source: git/Singular/LIB/teachstd.lib @ 592906

// $Id:
//GMG, last modified 28.9.01
///////////////////////////////////////////////////////////////////////////////
version="$Id: teachstd.lib,v 1.9 2009-01-14 16:07:05 Singular Exp $";
category="Teaching";
info="
LIBRARY:  teachstd.lib   Procedures for teaching standard bases
AUTHOR:                  G.-M. Greuel, greuel@mathematik.uni-kl.de

NOTE: The library is intended to be used for teaching purposes, but not
      for serious computations. Sufficiently high printlevel allows to
      control each step, thus illustrating the algorithms at work.
      The procedures are implemented exactly as described in the book
      'A SINGULAR Introduction to Commutative Algebra' by G.-M. Greuel and
      G. Pfister (Springer 2002).

PROCEDURES:
 ecart(f);              ecart of f
 tail(f);               tail of f
 sameComponent(f,g);    test for same module component of lead(f) and lead(g)
 leadmonomial(f);       leading monomial as poly (also for vectors)
 monomialLcm(m,n);      lcm of monomials m and n as poly (also for vectors)
 spoly(f[,1]);          s-polynomial of f [symmetric form]
 minEcart(T,h);         element g from T of minimal ecart s.t. LM(g)|LM(h)
 NFMora(i);             normal form of i w.r.t Mora algorithm
 prodcrit(f,g);         test for product criterion
 chaincrit(f,g,h);      test for chain criterion
 pairset(G);            pairs form G neither satifying prodcrit nor chaincrit
 updatePairs(P,S,h);    pairset P enlarded by not useless pairs (h,f), f in S
 standard(id);          standard basis of ideal/module
 localstd(id);          local standard basis of id using Lazard's method

              [parameters in square brackets are optional]
";

LIB "poly.lib";

///////////////////////////////////////////////////////////////////////////////
proc ecart(f)
"USAGE:   ecart(f); f poly or vector
RETURN:  the ecart e of f of type int
EXAMPLE: example ecart; shows an example
"
{
  int e = maxdeg1(f)-maxdeg1(lead(f));
  return(e);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),ls;
   ecart((y+z+x+xyz)**2);
   ring s=0,(x,y,z),dp;
   ecart((y+z+x+xyz)**2);
}
///////////////////////////////////////////////////////////////////////////////
proc leadmonomial(f)
"USAGE:   leadmonomial(f); f poly or vector
RETURN:  the leading monomial of f of type poly
NOTE:    if f is of type poly, leadmonomial(f)=leadmonom(f), if f is of type
         vector and if leadmonom(f)=m*gen(i) then leadmonomial(f)=m
EXAMPLE: example leadmonomial; shows an example
"
{
  int e;
  poly m;
  if(typeof(f) == "vector")
  {
    e=leadexp(f)[nvars(basering)+1];
    m=leadmonom(f)[e,1];
  }
  if(typeof(f) == "poly")
  {
    m=leadmonom(f);
  }
  return(m);
}
example
{ "EXAMPLE:"; echo = 2;
   ring s=0,(x,y,z),(c,dp);
   leadmonomial((y+z+x+xyz)**2);
   leadmonomial([(y+z+x+xyz)**2,xyz5]);
}
///////////////////////////////////////////////////////////////////////////////
proc tail(f)
"USAGE:   tail(f); f poly or vector
RETURN:  f-lead(f), the tail of f of type poly
EXAMPLE: example tail; shows an example
"
{
  def t = f-lead(f);
  return(t);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),ls;
   tail((y+z+x+xyz)**2);
   ring s=0,(x,y,z),dp;
   tail((y+z+x+xyz)**2);
}
///////////////////////////////////////////////////////////////////////////////
proc sameComponent(f,g)
"USAGE:   sameComponent(f,g); f,g poly or vector
RETURN:  1 if f and g are of type poly or if f and g are of type vector and
         their leading monomials involve the same module component,
         0 if not
EXAMPLE: example sameComponent; shows an example
"
{
  if(typeof(f) != typeof(g))
  {
    ERROR("** arguments must be of same type");
  }
  if(typeof(f) == "vector")
  {
     if( leadexp(f)[nvars(basering)+1] !=  leadexp(g)[nvars(basering)+1] )
     {
       return(0);
     }
  }
  return(1);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   sameComponent([y+z+x,xyz],[z2,xyz]);
   sameComponent([y+z+x,xyz],[z4,xyz]);
   sameComponent(y+z+x+xyz, xy+z5);
}
///////////////////////////////////////////////////////////////////////////////
proc monomialLcm(m,n)
"USAGE:   monomialLcm(m,n); m,n of type poly or vector
RETURN:  least common multiple of leading monomials of m and n, of type poly
NOTE:    if m = (x1...xr)^(a1,...,ar)*gen(i) (gen(i)=1 if m is of type poly)
         and n = (x1...xr)^(b1,...,br)*gen(j), then the proc returns
         (x1,...,xr)^(max(a1,b1),...,max(ar,br)) if i=j and 0 if i!=j.
EXAMPLE: example monomialLcm; shows an example
"
{
  if(typeof(n) != typeof(m))
  {
    ERROR("** arguments must be of same type");
  }

