source: git/Singular/LIB/primdecint.lib @ d69ff5

////////////////////////////////////////////////////////////////////////////////
version="$Id: primdecint.lib,v 1.16 2010/06/01 08:01:11 Singular Exp $";
category = "Commutative Algebra";
info="
LIBRARY:  primdecint.lib   primary decomposition over the integers

AUTHORS:  G. Pfister       pfister@mathematik.uni-kl.de
@*        A. Sadiq         afshanatiq@gmail.com
@*        S. Steidel       steidel@mathematik.uni-kl.de

OVERVIEW:

  A library for computing the primary decomposition of an ideal in the 
  polynomial ring over the integers, Z[x_1,...,x_n].

PROCEDURES:
 primdecZ(I);       compute the primary decomposition of I
 minAssZ(I);        compute the minimal associated primes of I
 radicalZ(I);       compute the radical of I
 heightZ(I);        compute the height of I
 equidimZ(I);       compute the equidimensional part of I
 intersectZ(I,J)    compute the intersection of I and J 
";

LIB "crypto.lib";

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

proc primdecZ(ideal I, list #)
"USAGE:  primdecZ(I); I ideal
NOTE:    If size(#) > 0, then #[1] is the number of available processors for
         the computation. Parallelization is just applicable using 32-bit 
         Singular version since MP-links are not compatible with 64-bit Singular 
         version.
RETURN:  a list pr of primary ideals and their associated primes:
@format
   pr[i][1]   the i-th primary component,
   pr[i][2]   the i-th prime component.
@end format
EXAMPLE: example primdecZ; shows an example
"
{
   if(size(I)==0){return(list(ideal(0),ideal(0)));}

//--------------------  Initialize optional parameters  ------------------------
   if(size(#) > 0)
   {
      if(size(#) == 1)
      {
         int n = #[1];
         if((n > 1) && (1 - system("with","MP")))
         {
     "========================================================================";
     "There is no MP available on your system. Since this is necessary to     ";
     "parallelize the algorithm, the computation will be done without forking.";
     "========================================================================";
            n = 1;
         }
         ideal TES = 1;
      }
      if(size(#) == 2)
      {
         int n = #[1];
         if((n > 1) && (1 - system("with","MP")))
         {
     "========================================================================";
     "There is no MP available on your system. Since this is necessary to     ";
     "parallelize the algorithm, the computation will be done without forking.";
     "========================================================================";
            n = 1;
         }
         ideal TES = #[2];
      }
   }
   else
   {
      int n = 1;
      ideal TES = 1;
   }


   if(deg(I[1])==0)
   {
      ideal J=I;
   }
   else
   {
      ideal J=stdZ(I);
   }

   ideal K,N;
   def R=basering;
   number s;
   list rl=ringlist(R);
   int i,j,p,m,ex,nu,k_link;
   list P,B,IS;
   ideal Q,JJ;
   ideal TQ=1;
   if(deg(J[1])==0)
   {
      //=== I intersected with Z is not zero
      list rp=rl;
      rp[1]=0;
      //=== q is generator of I intersect Z
      number q=leadcoef(J[1]);
      def Rhelp=ring(rp);
      setring Rhelp;
      number q=imap(R,q);
      //=== computes the primes occuring in a generator of I intersect Z

      list L = primefactors(q);

      list A;
      ideal J = imap(R,J);

      for(j=1;j<=size(L[2]);j++)
      {
         if(L[2][j] > 1){ ex = 1; break; }
      }

      if(printlevel >= 10)
      {
         "n = "+string(n);
         "size(L[2]) = "+string(size(L[2]));
      }

      int RT = rtimer;
      if((n > 1) && (n < size(L[2])))
      {

//-----  Create n1 links l(1),...,l(n1), open all of them and compute  ---------
//-----  standard basis for the primes L[1][2],...,L[1][n + 1].        ---------

         for(i = 1; i <= n; i++)
         {
            p=int(L[1][i + 1]);
            nu=int(L[2][i + 1]);
            link l(i) = "MPtcp:fork";
//            link l(i) = "ssi:fork";
            open(l(i));
            write(l(i), quote(modp(eval(J), eval(p), eval(nu))));
         }

         p = int(L[1][1]);
         nu = int(L[2][1]);
         int t = timer;
         A[size(A)+1] = modp(J, p, nu);
         t = timer - t;
         if(t > 60) { t = 60; }
         int i_sleep = system("sh", "sleep "+string(t));

         j = n + 2;

         while(j <= size(L[2]) + 1)
         {
            for(i = 1; i <= n; i++)
            {
               //=== ask if link l(i) is ready otherwise sleep for t seconds 
               if(status(l(i), "read", "ready"))                         
               {
                  //=== read the result from l(i) 
                  A[size(A)+1] = read(l(i));                             

                  if(j <= size(L[2]))
                  {
                     p=int(L[1][j]);
                     nu=int(L[2][j]);
                     write(l(i), quote(modp(eval(J), eval(p), eval(nu))));
                     j++;
                  }
                  else
                  {
                     k_link++;
                     close(l(i));
                  }
               }
            }
            //=== k_link describes the number of closed links
            if(k_link == n)                                              
            {
               j++;
            }
            i_sleep = system("sh", "sleep "+string(t));
         }

