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Diff for /OpenXM/src/k097/lib/minimal/minimal.k between version 1.17 and 1.18

version 1.17, 2000/07/26 12:56:36 version 1.18, 2000/07/30 02:26:25
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 /* $OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 1.16 2000/06/15 07:38:36 takayama Exp $ */  /* $OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 1.17 2000/07/26 12:56:36 takayama Exp $ */
 #define DEBUG 1  #define DEBUG 1
 /* #define ORDINARY 1 */  /* #define ORDINARY 1 */
 /* If you run this program on openxm version 1.1.2 (FreeBSD),  /* If you run this program on openxm version 1.1.2 (FreeBSD),
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Line 6 
    ln -s /usr/bin/cpp /lib/cpp     ln -s /usr/bin/cpp /lib/cpp
 */  */
 #define OFFSET 0  #define OFFSET 0
 #define TOTAL_STRATEGY 1  
 /* #define OFFSET 20*/  /* #define OFFSET 20*/
 /* Test sequences.  /* Test sequences.
    Use load["minimal.k"];;     Use load["minimal.k"];;
Line 367  def Sdegree(f,tower,level) {
Line 366  def Sdegree(f,tower,level) {
   f = Init(f);    f = Init(f);
   if (level <= 1) return(StotalDegree(f));    if (level <= 1) return(StotalDegree(f));
   i = Degree(f,es);    i = Degree(f,es);
 #ifdef TOTAL_STRATEGY  
   return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1));    return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1));
 #endif  
   /* Strategy must be compatible with ordering.  */  
   /* Weight vector must be non-negative, too.  */  
   /* See Sdegree, SgenerateTable, reductionTable. */  
   wd = Sord_w(f,ww);  
   return(wd+Sdegree(tower[level-2,i],tower,level-1));  
   
 }  }
   
Line 888  def Sbases_to_vec(bases,size) {
Line 880  def Sbases_to_vec(bases,size) {
 }  }
   
 HelpAdd(["Sminimal",  HelpAdd(["Sminimal",
 ["It constructs the V-minimal free resolution by LaScala-Stillman's algorithm",  ["It constructs the V-minimal free resolution by LaScala's algorithm",
  "option: \"homogenized\" (no automatic homogenization ",   "option: \"homogenized\" (no automatic homogenization ",
  "Example:  Sweyl(\"x,y\",[[\"x\",-1,\"y\",-1,\"Dx\",1,\"Dy\",1]]);",   "Example:  Sweyl(\"x,y\",[[\"x\",-1,\"y\",-1,\"Dx\",1,\"Dy\",1]]);",
  "          v=[[2*x*Dx + 3*y*Dy+6, 0],",   "          v=[[2*x*Dx + 3*y*Dy+6, 0],",
Line 1105  def Sannfs2(f) {
Line 1097  def Sannfs2(f) {
   local p,pp;    local p,pp;
   p = Sannfs(f,"x,y");    p = Sannfs(f,"x,y");
   sm1(" p 0 get { [(x) (y) (Dx) (Dy)] laplace0 } map /p set ");    sm1(" p 0 get { [(x) (y) (Dx) (Dy)] laplace0 } map /p set ");
 /*  
   Sweyl("x,y",[["x",1,"y",1,"Dx",1,"Dy",1,"h",1],  
                ["x",-1,"y",-1,"Dx",1,"Dy",1]]); */  
   /* Sweyl("x,y",[["x",1,"y",1,"Dx",1,"Dy",1,"h",1]]); */  
   
   Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);    Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
   pp = Map(p,"Spoly");    pp = Map(p,"Spoly");
   return(Sminimal_v(pp));    return(Sminimal(pp));
   /* return(Sminimal(pp)); */  
 }  }
   
