Index: Singular/LIB/primdecint.lib
===================================================================
--- Singular/LIB/primdecint.lib	(revision 22a129e7a48031d6420b3e495ec7f895ad49c0d2)
+++ Singular/LIB/primdecint.lib	(revision 7ef5b62b7463b2574e24d6d9b80966ea720ba93b)
@@ -29,6 +29,6 @@
 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 
+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:
@@ -42,5 +42,5 @@
    if(size(I)==0){return(list(ideal(0),ideal(0)));}
 
-   
+
    //--------------------  Initialize optional parameters  ------------------------
    if(size(#) > 0)
@@ -69,5 +69,5 @@
             "========================================================================";
             n = 1;
-         }  
+         }
          ideal TES = #[2];
       }
@@ -79,5 +79,5 @@
    }
 
-   
+
    if(deg(I[1])==0)
    {
@@ -88,5 +88,5 @@
       ideal J=stdZ(I);
    }
-   
+
    ideal K,N;
    def R=basering;
@@ -108,17 +108,17 @@
       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[3]);j++)
       {
@@ -126,5 +126,5 @@
 
       }
-      
+
       if(printlevel >= 10)
       {
@@ -157,7 +157,7 @@
          if(t > 60) { t = 60; }
          int i_sleep = system("sh", "sleep "+string(t));
-         
+
          j = n + 2;
- 
+
          while(j <= size(L[3]) + 1)
          {
@@ -167,5 +167,5 @@
                {
                   A[size(A)+1] = read(l(i));                             // read the result from l(i)
-                  
+
                   if(j <= size(L[3]))
                   {
@@ -198,9 +198,9 @@
       }
 