      }
      else
      {
         for(j=1;j<=size(L[2]);j++)
         {
            A[size(A)+1] = modp(J, L[1][j], L[2][j]);
         }
      }

      setring R;
      list A = imap(Rhelp,A);
      if(printlevel >= 10)
      {
         "A is computed in "+string(rtimer - RT)+" seconds.";
      }
      for(i=1;i<=size(A);i++)
      {
      //=== computes for all p in L the minimal associated primes of 
      //=== IZ/p[variables]
         p = int(A[i][2]);
         if(printlevel >= 10)
         {
            "p = "+string(p);
            RT = rtimer;
         }
         nu = int(A[i][3]);
         //=== maximal power of p dividing q, generator of I intersect Z
         s = p^nu;

         rp[1] = p;
         def S = ring(rp);
         setring S;
         ideal J = imap(R,J);
         setring R;

         if(nu>1)
         {
            //=== p is of multiplicity > 1 in q

            B = A[i][1];
            for(j=1;j<=size(B);j++)
            {
               //=== the minimal associated primes of I
               K=B[j],p;
               K=stdZ(K);
               B[j]=K;
            }
            for(j=1;j<=size(B);j++)
            {
               K=B[j];
               //=== compute maximal independent set for KZ/p[variables]

               setring S;
               J=imap(R,K);
               J=simplify(J,2);
               attrib(J,"isSB",1);
               IS=maxIndependSet(J);
               setring R;
               //=== computing the pseudo primary and extract it
               N=J,s;
               N=stdZ(N);
               Q=extractZ(N,j,IS,B);
               //=== test for useless primaries
               if(size(reduce(TES,Q))>0)
               {
                  TQ=intersectZ(TQ,Q);
                  P[size(P)+1]=list(Q,K);
               }
            }
         }
         else
         {
            //=== p is of multiplicity 1 in q we can compute the
            //=== primary decomposition directly

            B = A[i][1];
            for(j=1;j<=size(B);j++)
            {
               K=B[j][2],p;
               K=stdZ(K);
               Q=B[j][1],p;
               Q=stdZ(Q);
               if(size(reduce(TES,Q))>0)
               {
                  //TQ=intersectZ(TQ,Q);
                  P[size(P)+1]=list(Q,K);
               }
            }
            if(ex)
            {
               JJ=imap(S,J);
               JJ=JJ,p;
               JJ=stdZ(JJ);
               TQ=intersectZ(TQ,JJ);
            }
         }
         kill S;
         if(printlevel >= 10)
         {
            string(p)+" done in "+string(rtimer - RT)+" seconds.";
         }
      }

      setring R;
      if(!ex){return(P);}
      J=stdZ(J);
      TQ=intersectZ(TQ,TES);
      if(size(reduce(TQ,J))!=0)
      {
         //=== taking care about embedded components
         K=stdZ(quotientZ(J,TQ));
         ideal W=K;
         m++;
         while(size(reduce(intersectZ(W,TQ),J))!=0)
         {
            //W=stdZ(addIdealZ(I,K^m));
            W=stdZ(addIdealZ(I,specialPowerZ(K,m)));
            m++;
         }
         list E=primdecZ(W,n,TQ);
         for(i=1;i<=size(E);i++)
         {
            P[size(P)+1]=E[i];
         }
      }
      return(P);
   }

   //==== the ideal intersected with Z is zero
   rl[1]=0;
   def Rhelp=ring(rl);
   setring Rhelp;
   ideal J=imap(R,J);
   J=std(J);
   //=== the primary decomposition over Q which gives the primary 
   //=== decomposition of I:h for a suitable integer h
   list pr=primdecGTZ(J);
   for(i=1;i<=size(pr);i++)
   {
      pr[i]=list(std(pr[i][1]),std(pr[i][2]));
   }
   setring R;
   list pr=imap(Rhelp,pr);
   //=== intersection with Z[variables]
   for(i=1;i<=size(pr);i++)
   {
      pr[i]=list(coefZ(pr[i][1])[1],coefZ(pr[i][2])[1]);
   }
   //=== find h in Z such that I is the intersection of I:h and <I,h>
   //=== and I:h = IQ[variables] intersected with Z[varables]
   list H =coefZ(J);
   ideal Y=H[1];
   int h=H[2];
   J=J,h;
   //=== call primary decomposition over Z for <I,h>
   list M;
   if(h!=1)
   {
       M=primdecZ(J,n,Y);
       j=0;
       //=== remove useless primary ideals
       while(j<size(M))
       {
          j++;
          M[j][1]=stdZ(M[j][1]);
          for(i=1;i<=size(pr);i++)
          {
             if(size(reduce(pr[i][1],M[j][1]))==0)
             {
                M=delete(M,j);
                j--;
                break;
             }
          }
      }
      for(i=1;i<=size(M);i++)
      {
         pr[size(pr)+1]=M[i];
      }
   }
   return(pr);
}
example
{ "EXAMPLE:";  echo = 2;
   ring R=integer,(a,b,c,d),dp;
   ideal I1=9,a,b;
   ideal I2=3,c;
   ideal I3=11,2a,7b;
   ideal I4=13a2,17b4;
   ideal I5=9c5,6d5;
   ideal I6=17,a15,b15,c15,d15;
   ideal I=intersectZ(I1,I2);
   I=intersectZ(I,I3);
   I=intersectZ(I,I4);
   I=intersectZ(I,I5);
   I=intersectZ(I,I6);
   primdecZ(I);
   ideal J=intersectZ(ideal(17,a),ideal(17,a2,b));
   primdecZ(J);
   ideal K=intersectZ(ideal(9,a+3),ideal(9,b+3));
   primdecZ(K);
}