 HelpAdd(["Sannfs2",  HelpAdd(["Sannfs2",
 ["Sannfs2(f) constructs the V-minimal free resolution for the weight (-1,1)",  ["Sannfs2(f) constructs the V-minimal free resolution for the weight (-1,1)",
  "of the Laplace transform of the annihilating ideal of the polynomial f in x,y.",   "of the Laplace transform of the annihilating ideal of the polynomial f in x,y.",
  "See also Sminimal_v, Sannfs3.",   "See also Sminimal, Sannfs3.",
  "Example: a=Sannfs2(\"x^3-y^2\");",   "Example: a=Sannfs2(\"x^3-y^2\");",
  "         b=a[0]; sm1_pmat(b);",   "         b=a[0]; sm1_pmat(b);",
  "         b[1]*b[0]:",   "         b[1]*b[0]:",
Line 1127  HelpAdd(["Sannfs2",
Line 1113  HelpAdd(["Sannfs2",
  "         b=a[0]; sm1_pmat(b);",   "         b=a[0]; sm1_pmat(b);",
  "         b[1]*b[0]:"   "         b[1]*b[0]:"
 ]]);  ]]);
   /* Some samples.
     The betti numbers of most examples are 2,1. (0-th and 1-th).
     a=Sannfs2("x*y*(x+y-1)"); ==> The betti numbers are 3, 2.
     a=Sannfs2("x^3-y^2-x");
     a=Sannfs2("x*y*(x-y)");
   */
   
 /* Do not forget to turn on TOTAL_STRATEGY */  
 def Sannfs2_laScala(f) {  
   local p,pp;  
   p = Sannfs(f,"x,y");  
   /*   Do not make laplace transform.  
     sm1(" p 0 get { [(x) (y) (Dx) (Dy)] laplace0 } map /p set ");  
     p = [p];  
   */  
   Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);  
   pp = Map(p[0],"Spoly");  
   return(Sminimal(pp));  
 }  
   
 def Sannfs2_laScala2(f) {  
   local p,pp;  
   p = Sannfs(f,"x,y");  
   sm1(" p 0 get { [(x) (y) (Dx) (Dy)] laplace0 } map /p set ");  
   p = [p];  
   Sweyl("x,y",[["x",1,"y",1,"Dx",1,"Dy",1,"h",1],  
                ["x",-1,"y",-1,"Dx",1,"Dy",1]]);  
   pp = Map(p[0],"Spoly");  
   return(Sminimal(pp));  
 }  
   
 def Sannfs3(f) {  def Sannfs3(f) {
   local p,pp;    local p,pp;
   p = Sannfs(f,"x,y,z");    p = Sannfs(f,"x,y,z");
   sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");    sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");
   Sweyl("x,y,z",[["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]]);    Sweyl("x,y,z",[["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]]);
   pp = Map(p,"Spoly");    pp = Map(p,"Spoly");
   return(Sminimal_v(pp));    return(Sminimal(pp));
 }  }
   
 HelpAdd(["Sannfs3",  HelpAdd(["Sannfs3",
 ["Sannfs3(f) constructs the V-minimal free resolution for the weight (-1,1)",  ["Sannfs3(f) constructs the V-minimal free resolution for the weight (-1,1)",
  "of the Laplace transform of the annihilating ideal of the polynomial f in x,y,z.",   "of the Laplace transform of the annihilating ideal of the polynomial f in x,y,z.",
  "See also Sminimal_v, Sannfs2.",   "See also Sminimal, Sannfs2.",
  "Example: a=Sannfs3(\"x^3-y^2*z^2\");",   "Example: a=Sannfs3(\"x^3-y^2*z^2\");",
  "         b=a[0]; sm1_pmat(b);",   "         b=a[0]; sm1_pmat(b);",
  "         b[1]*b[0]: b[2]*b[1]:"]]);   "         b[1]*b[0]: b[2]*b[1]:"]]);
   
 /*  
   The betti numbers of most examples are 2,1. (0-th and 1-th).  
   a=Sannfs2("x*y*(x+y-1)"); ==> The betti numbers are 3, 2.  
   a=Sannfs2("x^3-y^2-x");    : it causes an error. It should be fixed.  
   a=Sannfs2("x*y*(x-y)");    : it causes an error. It should be fixed.  
   
 */  
   
 def Sannfs3_laScala2(f) {  
   local p,pp;  
   p = Sannfs(f,"x,y,z");  
   sm1(" p 0 get { [(x) (y) (z) (Dx) (Dy) (Dz)] laplace0 } map /p set ");  
   Sweyl("x,y,z",[["x",1,"y",1,"z",1,"Dx",1,"Dy",1,"Dz",1,"h",1],  
                  ["x",-1,"y",-1,"z",-1,"Dx",1,"Dy",1,"Dz",1]]);  
   pp = Map(p,"Spoly");  
   return(Sminimal(pp));  
 }  
   