  poly u ;
  if(sameComponent(m,n) == 0)   //leading term of vectors involve
  {                             //different module components
     return(u);
  }

  intvec v = leadexp(m);        //now start to compute lcm
  intvec w = leadexp(n);
  u=1;
  int i;
  for (i=1; i<=nvars(basering); i++)
  {
      if(v[i]>=w[i])
      {
         u = u*var(i)**v[i];
      }
      else
      {
         u = u*var(i)**w[i];
      }
   }
   return(u);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),ds;
   monomialLcm(xy2,yz3);
   monomialLcm([xy2,xz],[yz3]);
   monomialLcm([xy2,xz3],[yz3]);
}
///////////////////////////////////////////////////////////////////////////////
proc spoly(f,g,list #)
"USAGE:   spoly(f,g[,s]); f,g poly or vector, s int
RETURN:  the s-polynomial of f and g, of type poly or vector
         if s!=0 the symmetric s-polynomial (without division) is returned
EXAMPLE: example spoly; shows an example
"
{
  if(typeof(f) != typeof(g))
  {
    ERROR("** arguments must be of same type");
  }
  if(size(#) == 0)
  {
     #[1]=0;
  }

  int e;
  poly o = monomialLcm(f,g);

  if( o == 0)                    //can only happen, if vectors f and g involve
  {                              //different module components
    vector sp;
    return(sp);
  }

  poly m=(o/leadmonomial(f));         //compute the leading monomial as poly
  poly n=(o/leadmonomial(g));

  f = m * f;
  g = n * g; // now they have the same LM!

  if (#[1]==0)                    //the asymmetric s-poly
  {
    def sp = f - (leadcoef(f)/leadcoef(g))*g;
  }
  else                            //the symmetric s-poly, avoiding division
  {
    def sp = leadcoef(g)*f - leadcoef(f)*g;
  }
  return(sp);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),ls;
   spoly(2x2+x2y,3y3+xyz);
   ring s=0,(x,y,z),(c,dp);
   spoly(2x2+x2y,3y3+xyz);
   spoly(2x2+x2y,3y3+xyz,1);             //symmetric s-poly without division
   spoly([5x2+x2y,z5],[x2,y3,y4]);       //s-poly for vectors
}
///////////////////////////////////////////////////////////////////////////////
proc minEcart(T,h)
"USAGE:   minEcart(T,h); T ideal or module, h poly or vector
RETURN:  element g from T such that leadmonom(g) divides leadmonom(h)
         ecart(g) is minimal with this property (if T != 0);
         return 0 if T is 0 or h = 0
EXAMPLE: example minEcart; shows an example
"
{
  int i,k,e;
  if (size(T)==0 or h==0 )       //trivial cases
  {
     h = 0;
     return(h);
  }
//---- check whether some element in T has the same module component as h ----
  int v;
  intvec w;
  T = simplify(T,2);

  if (typeof(h) == "vector")
  {
     e=1;
     v = leadexp(h)[nvars(basering)+1];
     for( i=1; i<=size(T); i++)
     {
       w[i]=leadexp(T[i])[nvars(basering)+1];
       if(v == w[i])
       {
         e=0;         //some element in T involves the same component as h
       }
     }
  }
  if ( e == 1 )       //no element in T involves the same component as h
  {
     h = 0;
     return(h);
  }

  if (typeof(h) == "poly")   //for polys v=w[i] for all i
  {
    v = 1;
    w[size(T)]=0;
    w=w+1;
  }