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

proc minAssZ(ideal I)
"USAGE:  minAssZ(I); I ideal
RETURN:  a list pr of associated primes:
EXAMPLE: example minAssZ; shows an example
"
{
   if(size(I)==0){return(list(ideal(0)));}
   if(deg(I[1])==0)
   {
      ideal J=I;
   }
   else
   {
      ideal J=stdZ(I);
   }
   ideal K;
   def R=basering;
   list rl=ringlist(R);
   int i,j,p,m;
   list P,B;
   if(deg(J[1])==0)
   {
      //=== I intersected with Z is not zero
      list rp=rl;
      rp[1]=0;
      number q=leadcoef(J[1]);
      def Rhelp=ring(rp);
      setring Rhelp;
      number q=imap(R,q);
      //=== computes the primes occuring in a generator of I intersect Z
      list L=PollardRho(q,5000,1);
      for(i=1;i<=size(L);i++)
      {
      //=== computes for all p in L the minimal associated primes of 
      //=== IZ/p[variables]
         p=int(L[i]);
         setring R;
         rp[1]=p;
         def S=ring(rp);
         setring S;
         ideal J=imap(R,J);
         list A=minAssGTZ(J);
         setring R;
         B=imap(S,A);
         kill S;
         for(j=1;j<=size(B);j++)
         {
            //=== the minimal associated primes of I
            if(B[j][1]!=1)
            {
               K=B[j],p;
               K=stdZ(K);
               P[size(P)+1]=K;
            }
         }
         setring Rhelp;
      }
      setring R;
      return(P);
   }
   //==== the ideal intersected with Z is zero
   rl[1]=0;
   def Rhelp=ring(rl);
   setring Rhelp;
   ideal J=imap(R,J);
   J=std(J);
   //=== the primary decomposition over Q which gives the primary 
   //=== decomposition of I:h for a suitable integer h
   list pr=minAssGTZ(J);
   for(i=1;i<=size(pr);i++)
   {
      pr[i]=std(pr[i]);
   }
   setring R;
   list pr=imap(Rhelp,pr);
   //=== intersection with Z[variables]
   for(i=1;i<=size(pr);i++)
   {
      pr[i]=coefZ(pr[i])[1];
   }
   //=== find h in Z such that I is the intersection of I:h and I,h
   //=== and I:h =IQ[variables] intersected with Z[varables]
   list H=coefZ(J);
   int h=H[2];
   J=J,h;
   //=== call associated primes over Z for I,h
   list M;
   if(h!=1)
   {
       M=minAssZ(J);
       //=== remove non-minimal primes
       j=0;
       while(j<size(M))
       {
          j++;
          M[j]=stdZ(M[j]);
          for(i=1;i<=size(pr);i++)
          {
             if(size(reduce(pr[i],M[j]))==0)
             {
                M=delete(M,j);
                j--;
                break;
             }
          }
      }
      for(i=1;i<=size(M);i++)
      {
         pr[size(pr)+1]=M[i];
      }
   }
   return(pr);
}
example
{ "EXAMPLE:";  echo = 2;
   ring R=integer,(a,b,c,d),dp;
   ideal I1=9,a,b;
   ideal I2=3,c;
   ideal I3=11,2a,7b;
   ideal I4=13a2,17b4;
   ideal I5=9c5,6d5;
   ideal I6=17,a15,b15,c15,d15;
   ideal I=intersectZ(I1,I2);
   I=intersectZ(I,I3);
   I=intersectZ(I,I4);
   I=intersectZ(I,I5);
   I=intersectZ(I,I6);
   minAssZ(I);
   ideal J=intersectZ(ideal(17,a),ideal(17,a2,b));
   minAssZ(J);
   ideal K=intersectZ(ideal(9,a+3),ideal(9,b+3));
   minAssZ(K);
}

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

proc heightZ(ideal I)
"USAGE:  heightZ(I); I ideal
RETURN:  the height of the input ideal
EXAMPLE: example heightZ; shows an example
"
{
   if(size(I)==0){return(0);}
   if(deg(I[1])==0)
   {
      ideal J=I;
   }
   else
   {
      ideal J=stdZ(I);
   }
   ideal K=1;
   def R=basering;
   list rl=ringlist(R);
   int i,j,p,m;
   list P;
   ideal B;
   if(deg(J[1])==0)
   {
      //=== I intersected with Z is not zero
      m=nvars(R);
      list rp=rl;
      rp[1]=0;
      number q=leadcoef(J[1]);
      def Rhelp=ring(rp);
      setring Rhelp;
      number q=imap(R,q);
      //=== computes the primes occuring in a generator of I intersect Z
      list L=PollardRho(q,5000,1);
      for(i=1;i<=size(L);i++)
      {
      //=== computes for all p in L the std of IZ/p[variables]
         p=int(L[i]);
         setring R;
         rp[1]=p;
         def S=ring(rp);
         setring S;
         ideal J=imap(R,J);
         j=nvars(R)-dim(std(J));
         if(j<m){m=j;}
         setring Rhelp;
         kill S;
      }
      setring R;
      return(m+1);
   }
   //==== the ideal intersected with Z is zero
   rl[1]=0;
   def Rhelp=ring(rl);
   setring Rhelp;
   ideal J=imap(R,J);
   J=std(J);
   m=nvars(R)-dim(J);
   //=== the height over Q
   //=== of I:h for a suitable integer h
   setring R;
   //=== find h in Z such that I is the intersection of I:h and I,h
   //=== and I:h =IQ[variables] intersected with Z[varables]
   list H=coefZ(J);
   int h=H[2];
   J=J,h;
   //=== call height over Z for I,h
   if(h!=1)
   {
      j=heightZ(J);
      if(j<m){m=j;}
   }
   return(m);
}
example
{ "EXAMPLE:";  echo = 2;
   ring R=integer,(a,b,c,d),dp;
   ideal I1=9,a,b;
   ideal I2=3,c;
   ideal I3=11,2a,7b;
   ideal I4=13a2,17b4;
   ideal I5=9c5,6d5;
   ideal I6=17,a15,b15,c15,d15;
   ideal I=intersectZ(I1,I2);
   I=intersectZ(I,I3);
   I=intersectZ(I,I4);
   I=intersectZ(I,I5);
   I=intersectZ(I,I6);
   heightZ(I);
}