   
 /*  The below does not use LaScala-Stillman's algorithm. */  
 def Sschreyer(g) {  
   local rf, tower, reductionTable, skel, redundantTable, bases,  
         strategy, maxOfStrategy, height, level, n, i,  
         freeRes,place, f, reducer,pos, redundant_seq,bettiTable,freeResV,ww,  
         redundantTable_ordinary, redundant_seq_ordinary,  
         reductionTable_tmp,c2,ii,nn, m,ii, jj, reducerBase;  
   /* extern WeightOfSweyl; */  
   ww = WeightOfSweyl;  
   Print("WeghtOfSweyl="); Println(WeightOfSweyl);  
   rf = SresolutionFrameWithTower(g);  
   redundant_seq = 1;   redundant_seq_ordinary = 1;  
   tower = rf[1];  
   Println("Generating reduction table which gives an order of reduction.");  
   Println("But, you are in Sschreyer...., you may not use LaScala-Stillman");  
   Print("WeghtOfSweyl="); Println(WeightOfSweyl);  
   Print("tower"); Println(tower);  
   reductionTable = SgenerateTable(tower);  
   skel = rf[2];  
   redundantTable = SnewArrayOfFormat(rf[1]);  
   redundantTable_ordinary = SnewArrayOfFormat(rf[1]);  
   reducer = SnewArrayOfFormat(rf[1]);  
   freeRes = SnewArrayOfFormat(rf[1]);  
   bettiTable = SsetBettiTable(rf[1],g);  
   
   height = Length(reductionTable);  
   for (level = 0; level < height; level++) {  
       n = Length(reductionTable[level]);  
       for (i=0; i<n; i++) {  
            Println([level,i]);  
            Print("Processing "); Print([level,i]);  
            if (level == 0) {  
              if (IsNull(redundantTable[level,i])) {  
                bases = freeRes[level];  
                /* Println(["At floor : GB=",i,bases,tower[0,i]]); */  
                pos = SwhereInGB(tower[0,i],rf[3,0]);  
                bases[i] = rf[3,0,pos];  
                /* redundantTable[level,i] = 0;  
                redundantTable_ordinary[level,i] = 0; */  
                freeRes[level] = bases;  
                /* Println(["GB=",i,bases,tower[0,i]]); */  
              }  
            }else{ /* level >= 1 */  
              if (IsNull(redundantTable[level,i])) {  
                bases = freeRes[level];  
                f = SpairAndReduction2(skel,level,i,freeRes,tower,  
                                       ww,redundantTable);  
                if (f[0] != Poly("0")) {  
                   place = f[3];  
                   /* (level-1, place) is the place for f[0],  
                      which is a newly obtained  GB. */  
 #ifdef ORDINARY  
                   redundantTable[level-1,place] = redundant_seq;  
                   redundant_seq++;  
 #else  
                   if (f[4] > f[5]) {  
                     /* Zero in the gr-module */  
                     Print("v-degree of [org,remainder] = ");  
                     Println([f[4],f[5]]);  
                     Print("[level,i] = "); Println([level,i]);  
                     redundantTable[level-1,place] = 0;  
                   }else{  
                     redundantTable[level-1,place] = redundant_seq;  
                     redundant_seq++;  
                   }  
 #endif  
                   redundantTable_ordinary[level-1,place]  
                      =redundant_seq_ordinary;  
                   redundant_seq_ordinary++;  
                   bases[i] = SunitOfFormat(place,f[1])-f[1];  /* syzygy */  
                   /* redundantTable[level,i] = 0;  
                   redundantTable_ordinary[level,i] = 0; */  
                   /* i must be equal to f[2], I think. Double check. */  
   
                   /* Correction Of Constant */  
                   /* Correction of syzygy */  
                   c2 = f[6];  /* or -f[6]?  Double check. */  
                   Print("c2="); Println(c2);  
                   nn = Length(bases);  
                   for (ii=0; ii<nn;ii++) {  
                      if ((ii != i) && (! IsNull(bases[ii]))) {  
                        m = Length(bases[ii]);  
                        for (jj=0; jj<m; jj++) {  
                          if (jj != place) {  
                            bases[ii,jj] = bases[ii,jj]*c2;  
                          }  
                        }  
                      }  
                   }  
   
                   Print("Old freeRes[level] = "); sm1_pmat(freeRes[level]);  
                   freeRes[level] = bases;  
                   Print("New freeRes[level] = "); sm1_pmat(freeRes[level]);  
   
