
      vec = sort(L)[2];
      if ( n(kk) != 0 ) { vec = vec[size(vec)..1]; }
      blockvec[kk/2] = vec;
      vec = sort(v(kk),vec)[1];
      varvec = varvec,vec;
   }
   varvec = varvec[2..size(varvec)];
   list result = varvec,blockvec,LM;
   return(result);
}
example
{ "EXAMPLE:"; echo = 2;
   ring s=0,(x,y,z,a,b),dp;
   ideal i=ax2+ay3-b2x,abz+by2;
   valvars (i,0,xyz);
}
///////////////////////////////////////////////////////////////////////////////
proc idealSplit(ideal I,list #)
"USAGE:  idealSplit(id,timeF,timeS);  id ideal and optional
         timeF, timeS integers to bound the time which can be used
         for factorization resp. standard basis computation
RETURN:  a list of ideals such that their intersection
         has the same radical as id
EXAMPLE: example idealSplit; shows an example
"
{
   option(redSB);
   int j,k,e;
   int i=1;
   int l=attrib(I,"isSB");
   ideal J;
   int timeF;
   int timeS;
   list re,fac,te;

   if(size(#)==1)
   {
     if(typeof(#[1])=="ideal")
     {
        re=#;
     }
     else
     {
       timeF=#[1];
     }
   }
   if(size(#)==2)
   {
     if(typeof(#[1])=="list")
     {
        re=#[1];
        timeF=#[2];
     }
     else
     {
       timeF=#[1];
       timeS=#[2];
     }
   }
   if(size(#)==3){re=#[1];timeF=#[2];timeS=#[3];}

   fac=timeFactorize(I[1],timeF);

   while((size(fac[1])==2)&&(i<size(I)))
   {
      i++;
      fac=timeFactorize(I[i],timeF);
   }
   if(size(fac[1])>2)
   {
      for(j=2;j<=size(fac[1]);j++)
      {
         I[i]=fac[1][j];
         attrib(I,"isSB",1);
         e=1;
         k=0;
         while(k<size(re))
         {
            k++;
            if(size(reduce(re[k],I))==0){e=0;break;}
            attrib(re[k],"isSB",1);
            if(size(reduce(I,re[k]))==0){re=delete(re,k);k--;}
         }
         if(e)
         {
            if(l)
            {
               J=I;
               J[i]=0;
               J=simplify(J,2);
               attrib(J,"isSB",1);
               re=idealSplit(std(J,fac[1][j]),re,timeF,timeS);
            }
            else
            {
               re=idealSplit(timeStd(I,timeS),re,timeF,timeS);
            }
         }
      }
      return(re);
   }
   J=timeStd(I,timeS);
   attrib(I,"isSB",1);
   if(size(reduce(J,I))==0){return(re+list(I));}
   return(re+idealSplit(J,re,timeF,timeS));
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=32003,(b,s,t,u,v,w,x,y,z),dp;
   ideal i=
   bv+su,
   bw+tu,
   sw+tv,
   by+sx,
   bz+tx,
   sz+ty,
   uy+vx,
   uz+wx,
   vz+wy,
   bvz;
   idealSplit(i);
}
///////////////////////////////////////////////////////////////////////////////
proc idealSimplify(ideal J,list #)
"USAGE:  idealSimplify(id);  id ideal
RETURN:  ideal I = eliminate(Id,m) m is a product of variables
         which are only linearly involved in the generators of id
EXAMPLE: example idealSimplify; shows an example
"
{
   ideal I=J;
   if(size(#)!=0){I=#[1];}
   def R=basering;
   matrix M=jacob(I);
   ideal ma=maxideal(1);
   int i,j,k;
   map phi;

   for(i=1;i<=nrows(M);i++)
   {
      for(j=1;j<=ncols(M);j++)
      {
         if(deg(M[i,j])==0)
         {
            ma[j]=(-1/M[i,j])*(I[i]-M[i,j]*var(j));
            phi=R,ma;
            I=phi(I);
            J=phi(J);
            for(k=1;k<=ncols(I);k++){I[k]=cleardenom(I[k]);}
            M=jacob(I);
         }
      }
   }
   J=simplify(J,2);
   for(i=1;i<=size(J);i++){J[i]=cleardenom(J[i]);}
   return(J);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z,w,t),dp;
   ideal i=
   t,
   x3+y2+2z,
   x2+3y,
   x2+y2+z2,
   w2+z;
   ideal j=idealSimplify(i);
   ideal k=eliminate(i,zyt);
   reduce(k,std(j));
   reduce(j,std(k));
}

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

/*

 ring s=31991,(e,f,x,y,z,t,u,v,w,a,b,c,d),dp;
 ring s=31991,(x,y,z,t,u,v,w,a,b,c,d,f,e,h),dp; //standard
 ring s1=31991,(y,u,b,c,a,z,t,x,v,d,w,e,f,h),dp; //gut
v;
13,12,11,10,8,7,6,5,4,3,2,1,9,14
print(matrix(sort(maxideal(1),v)));
f,e,w,d,x,t,z,a,c,b,u,y,v,h
print(matrix(maxideal(1)));
y,u,b,c,a,z,t,x,v,d,w,e,f,h
v0;
14,9,12,11,10,8,7,6,5,4,3,2,1,13
print(matrix(sort(maxideal(1),v0)));
h,v,e,w,d,x,t,z,a,c,b,u,y,f
v1;v2;
9,12,11,10,8,7,6,5,4,3,2,1,13,14
13,12,11,10,8,7,6,5,4,3,2,1,9,14

Ev. Gute Ordnung fuer i:
========================
i=ad*x^d+ad-1*x^(d-1)+...+a1*x+a0, ad!=0
mit ar=(ar1,...,ark), k=size(i)
    arj in K[..x^..]
d=deg_x(i) := max{deg_x(i[k]) | k=1..size(i)}
size_x(i,deg_x(i)..0) := size(ad),...,size(a0)
x>y  <==
  1. deg_x(i)>deg_y(i)
  2. "=" in 1. und size_x lexikographisch

hier im Beispiel:
f: 5,1,0,1,2

u: 3,1,4

y: 3,1,3
b: 3,1,3
c: 3,1,3
a: 3,1,3
z: 3,1,3
t: 3,1,3

x: 3,1,2
v: 3,1,2
d: 3,1,2
w: 3,1,2
e: 3,1,2
probier mal:
 ring s=31991,(f,u,y,z,t,a,b,c,v,w,d,e,h),dp; //standard

*/