-      
+
       setring R;
       list A = imap(Rhelp,A);
-      if(printlevel >= 10) 
-      { 
+      if(printlevel >= 10)
+      {
          "A is computed in "+string(rtimer - RT)+" seconds.";
       }
@@ -209,7 +209,7 @@
       //=== 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); 
+         if(printlevel >= 10)
+         {
+            "p = "+string(p);
             RT = rtimer;
          }
@@ -240,5 +240,5 @@
                K=B[j];
                //=== compute maximal independent set for KZ/p[variables]
-             
+
                setring S;
                J=imap(R,K);
@@ -268,5 +268,5 @@
             {
                K=B[j][2],p;
-               K=stdZ(K);             
+               K=stdZ(K);
                Q=B[j][1],p;
                Q=stdZ(Q);
@@ -291,9 +291,9 @@
          }
       }
-     
+
       setring R;
       if(!ex){return(P);}
       J=stdZ(J);
-      TQ=intersectZ(TQ,TES);    
+      TQ=intersectZ(TQ,TES);
       if(size(reduce(TQ,J))!=0)
       {
@@ -304,5 +304,5 @@
          while(size(reduce(intersectZ(W,TQ),J))!=0)
          {
-            //W=stdZ(addIdealZ(I,K^m)); 
+            //W=stdZ(addIdealZ(I,K^m));
             W=stdZ(addIdealZ(I,specialPowerZ(K,m)));
             m++;
@@ -347,5 +347,5 @@
    if(h!=1)
    {
-       M=primdecZ(J,n,Y);        
+       M=primdecZ(J,n,Y);
        j=0;
        //=== remove useless primary ideals
@@ -401,5 +401,5 @@
 "
 {
-   if(size(I)==0){return(list(ideal(0)));} 
+   if(size(I)==0){return(list(ideal(0)));}
    if(deg(I[1])==0)
    {
@@ -425,5 +425,5 @@
       number q=imap(R,q);
       //=== computes the primes occuring in a generator of I intersect Z
-      list L=PollardRho(q,5000,1);  
+      list L=PollardRho(q,5000,1);
       for(i=1;i<=size(L);i++)
       {
@@ -435,5 +435,5 @@
          setring S;
          ideal J=imap(R,J);
-         list A=minAssGTZ(J);   
+         list A=minAssGTZ(J);
          setring R;
          B=imap(S,A);
@@ -460,5 +460,5 @@
    ideal J=imap(R,J);
    J=std(J);
-   //=== the primary decomposition over Q which gives the primary decomposition 
+   //=== the primary decomposition over Q which gives the primary decomposition
    //=== of I:h for a suitable integer h
    list pr=minAssGTZ(J);
@@ -536,5 +536,5 @@
 "
 {
-   if(size(I)==0){return(0);} 
+   if(size(I)==0){return(0);}
    if(deg(I[1])==0)
    {
@@ -562,5 +562,5 @@
       number q=imap(R,q);
       //=== computes the primes occuring in a generator of I intersect Z
-      list L=PollardRho(q,5000,1);  
+      list L=PollardRho(q,5000,1);
       for(i=1;i<=size(L);i++)
       {
@@ -573,5 +573,5 @@
          ideal J=imap(R,J);
          j=nvars(R)-dim(std(J));
-         if(j<m){m=j;}   
+         if(j<m){m=j;}
          setring Rhelp;
          kill S;
@@ -587,5 +587,5 @@
    J=std(J);
    m=nvars(R)-dim(J);
-   //=== the height over Q 
+   //=== the height over Q
    //=== of I:h for a suitable integer h
    setring R;
@@ -628,5 +628,5 @@
 "
 {
-   if(size(I)==0){return(ideal(0));} 
+   if(size(I)==0){return(ideal(0));}
    if(deg(I[1])==0)
    {
@@ -653,5 +653,5 @@
       number q=imap(R,q);
       //=== computes the primes occuring in a generator of I intersect Z
-      list L=PollardRho(q,5000,1);  
+      list L=PollardRho(q,5000,1);
       for(i=1;i<=size(L);i++)
       {
@@ -663,5 +663,5 @@
          setring S;
          ideal J=imap(R,J);
-         ideal A=radical(J);   
+         ideal A=radical(J);
          setring R;
          B=imap(S,A);
@@ -758,5 +758,5 @@
       number s;
       //=== computes the primes occuring in a generator of I intersect Z
-      list L=PollardRho(q,5000,1);  
+      list L=PollardRho(q,5000,1);
       list Le;
       for(i=1;i<=size(L);i++)
@@ -791,5 +791,5 @@
             {
                //=== p is of multiplicity >1 in q
-               list A=minAssGTZ(J);  
+               list A=minAssGTZ(J);
                j=0;
                while(j<size(A))
@@ -825,5 +825,5 @@
                   N=stdZ(N);
                   Q=extractZ(N,j,IS,B);
-                  TQ=intersectZ(TQ,Q);          
+                  TQ=intersectZ(TQ,Q);
                }
                setring Rhelp;
@@ -831,5 +831,5 @@
             else
             {
-               //=== p is of multiplicity 1 in q we can compute the 
+               //=== p is of multiplicity 1 in q we can compute the
                //=== equidimensional part directly
                ideal E=equidimMax(J);
@@ -853,5 +853,5 @@
    ideal J=imap(R,J);
    J=std(J);
-   //=== the equidimensional part over Q which gives the equdimensional part 
+   //=== the equidimensional part over Q which gives the equdimensional part
    //=== of I:h for a suitable integer h
    ideal E=1;
@@ -866,5 +866,5 @@
    //=== 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];         
+   int h =coefZ(J)[2];
    J=J,h;
    //=== call equidimensional part over Z for I,h
@@ -872,9 +872,9 @@
    if(h!=1)
    {
-       M=equidimZ(J);  
+       M=equidimZ(J);
        if(he==heightZ(M))
        {
           E=intersectZ(M,E);
-       }    
+       }
    }
    return(E);
@@ -902,8 +902,8 @@
 