                  /* Update the freeRes[level-1] */  
                   Print("Old freeRes[level-1] = "); sm1_pmat(freeRes[level-1]);  
                   bases = freeRes[level-1];  
                   bases[place] = f[0];  
                   freeRes[level-1] = bases;  
                   Print("New freeRes[level-1] = "); sm1_pmat(freeRes[level-1]);  
   
                   reducer[level-1,place] = f[1]-SunitOfFormat(place,f[1]);  
                    /* This reducer is different from that of SlaScala(). */  
   
                   reducerBasis = reducer[level-1];  
                   nn = Length(reducerBasis);  
                   for (ii=0; ii<nn;ii++) {  
                      if ((ii != place) && (! IsNull(reducerBasis[ii]))) {  
                        m = Length(reducerBasis[ii]);  
                        for (jj=0; jj<m; jj++) {  
                          if (jj != place) {  
                            reducerBasis[ii,jj] = reducerBasis[ii,jj]*c2;  
                          }  
                        }  
                      }  
                   }  
                   reducer[level-1] = reducerBasis;  
   
                }else{  
                   /* redundantTable[level,i] = 0; */  
                   bases = freeRes[level];  
                   bases[i] = f[1];  /* Put the syzygy. */  
                   freeRes[level] = bases;  
                }  
              }  /* end of level >= 1 */  
           }  
     } /* i loop */  
   
     /* Triangulate reducer */  
     if (level >= 1) {  
       Println(" ");  
       Print("Triangulating reducer at level "); Println(level-1);  
       Println("freeRes[level]="); sm1_pmat(freeRes[level]);  
       reducerBase = reducer[level-1];  
       Print("reducerBase=");  Println(reducerBase);  
       Println("Compare freeRes[level] and reducerBase (put -1)");  
       m = Length(reducerBase);  
       for (ii=m-1; ii>=0; ii--) {  
         if (!IsNull(reducerBase[ii])) {  
            for (jj=ii-1; jj>=0; jj--) {  
              if (!IsNull(reducerBase[jj])) {  
               if (!IsZero(reducerBase[jj,ii])) {  
                 /* reducerBase[ii,ii] should be always constant. */  
                 reducerBase[jj] = reducerBase[ii,ii]*reducerBase[jj]-reducerBase[jj,ii]*reducerBase[ii];  
               }  
              }  
            }  
          }  
        }  
        Println("New reducer");  
        sm1_pmat(reducerBase);  
        reducer[level-1] = reducerBase;  
     }  
   
   } /* level loop */  
   n = Length(freeRes);  
   freeResV = SnewArrayOfFormat(freeRes);  
   for (i=0; i<n; i++) {  
     bases = freeRes[i];  
     bases = Sbases_to_vec(bases,bettiTable[i]);  
     freeResV[i] = bases;  
   }  
   
   /* Mark the non-redundant elements. */  
   for (i=0; i<n; i++) {  
     m = Length(redundantTable[i]);  
     for (jj=0; jj<m; jj++) {  
       if (IsNull(redundantTable[i,jj])) {  
         redundantTable[i,jj] = 0;  
       }  
     }  
   }  
   
   
   return([freeResV, redundantTable,reducer,bettiTable,redundantTable_ordinary]);  
 }  
   
 def SpairAndReduction2(skel,level,ii,freeRes,tower,ww,redundantTable) {  
   local i, j, myindex, p, bases, tower2, gi, gj,  
        si, sj, tmp, t_syz, pos, ans, ssp, syzHead,pos2,  
        vdeg,vdeg_reduced,n,c2;  
   Println("SpairAndReduction2 : -------------------------");  
   
   if (level < 1) Error("level should be >= 1 in SpairAndReduction.");  
   p = skel[level,ii];  
   myindex = p[0];  
   i = myindex[0]; j = myindex[1];  
   bases = freeRes[level-1];  
   Println(["p and bases ",p,bases]);  
   if (IsNull(bases[i]) || IsNull(bases[j])) {  
     Println([level,i,j,bases[i],bases[j]]);  
     Error("level, i, j : bases[i], bases[j]  must not be NULL.");  
   }  
   
   tower2 = StowerOf(tower,level-1);  
   SsetTower(tower2);  
   Println(["level=",level]);  
   Println(["tower2=",tower2]);  
   /** sm1(" show_ring ");   */  
   
   gi = Stoes_vec(bases[i]);  
   gj = Stoes_vec(bases[j]);  
   
   ssp = Sspolynomial(gi,gj);  
   si = ssp[0,0];  
   sj = ssp[0,1];  
   syzHead = si*es^i;  
   /* This will be the head term, I think. But, double check. */  
   Println([si*es^i,sj*es^j]);  
   