//------ check whether for some g in T leadmonom(g) divides leadmonom(h) -----
  def g = T[1];
  poly f = leadmonomial(h);

  for( i=1; i<=size(T); i++)
  {
    if( f/leadmonomial(T[i]) != 0 and v==w[i] )
    {
       g=T[i];
       k=i;
       break;
    }
  }
  if (k == 0)        //no leadmonom(g) divides leadmonom(h)
  {
     g = 0;
     return(g);
  }
//--look for T[i] with minimal ecart s.t.leadmonom(T[i]) divides leadmonom(h)--

  for( i=k+1; i<=size(T); i++)
  {
    if( f/leadmonomial(T[i]) != 0 and v==w[i] )
    {
       if (ecart(T[i]) < ecart(g))
       {
         g=T[i];
        }
     }
  }
  return(g);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=0,(x,y,z),dp;
   ideal T = x2y+x2,y3+xyz,xyz2+z4;
   poly h = x2y2z2+x5+yx3+z6;
   minEcart(T,h);"";
   ring S=0,(x,y,z),(c,ds);
   module T = [x2+x2y,y2],[y3+xyz,x3-z3],[x3y+z4,0,x2];
   vector h = [x3y+x5+x2y2z2+z6,x3];
   minEcart(T,h);
}
///////////////////////////////////////////////////////////////////////////////
proc NFMora(f,G,list #)
"USAGE:   NFMora(f,G[,s]); f poly or vector,G ideal or module, s int
RETURN:  the Mora normal form of f w.r.t. G, same type as f
         if s!=0 the symmetric s-polynomial (without division) is used
NOTE:    Show comments if printlevel > 0, pauses computation if printlevel > 1
EXAMPLE: example NFMora; shows an example
"
{
  if(size(#) == 0)
  {
     #[1]=0;
  }

  int y = printlevel - voice + 2;
  if( f==0 or size(G) ==0 )
  {
     if (y>0)
     {
        "// 1st or 2nd argument 0";
     }
     return(f);
  }

  int i,e;
  def h = f;
  def T = G;
// -------------------- start with f to be reduced by G --------------------
  if (y>0)
  {"";
   "// Input for NFMora is (f,T):";
   "// f:"; f;
   "// T:"; T;
   "// Reduce f with T, eventually enlarging T for local ordering";
  }

// ----------------------- enter the reduction loop ------------------------
  def g = minEcart(T,h);
  while (h!=0 and g!=0)
  {
     if (y>0)
     {  "";
        "//                Reduction-step in NFMora:",i;
        "// h = (f after",i,"reductions) reduction with g from T:";
        "// g = element of minimal ecart in T s.t. LM(g)|LM(h):";
        "// h:";h;
        "// g:";g;
     }
     if (y>1)
     {
        pause("press <return> to continue");
        "// T, set used for reduction:"; T;
        pause("press <return> to continue");
     }
     e=0;
     if( ecart(g) > ecart(h) )
     {
        T=T,h;  e=1;
     }
     if (y>0 )
     {
        "// T-set enlarged for next reduction? (yes/no = 1/0):  ", e;
        if( e==1 )
        {
          "// T-set for next reduction got enlarged by h:";
          "// h:";h;
          if (y>1)
          {
             pause("press <return> to continue");
          }
        }
     }
     h = spoly(h,g,#[1]);
     g = minEcart(T,h);
     i=i+1;
  }
  if(y>0)
  { "";
    "// normal form is:";
  }
  return(h);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   poly f = x2y2z2+x5+yx3+z6-3y3;
   ideal G = x2y+x2,y3+xyz,xyz2+z6;
   NFMora(f,G);"";
   ring s=0,(x,y,z),ds;
   poly f = x3y+x5+x2y2z2+z6;
   ideal G = x2+x2y,y3+xyz,x3y2+z4;
   NFMora(f,G);"";
   vector v = [f,x2+x2y];
   module M = [x2+x2y,f],[y3+xyz,y3],[x3y2+z4,z2];
   NFMora(v,M);
}
///////////////////////////////////////////////////////////////////////////////
proc prodcrit(f,g)
"USAGE:   prodcrit(f,g); f,g poly or vector
RETURN:  1 if product criterion applies in the same module component,
         2 if lead(f) and lead(g) involve different components, 0 else
NOTE:    if product criterion applies we can delete (f,g) from pairset
EXAMPLE: example prodcrit; shows an example
"
{
  if( defined(@@@NOPC) )
  {
    if( @@@NOPC == 1) { return(0); }
  }