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

proc radicalZ(ideal I)
"USAGE:  radicalZ(I); I ideal
RETURN:  the radcal of the input ideal
EXAMPLE: example radicalZ; shows an example
"
{
   if(size(I)==0){return(ideal(0));}
   if(deg(I[1])==0)
   {
      ideal J=I;
   }
   else
   {
      ideal J=stdZ(I);
   }
   ideal K=1;
   def R=basering;
   list rl=ringlist(R);
   int i,j,p,m;
   list P;
   ideal B;
   if(deg(J[1])==0)
   {
      //=== I intersected with Z is not zero
      list rp=rl;
      rp[1]=0;
      number q=leadcoef(J[1]);
      def Rhelp=ring(rp);
      setring Rhelp;
      number q=imap(R,q);
      //=== computes the primes occuring in a generator of I intersect Z
      list L=PollardRho(q,5000,1);
      for(i=1;i<=size(L);i++)
      {
      //=== computes for all p in L the radical of IZ/p[variables]
         p=int(L[i]);
         setring R;
         rp[1]=p;
         def S=ring(rp);
         setring S;
         ideal J=imap(R,J);
         ideal A=radical(J);
         setring R;
         B=imap(S,A);
         kill S;
         B=B,p;
         B=stdZ(B);
         K=stdZ(intersectZ(K,B));
         setring Rhelp;
      }
      setring R;
      return(K);
   }
   //==== the ideal intersected with Z is zero
   rl[1]=0;
   def Rhelp=ring(rl);
   setring Rhelp;
   ideal J=imap(R,J);
   J=std(J);
   //=== the radical over Q which gives the radical
   //=== of I:h for a suitable integer h
   ideal K=std(radical(J));
   setring R;
   K=imap(Rhelp,K);
   //=== intersection with Z[variables]
   K=coefZ(K)[1];
   //=== find h in Z such that I is the intersection of I:h and I,h
   //=== and I:h =IQ[variables] intersected with Z[varables]
   list H=coefZ(J);
   int h=H[2];
   J=J,h;
   //=== call radical over Z for I,h
   if(h!=1)
   {
      ideal M=radicalZ(J);
      K=intersectZ(K,M);
   }
   return(K);
}
example
{ "EXAMPLE:";  echo = 2;
   ring R=integer,(a,b,c,d),dp;
   ideal I1=9,a,b;
   ideal I2=3,c;
   ideal I3=11,2a,7b;
   ideal I4=13a2,17b4;
   ideal I5=9c5,6d5;
   ideal I6=17,a15,b15,c15,d15;
   ideal I=intersectZ(I1,I2);
   I=intersectZ(I,I3);
   I=intersectZ(I,I4);
   I=intersectZ(I,I5);
   I=intersectZ(I,I6);
   radicalZ(I);
   ideal J=intersectZ(ideal(17,a),ideal(17,a2,b));
   radicalZ(J);
}

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

proc equidimZ(ideal I)
"USAGE:  equidimZ(I); I ideal
RETURN:  the part of minimal height
EXAMPLE: example equidimZ; shows an example
"
{
   if(size(I)==0){return(ideal(0));}
   if(deg(I[1])==0)
   {
      ideal J=I;
   }
   else
   {
      ideal J=stdZ(I);
   }
   int he=heightZ(J);
   ideal K,N;
   def R=basering;
   number s;
   list rl=ringlist(R);
   int i,j,p,m,ex;
   list P,IS,B;
   ideal Q,JJ,E;
   ideal TQ=1;
   if(deg(J[1])==0)
   {
      //=== I intersected with Z is not zero
      list rp=rl;
      rp[1]=0;
      //=== generator of I intersect Z
      number q=leadcoef(J[1]);
      def Rhelp=ring(rp);
      setring Rhelp;
      number q=imap(R,q);
      number s;
      //=== computes the primes occuring in a generator of I intersect Z
      list L=PollardRho(q,5000,1);
      list Le;
      for(i=1;i<=size(L);i++)
      {
         L[i]=int(L[i]);
         p=int(L[i]);
         j=0;
         s=q;
         while((s mod p)==0)
         {
            j++;
            s=s/p;
         }
         Le[i]=j;
      }
      for(i=1;i<=size(L);i++)
      {
      //=== computes for all p in L the minimal associated primes of 
      //=== IZ/p[variables]
         p=int(L[i]);
         j=Le[i];
         setring R;
         //=== maximal power of p dividing q, generator of I intersect Z
         s=p^j;
         rp[1]=p;
         def S=ring(rp);
         setring S;
         ideal J=imap(R,J);
         J=std(J);
         if(nvars(R)-dim(J)+1==he)
         {
            if(j>1)
            {
               //=== p is of multiplicity >1 in q
               list A=minAssGTZ(J);
               j=0;
               while(j<size(A))
               {
                  j++;
                  if(dim(std(A[j]))!=nvars(R)-he+1)
                  {
                     A=delete(A,j);
                     j--;
                  }
               }
               setring R;
               B=imap(S,A);
               for(j=1;j<=size(B);j++)
               {
                  //=== the minimal associated primes of I
                  K=B[j],p;
                  K=stdZ(K);
                  B[j]=K;
               }
               for(j=1;j<=size(B);j++)
               {
                  K=B[j];
                  //=== compute maximal independent set for KZ/p[variables]
                  setring S;
                  J=imap(R,K);
                  J=simplify(J,2);
                  attrib(J,"isSB",1);
                  IS=maxIndependSet(J);
                  setring R;
                  //=== computing the pseudo primary and extract it
                  N=J,s;
                  N=stdZ(N);
                  Q=extractZ(N,j,IS,B);
                  TQ=intersectZ(TQ,Q);
               }
               setring Rhelp;
            }
            else
            {
               //=== p is of multiplicity 1 in q we can compute the
               //=== equidimensional part directly
               ideal E=equidimMax(J);
               setring R;
               E=imap(S,E);
               E=E,p;
               E=stdZ(E);
               TQ=intersectZ(TQ,E);
            }
            kill S;
            setring Rhelp;
         }
      }
      setring R;
      return(TQ);
   }
   //==== the ideal intersected with Z is zero
   rl[1]=0;
   def Rhelp=ring(rl);
   setring Rhelp;
   ideal J=imap(R,J);
   J=std(J);
   //=== the equidimensional part over Q which gives the equdimensional 
   //=== part of I:h for a suitable integer h
   ideal E=1;
   if(nvars(R)-he==dim(J))
   {
      E=std(equidimMax(J));
   }
   setring R;
   ideal  E =imap(Rhelp,E);
   //=== intersection with Z[variables]
   E=coefZ(E)[1];
   //=== find h in Z such that I is the intersection of I:h and I,h
   //=== and I:h =IQ[variables] intersected with Z[varables]
   int h =coefZ(J)[2];
   J=J,h;
   //=== call equidimensional part over Z for I,h
   ideal M;
   if(h!=1)
   {
       M=equidimZ(J);
       if(he==heightZ(M))
       {
          E=intersectZ(M,E);
       }
   }
   return(E);
}
example
{ "EXAMPLE:";  echo = 2;
   ring R=integer,(a,b,c,d),dp;
   ideal I1=9,a,b;
   ideal I2=3,c;
   ideal I3=11,2a,7b;
   ideal I4=13a2,17b4;
   ideal I5=9c5,6d5;
   ideal I6=17,a15,b15,c15,d15;
   ideal I=intersectZ(I1,I2);
   I=intersectZ(I,I3);
   I=intersectZ(I,I4);
   I=intersectZ(I,I5);
   I=intersectZ(I,I6);
   equidimZ(I);
}