   Print("[gi, gj] = "); Println([gi,gj]);  
   sm1(" [(Homogenize)] system_variable message ");  
   Print("Reduce the element "); Println(si*gi+sj*gj);  
   Print("by  "); Println(bases);  
   
   tmp = Sreduction(si*gi+sj*gj, bases);  
   
   Print("result is "); Println(tmp);  
   if (!IsZero(tmp[0])) {  
     Print("Error: base = ");  
     Println(Map(bases,"Stoes_vec"));  
     Error("SpairAndReduction2: the remainder should be zero. See tmp. tower2. show_ring.");  
   }  
   t_syz = tmp[2];  
   si = si*tmp[1]+t_syz[i];  
   sj = sj*tmp[1]+t_syz[j];  
   t_syz[i] = si;  
   t_syz[j] = sj;  
   
   c2 = null;  
   /* tmp[0] must be zero */  
   n = Length(t_syz);  
   for (i=0; i<n; i++) {  
      if (IsConstant(t_syz[i])){  
       if (!IsZero(t_syz[i])) {  
        if (IsNull(redundantTable[level-1,i])) {  
          /* i must equal to pos2 below. */  
          c2 = -t_syz[i];  
          tmp[0] = c2*Stoes_vec(freeRes[level-1,i]);  
          t_syz[i] = 0;  
          /* tmp[0] = t_syz . g */  
          /* break; does not work. Use */  
          i = n;  
        }  
       }  
      }  
   }  
   
   /* This is essential part for V-minimal resolution. */  
   /* vdeg = SvDegree(si*gi+sj*gj,tower,level-1,ww); */  
   vdeg = SvDegree(si*gi,tower,level-1,ww);  
   vdeg_reduced = SvDegree(tmp[0],tower,level-1,ww);  
   Print("vdegree of the original = "); Println(vdeg);  
   Print("vdegree of the remainder = "); Println(vdeg_reduced);  
   
   if (!IsNull(vdeg_reduced)) {  
     if (vdeg_reduced < vdeg) {  
       Println("--- Special in V-minimal!");  
       Println(tmp[0]);  
       Println("syzygy="); sm1_pmat(t_syz);  
       Print("[vdeg, vdeg_reduced] = "); Println([vdeg,vdeg_reduced]);  
     }  
   }  
   
   SsetTower(StowerOf(tower,level));  
   pos = SwhereInTower(syzHead,tower[level]);  
   
   SsetTower(StowerOf(tower,level-1));  
   pos2 = SwhereInTower(tmp[0],tower[level-1]);  
   ans = [tmp[0],t_syz,pos,pos2,vdeg,vdeg_reduced,c2];  
   /* pos is the place to put syzygy at level. */  
   /* pos2 is the place to put a new GB at level-1. */  
   Println(ans);  
   Println("--- end of SpairAndReduction2  ");  
   return(ans);  
 }  
   
 HelpAdd(["Sminimal_v",  
 ["It constructs the V-minimal free resolution from the Schreyer resolution",  
  "step by step.",  
  "This code still contains bugs. It sometimes outputs wrong answer.",  
  "Example:   Sweyl(\"x,y\",[[\"x\",-1,\"y\",-1,\"Dx\",1,\"Dy\",1]]);",  
  "          v=[[2*x*Dx + 3*y*Dy+6, 0],",  
  "             [3*x^2*Dy + 2*y*Dx, 0],",  
  "             [0,  x^2+y^2],",  
  "             [0,  x*y]];",  
  "         a=Sminimal_v(v);",  
  "         sm1_pmat(a[0]); b=a[0]; b[1]*b[0]:",  
  "Note:  a[0] is the V-minimal resolution. a[3] is the Schreyer resolution."]]);  
   