 {
-   
+
    def R = basering;
-  
-   list rp = ringlist(R); 
+
+   list rp = ringlist(R);
    rp[1] = p;
    def Rp = ring(rp);
@@ -915,24 +915,24 @@
       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));
-   
-   } 
+
+   }
 
 }
@@ -963,6 +963,6 @@
 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 
+   //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
@@ -1045,11 +1045,11 @@
    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));  
- 
+   //ideal I=addIdealZ(J,specialPowerZ(P,20));
+
    //=== method of Shimoyama-Yokoyama
-   poly s=separatorsZ(j,B);   
+   poly s=separatorsZ(j,B);
    ideal I=satZ(J,s);
    //=== size(L)=0 means P is maximal ideal and I is primary
@@ -1068,6 +1068,6 @@
          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 
+         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]);
@@ -1104,13 +1104,13 @@
       {
          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);      
+      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=satZ(I,fac[i]);
       }
    }
@@ -1190,8 +1190,8 @@
 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)
@@ -1421,5 +1421,5 @@
          y2z2w+(-y*4+2*y^3-y^2)*z3;
 
-ring R4=integer,(w,z,y,x),dp; 
+ring R4=integer,(w,z,y,x),dp;
 ideal I=-2*yxzw+(-yx-y^2+y)*z^2,
          xw^2-yz^2,
@@ -1452,29 +1452,29 @@
         -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; 
+
+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; 
+        (-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; 
+        (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, 
-Â         (x+y+1)*zw+(-4*y^2*x-4*y^3-4*y^2)*z^2, 
-Â         (-x-2*y^2 - 2*y - 1)*zw +Â (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=(4*y^2*x^2+(4*y^3+4*y^2-y)*x-y^2-y)*z^2,
+Â         (x+y+1)*zw+(-4*y^2*x-4*y^3-4*y^2)*z^2,
+Â         (-x-2*y^2 - 2*y - 1)*zw +Â (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,
@@ -1482,71 +1482,71 @@
         (x + 1)*zw + (-y^2 -y)*z^2, (x^2 + x)*w^2 + (-yx - y)*zw;
 
-ring R13=integer,(w,z,y,x),dp; 
+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,Â  
-        (-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; 
+        -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,Â 
+        (-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(4)^2,
          x(2)^2*x(3)*x(4),
-         x(1)*x(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); 
-
- 
+         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, 
+        (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, 
+        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,Â  
-        x(3)^3*x(5)^2, 
+        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,Â 
+        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),Â  
+        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),Â 
         x(1)*x(2)*x(4)^2*x(5),
-        x(1)*x(4)^3*x(5), 
-        x(4)^4*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, 
-Â Â Â Â     17y*(z+2)^4-374*y*(z+2)^2+221y+2z7-281z5+5240z3-3081z, 
-Â Â Â    Â  204y2+136yz3-3128yz+z6-149z4+2739z2+117, 
-Â Â Â Â     17xz4-374xz2+221x+2z7-281z5+5240z3-3081z, 
-Â Â Â Â     136xy-136xz-136yz+2z6-281z4+5376z2-3081, 
-Â Â Â Â     204x2+136xz3-3128xz+z6-149z4+2739z2+117; 
-
-ring R18=integer,(B,D,F,b,d,f),dp; 
+        (z+2)^8-140z6+2622*(z+2)^4-1820*(z+2)^2+169,
+Â Â Â Â     17y*(z+2)^4-374*y*(z+2)^2+221y+2z7-281z5+5240z3-3081z,
+Â Â Â    Â  204y2+136yz3-3128yz+z6-149z4+2739z2+117,
+Â Â Â Â     17xz4-374xz2+221x+2z7-281z5+5240z3-3081z,
+Â Â Â Â     136xy-136xz-136yz+2z6-281z4+5376z2-3081,
+Â Â Â Â     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-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; 
+        F^2*f^3-1;
 
 ring R19=integer,(a,b,c,d,e,f),dp;
@@ -1579,9 +1579,9 @@
 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; 
+        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;
 */