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

proc intersectZ(ideal I, ideal J)
"USAGE:  intersectZ(I,J); I,J ideals
RETURN:  the intersection of the input ideals
NOTE:    this is needed because intersect(I,J) does not work, should be replaced
         by intersect later 
EXAMPLE: example intersectZ; shows an example
{
   def R = basering;
   execute("ring S=integer,(t,"+varstr(R)+"),(dp(1),dp(nvars(R)));");
   ideal I=imap(R,I);
   ideal J=imap(R,J);
   ideal K=addIdealZ(t*I,(1-t)*J);
   K=stdZ(K);
   int i;
   ideal L;
   for(i=1;i<=size(K);i++)
   {
      if(lead(K[i])/t==0){L[size(L)+1]=K[i];}
   }
   setring R;
   ideal L=imap(S,L);
   return(L);
}
example
{ "EXAMPLE:";  echo = 2;
   ring R=integer,(a,b,c,d),dp;
   ideal I1=9,a,b;
   ideal I2=3,c;
   ideal I3=11,2a,7b;
   ideal I4=13a2,17b4;
   ideal I5=9c5,6d5;
   ideal I6=17,a15,b15,c15,d15;
   ideal I=intersectZ(I1,I2); I;
   I=intersectZ(I,I3); I;
   I=intersectZ(I,I4); I;
   I=intersectZ(I,I5); I;
   I=intersectZ(I,I6); I;
}

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

static proc modp(ideal J, int p, int nu)
{
//=== computes the minimal associated primes (if nu > 1) resp. the primary 
//=== decomposition (else) of J in Z/p and maps the result back to the basering
   def R = basering;
   list rp = ringlist(R);
   rp[1] = p;
   def Rp = ring(rp);
   setring Rp;
   ideal J = imap(R,J);
   if(nu > 1)
   {
      //=== p is of multiplicity > 1 in q
      list A = minAssGTZ(J);
      setring R;
      list A = imap(Rp,A);
      return(list(A,p,nu));
   }
   else
   {
      list A = primdecGTZ(J);
      setring R;
      list A = imap(Rp,A);
      return(list(A,p,nu));
   }
}

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

static proc coefPrimeZ(ideal I)
{
//=== computes the primes occuring in the product of the leading coefficients 
//=== of I
   number h=1;
   int i;
   for(i=1;i<=size(I);i++)
   {
      h=h*leadcoef(I[i]);  // besser machen (gleich zerlegen, 
                           // nicht ausmultiplizieren)
   }
   def R=basering;
   ring Rhelp=0,x,dp;
   number h=imap(R,h);
   list L=PollardRho(h,5000,1);
   for(i=1;i<=size(L);i++){L[i]=int(L[i]);}
   setring R;
   return(L);
}

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

static proc coefZ(ideal I)
{
//=== assume IQ[variables]=<g_1,...,g_s>, Groebner basis, g_i in Z[variables]
//=== computes an integer h such that
//===   <g_1,...,g_s>Z[variables]:h^infinity = IQ[variables] intersected 
//===                                          with Z[variables]
//=== returns a list with IQ[variables] intersected with Z[variables] and h
   int h=1;
   int i,e;
   ideal K=1;
   list L=coefPrimeZ(I);
   if(size(L)==0){return(list(I,1));}
   int d=1;
   while(d!=0)
   {
      i++;
      K=quotientOneZ(I,L[i]);
      if(size(reduce(K,I))!=0)
      {
         h=h*L[i];
         I=stdZ(K);
         e=1;
      }
      if(i==size(L))
      {
         i=0;
         if(e)
         {
            e=0;
         }
         else
         {
           d=0;
         }
      }
   }
   if(h<0){h=-h;}
   return(list(K,h));
}

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

static proc specialPowerZ(ideal I, int m)
{
//=== computes the ideal generated by the m-th power of the generators of I
   int i;
   for(i=1;i<=size(I);i++)
   {
      I[i]=I[i]^m;
   }
   return(I);
}

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

static proc separatorsZ(int j, list B)
{
//=== computes s such that s is not in B[j] but s is in B[i] for all i!=j
   int i,k;
   poly s=1;
   for(i=1;i<=size(B);i++)
   {
      if(i!=j)
      {
         for(k=1;k<=size(B[i]);k++)
         {
            if(reduce(B[i][k],B[j])!=0)
            {
               s=s*B[i][k];
               break;
            }
         }
      }
   }
   return(s);
}

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

static proc extractZ(ideal J, int j, list L, list B)
{
   //=== P is an associated prime of J, the corresponding primary ideal is 
   //=== computed,
   //=== L is a list of maximal independent sets for P in Z/p[variables]
   def R=basering;
   ideal P=B[j];

   //=== first compute a pseudo primary ideal I, radical of I is P
   //=== method of Eisenbud
   //ideal I=addIdealZ(J,specialPowerZ(P,20));