  if(typeof(f)=="poly")    //product criterion for polynomials
  {
    if( monomialLcm(f,g)==leadmonom(f)*leadmonom(g))
    {
      return(1);
    }
    return(0);
  }
// ------------------- product criterion for modules ---------------------
  if(sameComponent(f,g)==1)
  {
     if( monomialLcm(f,g)==leadmonomial(f)*leadmonomial(g) )
     {
         int c = leadexp(f)[nvars(basering)+1];  //component involving lead(f)
         if((f-f[c]*gen(c))-(g-g[c]*gen(c))==0)  //other components are 0
         {
           return(1);
         }
     }
     return(0);
  }
  return(2);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   poly f = y3z3+x5+yx3+z6;
   poly g = x5+yx3;
   prodcrit(f,g);
   vector v = x3z2*gen(1)+x3y*gen(1)+x2y*gen(2);
   vector w = y4*gen(1)+y3*gen(2)+xyz*gen(1);
   prodcrit(v,w);
}
///////////////////////////////////////////////////////////////////////////////
proc chaincrit(f,g,h)
"USAGE:   chaincrit(f,g,h); f,g,h poly or module
RETURN:  1 if chain criterion applies, 0 else
NOTE:    if chain criterion applies to f,g,h we can delete (g,h) from pairset
EXAMPLE: example chaincrit; shows an example
"
{
  if(sameComponent(f,g) and sameComponent(f,h))
  {
    if( monomialLcm(g,h)/leadmonomial(f) !=0 )
    {
      return(1);
    }
  }
  return(0);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   poly f = x2y2z2+x5+yx3+z6;
   poly g = x5+yx3;
   poly h = y2z5+x5+yx3;
   chaincrit(f,g,h);
   vector u = [x2y3-z2,x2y];
   vector v = [x2y2+z2,x2-y2];
   vector w = [x2y4+z3,x2+y2];
   chaincrit(u,v,w);
}
///////////////////////////////////////////////////////////////////////////////
proc pairset(G)
"USAGE:   pairset(G); G ideal or module
RETURN:  list L,
         L[1] = the pairset of G as list (not containing pairs for
         which the product or the chain criterion applies)
         L[2] = intvec v, v[1]= # product criterion, v[2]= # chain criterion
EXAMPLE: example pairset; shows an example
"
{
   int i,j,k,s,c,ccrit,pcrit,pr;
   int y = printlevel - voice + 2;
   G = simplify(G,10);
   def g = G;
   ideal pair;
   list P,I;      //P=pairlist of G, I=list of corresponding indices of pairs
   for (i=1; i<=size(G); i++)
   {
      for(j = i+1; j<=size(G); j++)
      {
         pr = prodcrit(G[i],G[j]);       //check first product criterion
         if( pr != 0 )
         {
            pcrit=pcrit+(pr==1);
         }
         else
         {
            s = size(P);                //now check chain criterion
            for(k=1; k<=s; k++)
            {
              if( i==I[k][2] )
              {
                if ( chaincrit(P[k][1],P[k][2],G[j]) )
                {                     //need not include (G[i],G[j]) in P
                  c=1; ccrit=ccrit+1;
                  break;
                }
              }
              if( j==I[k][1] and c==0 )
              {
                "########### enter pairset2 #############";
                if ( chaincrit(G[i],P[k][1],P[k][2]) )
                {                     //can delete P[k]=(P[k][1],P[k][2])
                  ccrit=ccrit+1;
                  P = delete(P,k);
                  s = s-1;
                }
              }
            }
            if ( c==0 )
            {
               g = G[i],G[j];
               P[s+1]=g;
               I[s+1]=intvec(i,j);
            }
            c=0;
         }
      }
   }
   if (y>0)
   {  "";
      "// product criterion:",pcrit," chain criterion:",ccrit;
   }
   intvec v = pcrit,ccrit;
   P=P,v;
   return(P);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   ideal G = x2y+x2,y3+xyz,xyz2+z4;
   pairset(G);"";
   module T = [x2y3-z2,x2y],[x2y2+z2,x2-y2],[x2y4+z3,x2+y2];
   pairset(T);
}
///////////////////////////////////////////////////////////////////////////////
proc updatePairs(P,S,h)
"USAGE:   updatePairs(P,S,h); P list, S ideal or module, h poly or vector
         P a list of pairs of polys or vectors (obtained from pairset)
RETURN:  list Q,
         Q[1] = the pairset P enlarged  by all pairs (f,h), f from S,
         without pairs for which the product or the chain criterion applies
         Q[2] = intvec v, v[1]= # product criterion, v[2]= # chain criterion
EXAMPLE: example updatePairs; shows an example
"
{
   int i,j,k,s,r,c,ccrit,pcrit,pr;
   int y = printlevel - voice + 2;
   ideal pair;
   list Q = P;           //Q will become enlarged pairset
   s = size(P);
   r = size(Q);          //r will grow with Q
   list R;
   def g = S;            //give g the correct type ideal/module
   for (i=1; i<=size(S); i++)
   {
      pr =  prodcrit(h,S[i]);
      if( pr != 0 )     //check product criterion
      {
        pcrit=pcrit+(pr==1);       //count product criterion in same component
      }
      else
      {                 //prodcrit did not apply, check for chaincrit
         r=size(Q);
         for(k=1; k<=r; k++)
         {
           if( Q[k][2]==S[i] )    //S[i]=Q[k][2]
           {
             if( chaincrit(Q[k][1],S[i],h) )
             {                   //can forget (S[i],h)
                c=1; ccrit=ccrit+1;
                break;
             }
           }
         }
         if ( c==0 )
         {
            g = S[i],h;        //add pair (S[i],h)
            Q[r+1] = g;
         }
         c=0;
      }
   }
   if (y>0)
   {  "";;
      "// product criterion:",pcrit," chain criterion:",ccrit;
   }
   intvec v = pcrit,ccrit;
   Q = Q,v;
   return(Q);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R1=0,(x,y,z),(c,dp);
   ideal S = x2y+x2,y3+xyz;
   poly h = x2y+xyz;
   list P = pairset(S)[1];
   P;"";
   updatePairs(P,S,h);"";
   module T = [x2y3-z2,x2y],[x2y4+z3,x2+y2];
   P = pairset(T)[1];
   P;"";
   updatePairs(P,T,[x2+x2y,y3+xyz]);
}
///////////////////////////////////////////////////////////////////////////////
proc standard(id, list #)
"USAGE:   standard(i[,s]); id ideal or module, s int
RETURN:  a standard basis of id, using generalized Mora's algorithm
         which is Buchberger's algorithm for global monomial orderings.
         If s!=0 the symmetric s-polynomial (without division) is used
NOTE:    Show comments if printlevel > 0, pauses computation if printlevel > 1
EXAMPLE: example standard; shows an example
"
{
  if(size(#) == 0)
  {
     #[1]=0;
  }