   //=== method of Shimoyama-Yokoyama
   poly s=separatorsZ(j,B);
   ideal I=satZ(J,s);
   //=== size(L)=0 means P is maximal ideal and I is primary
   if(size(L)>0)
   {
      if(L[1][3]!=0)
      {
         //=== if u in x is an independent set of L then we compute a Groebner
         //=== Basis in Z[u][x-u]
         execute("ring S=integer,("+L[1][1]+"),lp;");
         ideal I=imap(R,I);
         I=stdZ(I);
         list rl=ringlist(S);
         rl[1]=0;
         def Shelp =ring(rl);
         setring Shelp;
         ideal I=imap(S,I);
         I[1]=0;
         I=simplify(I,2);
         //=== this is our way to obtain the coefficients in Z[u] of the
         //=== leading terms of the Groebner basis above
         string quotring=prepareQuotientring(nvars(basering)-L[1][3]);
         execute(quotring);
         ideal I=imap(Shelp,I);
         list C;
         int i;
         for(i=1;i<=size(I);i++)
         {
            C[i]=leadcoef(I[i]);
         }
         setring Shelp;
         list C=imap(quring,C);

         setring R;
         list C=imap(Shelp,C);
      }
      else
      {
         I=stdZ(I);
         list C;
         int i;
         for(i=1;i<=size(I);i++)
         {
            C[i]=I[i];
         }
         list rl=ringlist(R);
         rl[1]=0;
         def Shelp =ring(rl);
      }
      poly h=1;
      for(i=1;i<=size(C);i++)
      {
         if(deg(C[i])>0){h=h*C[i];}  // das muss noch besser gemacht werden,
                                     // nicht ausmultiplizieren!
      }
      setring Shelp;
      poly h=imap(R,h);
      ideal fac=factorize(h,1);
      setring R;
      ideal fac=imap(Shelp,fac);
      for(i=1;i<=size(fac);i++)
      {
         I=satZ(I,fac[i]);
      }
   }
   I=stdZ(I);
   return(I);
}
////////////////////////////////////////////////////////////////////////////////

static proc normalizeZ(ideal I)
{
//=== if I[1]=q in Z, it replaces all other coeffs of polys in I by there value 
//=== mod q, std should do this automatically and then this procedure should be
//=== removed
   if(deg(I[1])>0){return(I);}
   int i,j;
   number n;
   poly p;
   for(i=2;i<=size(I);i++)
   {
      j=1;
      while(j<=size(I[i]))
      {
         n=leadcoef(I[i][j]) mod leadcoef(I[1]);
         p=n*leadmonom(I[i][j]);
         I[i]=I[i]-I[i][j]+p;
         if(p!=0){j++;}
      }
   }
   return(I);
}

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

static proc satZ(ideal I,poly h)
{
//=== saturates I by h
   ideal J=quotientOneZ(I,h);
   while(size(reduce(J,stdZ(I)))!=0)
   {
      I=J;
      J=quotientOneZ(I,h);
      J=normalizeZ(J);
   }
   return(J);
}

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

static proc prepareQuotientring (int nnp)
{
//=== this is from primdec.lib, it is static there, should be imported later 
//=== if it is no more static
  ideal @ih,@jh;
  int npar=npars(basering);
  int @n;

  string quotring= "ring quring = ("+charstr(basering);
  for(@n=nnp+1;@n<=nvars(basering);@n++)
  {
     quotring=quotring+",var("+string(@n)+")";
     @ih=@ih+var(@n);
  }

  quotring=quotring+"),(var(1)";
  @jh=@jh+var(1);
  for(@n=2;@n<=nnp;@n++)
  {
     quotring=quotring+",var("+string(@n)+")";
     @jh=@jh+var(@n);
  }
  quotring=quotring+"),(C,lp);";

  return(quotring);
}

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

static proc maxIndependSet (ideal j)
{
//=== this is from primdec.lib, it is static there, should be imported later 
//=== if it is no more static
  int n,k,di;

  list resu,hilf;
  if(size(j)==0)
  {
     resu[1]=varstr(basering);
     resu[2]=ordstr(basering);
     resu[3]=0;
     return(list(resu));
  }
  string var1,var2;
  list v=indepSet(j,0);

  for(n=1;n<=size(v);n++)
  {
    di=0;
    var1="";
    var2="";
    for(k=1;k<=size(v[n]);k++)
    {
      if(v[n][k]!=0)
      {
        di++;
        var2=var2+"var("+string(k)+"),";
      }
      else
      {
        var1=var1+"var("+string(k)+"),";
      }
    }
    if(di>0)
    {
      var1=var1+var2;
      var1=var1[1..size(var1)-1];
      hilf[1]=var1;
      hilf[2]="lp";
      hilf[3]=di;
      resu[n]=hilf;
    }
    else
    {
      resu[n]=varstr(basering),ordstr(basering),0;
    }
  }
  return(resu);
}

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

static proc quotientOneZ(ideal I, poly f)
{
//=== this is needed because quotient(I,f) does not work properly, should be 
//=== replaced by quotient later
   def R=basering;
   int i;
   ideal K=intersectZ(I,ideal(f));
   //=== K[i]/f; does not work in rings with integer! This should be replaced 
   //=== later
   execute("ring Rhelp=0,("+varstr(R)+"),dp;");
   ideal K=imap(R,K);
   poly f=imap(R,f);
   for(i=1;i<=size(K);i++)
   {
      K[i]=K[i]/f;
   }
   setring R;
   K=imap(Rhelp,K);
   return(K);
}

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

static proc quotientZ(ideal I, ideal J)
{
//=== this is needed because quotient(I,J) does not work properly, should be 
//=== replaced by quotient later
   int i;
   ideal K=quotientOneZ(I,J[1]);
   for(i=2;i<=size(J);i++)
   {
      K=intersectZ(K,quotientOneZ(I,J[i]));
   }
   return(K);
}

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

static proc reduceZ(poly f, ideal I)
{
//=== this is needed because reduce(f,I) does not work properly, should be 
//=== replaced by reduce later
   if(f==0){return(f);}
   def R=basering;
   execute("ring Rhelp=0,("+varstr(R)+"),dp;");
   ideal I=imap(R,I);
   poly f=imap(R,f);
   int i,j;
   poly m;
   number n;
   while(!i)
   {
      i=1;
      j=0;
      while(j<size(I))
      {
         j++;
         m=leadmonom(f)/leadmonom(I[j]);
         if(m!=0)
         {
            n=leadcoef(f) mod leadcoef(I[j]);
            if(n==0)
            {
               f=f-leadcoef(f)/leadcoef(I[j])*m*I[j];
               if(f==0){setring R;return(0);}
               i=0;
               break;
            }
            if(n!=leadcoef(f))
            {
               f=f+(n-leadcoef(f))/leadcoef(I[j])*m*I[j];
               i=0;
               break;
            }
         }
      }
   }
   setring R;
   f=imap(Rhelp,f);
   return(lead(f)+reduceZ(f-lead(f),I));
}

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

static proc stdZ(ideal I)
{
//=== this is needed because we want the leading coefficients to be positive
//=== otherwhise reduce gives wrong results! should be replaced later by std
   I=simplify(I,2);
   I=normalizeZ(I);
   ideal J=std(I);
   int i;
   for(i=1;i<=size(J);i++)
   {
      if(leadcoef(J[i])<0){J[i]=-J[i];}
   }
   J=normalizeZ(J);
   attrib(J,"isSB",1);
   return(J);
}

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

static proc addIdealZ(ideal I,ideal J)
{
//=== this is needed because I+J does not work, should be replaced by + later
   int i;
   for(i=1;i<=size(J);i++)
   {
      I[size(I)+1]=J[i];
   }
   return(I);
}

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

static proc testPrimaryZ(ideal I, list L)
{
//=== test whether I is the intersection of the primary ideals in L
   int i;
   ideal K=L[1][1];
   for(i=2;i<=size(L);i++)
   {
      K=intersectZ(K,L[i][1]);
   }
   i=size(reduce(K,stdZ(I)))+size(reduce(I,stdZ(K)));
   if(!i){return(1);}
   return(0);
}

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

/*
Examples:

//=== IQ[a,b,c,d,e,f,g] intersect Z[a,b,c,d,e,f,g] = I  (takes some time)
ring R1=integer,(a,b,c,d,e,f,g),dp;
ideal I=a2+2de+2cf+2bg+a,
        2ab+e2+2df+2cg+b,
        b2+2ac+2ef+2dg+c,
        2bc+2ad+f2+2eg+d,
        c2+2bd+2ae+2fg+e,
        2cd+2be+2af+g2+f,
        d2+2ce+2bf+2ag+g;

ring R2=integer,(a,b,c,d,e,f,g),dp;
ideal I=181*32003,
        a2+2de+2cf+2bg+a,
        2ab+e2+2df+2cg+b,
        b2+2ac+2ef+2dg+c,
        2bc+2ad+f2+2eg+d,
        c2+2bd+2ae+2fg+e,
        2cd+2be+2af+g2+f,
        d2+2ce+2bf+2ag+g;
 
ring R3=integer,(w,z,y,x),dp;
ideal I=xzw+(-y^2+y)*z^2,
        (-x^2+x)*w^2+yzw,
        ((y^4-2*y^3+y^2)*x-y^4+y^3)*z^3,
        y2z2w+(-y*4+2*y^3-y^2)*z3;

ring R4=integer,(w,z,y,x),dp;
ideal I=-2*yxzw+(-yx-y^2+y)*z^2,
        xw^2-yz^2,
        (yx^2-(2*y^2+2*y)*x+y^3-2*y^2+y)*z^3,
        (-2*y^2+2*y)*z^2*w+(yx-3*y^2-y)*z^3;

ring R5=integer,(x,y,z),dp;
ideal I=x2-y2-z2,
        xy-z2,
        y3+xz2-yz2+2z3+xy-z2,
        -y2z2+2z4+x2-y2+z2,
        y3z9+3y2z10+3yz11+z12-y2z2+2z4;

ring R6=integer,(h, l, s, x, y, z),dp;  //takes some time
ideal I=hl-l2-4ls+hy,
        h2s-6ls3+h2z,
        xh2-l2s-h3;

ring R7=integer,(x,y,z),dp;
ideal I=x2-y2-(z+2)^2,
        xy-(z+2)^2,
        y3+x*(z+2)^2-y*(z+2)^2+2*(z+2)^3+xy-(z+2)^2,
        -y^2*(z+2)^2+2*(z+2)^4+x2-y2+(z+2)^2,
        y3z9+3y2z10+3yz11+z12-y2z2+2z4;

ring R8=integer,(x,y,z),dp;
ideal I=x2-y2-(z+2)^2,
        xy-(z+2)^2,
        y3+x*(z+2)^2-y*(z+2)^2+2*(z+2)^3+xy-(z+2)^2,
        -y^2*(z+2)^2+2*(z+2)^4+x2-y2+(z+2)^2,
        y3z9+3y2z10+3yz11+z12-y2z2+2z4;

ring R9=integer,(w,z,y,x),dp;
ideal I=630,
        ((y^2-y)*x-y^3+y^2)*z^2,
        (x-y)*zw,
        (x-y^2)*zw+(-y^2+y)*z^2,
        (-x^2+x)*w^2+(-yx+y)*zw;

ring R10=integer,(w,z,y,x),dp;
ideal I=1260,
        -yxzw+(-y^2+y)*z^2,
        (-x^2+x)*w^2-yxzw,
        ((-y^2+y)*x-y^3+2*y^2-y)*z^3,
        (y^2-y)*z^2*w+(-y^2+y)*z^2*w+(-y^2+y)*z^3;

ring R11=integer,(w,z,y,x),dp;
ideal I=(4*y^2*x^2+(4*y^3+4*y^2-y)*x-y^2-y)*z^2,
        (x+y+1)*zw+(-4*y^2*x-4*y^3-4*y^2)*z^2,
        (-x-2*y^2 - 2*y - 1)*zw + (8*y^3*x + 8*y^4 + 8*y^3 + 2*y^2+y)*z^2,
        ((y^3 + y^2)*x - y^2 - y)*z^2,
        (y +1)*zw + (-y^3 -y^2)*z^2,
        (x + 1)*zw +(- y^2 -y)*z^2,
        (x^2 +x)*w^2 + (-yx - y)*zw;

ring R12=integer,(w,z,y,x),dp;
ideal I=72,
        ((y^3 + y^2)*x - y^2 - y)*z^2,
        (y + 1)*zw + (-y^3 -y^2)*z^2,
        (x + 1)*zw + (-y^2 -y)*z^2, (x^2 + x)*w^2 + (-yx - y)*zw;