   def S = id;                 //S will become the standard basis of id
   def h = S[1];
   int i,z;
   int y = printlevel - voice + 2;
   if(y>0)
   {  "";
      "// the set S, to become a standard basis:"; S;
      if(y>1)
      {
         "// create pairset, i.e. pairs from S,";
         "// after application of product and chain criterion";
      }
   }
   list P = pairset(S);        //create pairset of S=id
   intvec v = P[2];
   P = P[1];
//-------------------------- Main loop in SB lgorithm ----------------------
   while (size(P) !=0)
   {  z=z+1;
      if(y>0)
      {  "";
         "//              Enter NFMora for next pair, count",z;
         "// size of partial standard basis S:    (",size(S),")";
         "// number of pairs of S after updating: (",size(P),")";
         if(y>1)
         {
            "// 1st pair of new pairset:"; P[1];
            "// set T=S used for reduction:";S;
            "// apply NFMora to (spoly,S), spoly = spoly(1st pair)";
         }
      }
      //-------------------- apply NFMora = Mora's normal form -------------
      h = spoly(P[1][1],P[1][2],#[1]);
      if(y>1)
      {
         "// spoly:";h;
      }
      h = NFMora(h,S,#[1]);
      if(h==0)           //normal form is 0
      {
         if(y==1)
         {
            "// pair has reduced to 0";
         }
          if(y>1)
         { h;"";
           pause("press <return> to continue");
         }
     }
      P = delete(P,1);   //spoly of pair reduced to 0, pair can be deleted
      //--- spoly of pair did not reduce to 0, update S and paiset of S ----
      if( h != 0)
      {
         if(y==1)
         {
            "// ** new spoly in degree **:", deg(h);
         }
         if(y>1)
         { h;"";
           pause("press <return> to continue");
           "// update pairset";
         }
        P=updatePairs(P,S,h);    //update P (=paisert of S)
        v=v+P[2];                //with useful pairs (g,h), g from S
         P=P[1];
         S=S,h;                  //update S, will become the standard basis
      }
   }
//------------------------------ finished ---------------------------------
   if( find(option(),"prot") or y>0 )
   {  "";                //show how often prodcrit and chaincrit applied
      "// product criterion:",v[1]," chain criterion:",v[2];
      "";
      "// Final standard basis:";
   }
   return(S);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),dp;
   ideal G = x2y+x2,y3+xyz,xyz2+z4;
   standard(G);"";
   ring s=0,(x,y,z),(c,ds);
   ideal G = 2x2+x2y,y3+xyz,3x3y+z4;
   standard(G);"";
   standard(G,1);"";           //use symmetric s-poly without division
   module M = [2x2,x3y+z4],[3y3+xyz,y3],[5z4,z2];
   standard(M);
}
///////////////////////////////////////////////////////////////////////////////
proc localstd (id)
"USAGE:   localstd (id); id = ideal
RETURN:  A standard basis for a local degree ordering, using Lazard's method.
NOTE:    The procedure homogenizes id w.r.t. a new 1st variable local@t,
         computes a SB w.r.t. (dp(1),dp) and substitutes local@t by 1.
         Hence the result is a SB with respect to an ordering which sorts
         first w.r.t. the subdegree of the original variables and then refines
         it with dp. This is the local degree ordering ds.
         localstd may be used in order to avoid cancellation of units and thus
         to be able to use option(contentSB) also for local orderings.
EXAMPLE: example localstd; shows an example
"
{
  int ii;
  def bas = basering;
  execute("ring  @r_locstd
     =("+charstr(bas)+"),(local@t,"+varstr(bas)+"),(dp(1),dp);");
  ideal id = imap(bas,id);
  ideal hid = homog(id,local@t);
  hid = std(hid);
  hid = subst(hid,local@t,1);
  setring bas;
  def hid = imap(@r_locstd,hid);
  attrib(hid,"isSB",1);
  kill @r_locstd;
  return(hid);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,(x,y,z),ds;
   ideal i  = xyz+z5,2x2+y3+z7,3z5+y5;
   localstd(i);
}
///////////////////////////////////////////////////////////////////////////////

/*
// some examples:
LIB"teachstd.lib";
option(prot); printlevel=3;
ring r0 = 0,(t,x,y),lp;
ideal i = x-t2,y-t3;
standard(i);

printlevel=1;
standard(i);

option(prot); printlevel =1;
ring r1 = (0,a,b),(x,y,z),(c,ds);
module M = [ax2,bx3y+z4],[a3y3+xyz,by3],[5az4,(a+b)*z2];
module N1= std(M);
module N2 = standard(M,1);
NF(lead(N2),lead(N1));
NF(lead(N1),lead(N2));rom T
ring r2 = 0,(x,y,z),dp;
ideal I = x2y+x2,y3+xyz,xyz2+z4;
option(prot);
int tt = timer;
ideal J = standard(I);
timer -tt;                 //4sec, product criterion: 9  chain criterion: 6
ideal J1 = std(I);
NF(lead(J),lead(J1));
NF(lead(J1),lead(J));

ring r3 = 0,(x,y,z),ds;
poly f = x3*y4+z5+xyz;
ideal I = f,jacob(f);
option(prot);
int tt = timer;
ideal J = standard(I);
timer -tt;                 //3sec, product criterion: 1  chain criterion: 3
ideal J1 = std(I);
NF(lead(J),lead(J1));
NF(lead(J1),lead(J));

//Becker:
ring r4 = 32003,(x,y,z),lp;
ideal I = x3-1, y3-1,
-27x3-243x2y+27x2z-729xy2+162xyz-9xz2-729y3+243y2z-27yz2+z3-27;
option(prot);
int tt = timer;
ideal J = standard(I);
timer -tt;                 //201sec, product criterion: 42  chain criterion: 33
ideal J1 = std(I);
NF(lead(J),lead(J1));
NF(lead(J1),lead(J));

//Alex
ring r5 = 32003,(x,y,z,t),dp;
ideal I =
2t3xy2z+x2ty+2x2y,
2tz+y3x2t+z2t3y2x;
option(prot); printlevel =1;
ideal J1 = std(I);
int tt = timer;
ideal J = standard(I);
timer -tt;                 //15sec product criterion: 0  chain criterion: 12
NF(lead(J),lead(J1));
NF(lead(J1),lead(J));

ring r6 = 32003,(x,y,z,t),dp; //is already SB for ds, for dp too long
ideal I=
9x8+y7t3z4+5x4y2t2+2xy2z3t2,
9y8+7xy6t+2x5y4t2+2x2yz3t2,
9z8+3x2y3z2t4;
option(prot);
int tt = timer;
ideal J = standard(I);
timer -tt;                 //0sec, product criterion: 3  chain criterion: 0
ideal J1 = std(I);
NF(lead(J),lead(J1));
NF(lead(J1),lead(J));


*/