ring R13=integer,(w,z,y,x),dp;
ideal I=(((12*y+8)*x^2 +(2*y+2)*x)*zw +((-15*y^2 -4*y)*x-4*y^2 -y)*z^2,
        -x*w^2 +((-12*y -8)*x+2*y)*zw +(15*y^2+4*y)*z^2,
        (81*y^4*x^2 +(-54*y^3 -12*y^2)*x-12*y^3 -3*y^2)*z^3, 
        (-24*yx+6*y^2-6*y)*z^2*w + (-81*y^4*x + 81*y^3 + 24*y^2)*z^3,
        (48*x^2 + (-30*y + 12)*x - 6*y)*z^2*w +Â ((81*y^3 -54*y^2 -24*y)*x
        -21*y^2 -6*y)*z^3,
        (-96*yx-18*y^3 +18*y^2-24*y)*z^2*w +(243*y^5*x-243*y^4 +72*y^3
        +48*y^2)*z^3,
        6*y*z^2*w^2 +((576*y+384)*x^2 + (-81*y^3 -306*y^2 -168*y+96)*x+81*y^2
        -18*y)*z^3*w +((-720*y^2 - 192*y)*x + 450*y^3 - 60*y^2 - 48*y)*z^4);

ring R14=integer,(x(1),x(2),x(3),x(4)),dp;
ideal I=181*49^2,
        x(4)^4,
        x(1)*x(4)^3,
        x(1)*x(2)*x(4)^2,
        x(2)^2*x(4)^2,
        x(2)^2*x(3)*x(4),
        x(1)*x(2)*x(3)*x(4),
        x(1)*x(3)^2*x(4),
        x(3)^3*x(4);


ring R15=integer,(x,y,z),dp; Â 
ideal I=32003*181*64,
        ((z^2-z)*y^2 + (z^2 -z)*y)*x; (z*y^3 + z*y^2)*x,
        (y^4 - y^2)*x, (z^2 - z)*y*x^2, (y^3 - y^2)*x^2,
        (z^3 - z^2)*x^4 + (2*z^3 -2*z^2)*x^3 + (z^3 -z^2)*x^2,
        z*y^2*x^2, z*y*x^4 +z*y*x^3,
        2*y^2*x^4 +6*y^2*x^3 +6*y^2*x^2 + (y^3 +y^2)*x, z*x^5 + (z^2 +z)*x^4 
        + (2*z^2 -z)*x^3 + (z^2 -z)*x^2,
        y*x^6 + 3*y*x^5 + 3*y*x^4 + y*x^3;


ring R16=integer,(x(1),x(2),x(3),x(4),x(5)),dp;
ideal I=x(5)^5,
        x(1)*x(5)^4,
        x(1)*x(2)*x(5)^3,
        x(2)^2*x(5)^3,
        x(2)^2*x(3)*x(5)^2,
        x(1)*x(2)*x(3)*x(5)^2,
        x(1)*x(3)^2*x(5)^2, 
        x(3)^3*x(5)^2,
        x(3)^3*x(4)*x(5),
        x(1)*x(3)^2*x(4)*x(5),
        x(1)*x(2)*x(3)*x(4)*x(5),
        x(2)^2*x(3)*x(4)*x(5),
        x(2)^2*x(4)^2*x(5), 
        x(1)*x(2)*x(4)^2*x(5),
        x(1)*x(4)^3*x(5),
        x(4)^4*x(5);
      I=intersectZ(I,ideal(64*181,x(1)^2));

ring R17=integer,(x,y,z),dp; Â 
ideal I=374,
        (z+2)^8-140z6+2622*(z+2)^4-1820*(z+2)^2+169,
        17y*(z+2)^4-374*y*(z+2)^2+221y+2z7-281z5+5240z3-3081z,
        204y2+136yz3-3128yz+z6-149z4+2739z2+117,
        17xz4-374xz2+221x+2z7-281z5+5240z3-3081z,
        136xy-136xz-136yz+2z6-281z4+5376z2-3081,
        204x2+136xz3-3128xz+z6-149z4+2739z2+117;

ring R18=integer,(B,D,F,b,d,f),dp;
ideal I=6,
        (b-d)*(B-D)-2*F+2,
        (b-d)*(B+D-2*F)+2*(B-D),
        (b-d)^2-2*(b+d)+f+1,
        B^2*b^3-1,
        D^2*d^3-1,
        F^2*f^3-1;

ring R19=integer,(a,b,c,d,e,f),dp;
ideal I=24,
        2*(f+2)*b+2ec+d2+a2+a,
        2*(f+2)*c+2ed+2ba+b,
        2*(f+2)*d+e2+2ca+c+b2,
        2*(f+2)*e+2da+d+2cb,
        (f+2)^2+2ea+e+2db+c2,
        2*(f+2)*a+f+2eb+2dc;

ring R20=integer,(x,y,z,w,u),dp;
ideal I=24,
         2x2-2y2+2z2-2w2+2u2-1,
         2x3-2y3+2z3-2w3+2u3-1,
         2x4-2y4+2z4-2w4+2u4-1,
         2x5-2y5+2z5-2w5+2u5-1,
         2x6-2y6+2z6-2w6+2u6-1;

ring R21=integer,(x,y,z,t,u,v,h),dp;
ideal I=66,
        2x2+2y2+2z2+2t2+2u2+v2-vh,
        xy+yz+2zt+2tu+2uv-uh,
        2xz+2yt+2zu+u2+2tv-th,
        2xt+2yu+2tu+2zv-zh,
        t2+2xv+2yv+2zv-yh,
        2x+2y+2z+2t+2u+v-h,
        x3+y3+z3+t3+u3+v3;

ring R22=integer,(s,p,S,P,T,F,f),dp; Â 
ideal I=35,
        2*T-S*s-2*F+2,
        8*F*p-4*p*S-2*F*s^2+S*s^2+4*T-2*S*s,
        -2*s-4*p+s^2+f+1,
        s*T^2-p*s*P-p*S*T-2,
        p^3*P^2-1,
        F^2*f^3-1;
*/