 /* This code still contains bugs. It sometimes outputs wrong answer. */  
 /* See test12() in minimal-test.k.  */  
 /* There may be remaining 1, too */  
 def Sminimal_v(g) {  
   local r, freeRes, redundantTable, reducer, maxLevel,  
         minRes, seq, maxSeq, level, betti, q, bases, dr,  
         betti_levelplus, newbases, i, j,qq,tminRes;  
   r = Sschreyer(g);  
   sm1_pmat(r);  
   Debug_Sminimal_v = r;  
   Println(" Return value of Schreyer(g) is set to Debug_Sminimal_v");  
   /* Should I turn off the tower?? */  
   freeRes = r[0];  
   redundantTable = r[1];  
   reducer = r[2];  
   minRes = SnewArrayOfFormat(freeRes);  
   seq = 0;  
   maxSeq = SgetMaxSeq(redundantTable);  
   maxLevel = Length(freeRes);  
   for (level = 0; level < maxLevel; level++) {  
     minRes[level] = freeRes[level];  
   }  
   for (level = 0; level < maxLevel; level++) {  
       betti = Length(freeRes[level]);  
       for (q = betti-1; q>=0; q--) {  
         if (redundantTable[level,q] > 0) {  
           Print("[seq,level,q]="); Println([seq,level,q]);  
           if (level < maxLevel-1) {  
             bases = freeRes[level+1];  
             dr = reducer[level,q];  
             /* dr[q] = -1;  We do not need this in our reducer format. */  
             /* dr[q] should be a non-zero constant. */  
             newbases = SnewArrayOfFormat(bases);  
             betti_levelplus = Length(bases);  
             /*  
                bases[i,j] ---> bases[i,j]+bases[i,q]*dr[j]  
             */  
             for (i=0; i<betti_levelplus; i++) {  
               newbases[i] = dr[q]*bases[i] - bases[i,q]*dr;  
             }  
             Println(["level, q =", level,q]);  
             Println("bases="); sm1_pmat(bases);  
             Println("dr="); sm1_pmat(dr);  
             Println("newbases="); sm1_pmat(newbases);  
             minRes[level+1] = newbases;  
             freeRes = minRes;  
 #ifdef DEBUG  
             for (qq=q; qq<betti; qq++) {  
                 for (i=0; i<betti_levelplus; i++) {  
                   if ((!IsZero(newbases[i,qq])) && (redundantTable[level,qq] >0)) {  
                     Println(["[i,qq]=",[i,qq]," is not zero in newbases."]);  
                     Print("redundantTable ="); sm1_pmat(redundantTable[level]);  
                     Error("Stop in Sminimal for debugging.");  
                   }  
                 }  
             }  
 #endif  
           }  
         }  
       }  
    }  
    tminRes = Stetris(minRes,redundantTable);  
    return([SpruneZeroRow(tminRes), tminRes,  
           [ minRes, redundantTable, reducer,r[3],r[4]],r[0]]);  
   /* r[4] is the redundantTable_ordinary */  
   /* r[0] is the freeResolution */  
 }  
   
 /* Sannfs2("x*y*(x-y)*(x+y)"); is a test problem */  /* Sannfs2("x*y*(x-y)*(x+y)"); is a test problem */
 /* x y (x+y-1)(x-2),  x^3-y^2, x^3 - y^2 z^2,  /* x y (x+y-1)(x-2),  x^3-y^2, x^3 - y^2 z^2,
    x y z (x+y+z-1) seems to be interesting, because the first syzygy     x y z (x+y+z-1) seems to be interesting, because the first syzygy
Line 1688  def Skernel(m,v) {
Line 1282  def Skernel(m,v) {
   sm1(" [ m v ] syz /FunctionValue set ");    sm1(" [ m v ] syz /FunctionValue set ");
 }  }
   
 def test3() {  
   local a1,a2,b1,b2;  
   a1 = Sannfs3("x^3-y^2*z^2");  
   a1 = a1[0];  
   a2 = Sannfs3_laScala2("x^3-y^2*z^2");  
   a2 = a2[0];  
   b1 = a1[1];  
   b2 = a2[1];  
   sm1_pmat(b2);  
   Println("  OVER ");  
   sm1_pmat(b1);  
   return([sm1_res_div(b2,b1,["x","y","z"]),b2,b1,a2,a1]);  
 }  
   
 def test4() {  
   local a,b;  
   a = Sannfs3_laScala2("x^3-y^2*z^2");  
   b = a[0];  
   sm1_pmat( sm1_res_kernel_image(b[0],b[1],[x,y,z]));  
   sm1_pmat( sm1_res_kernel_image(b[1],b[2],[x,y,z]));  
   return(a);  
 }  
   
 def sm1_gb(f,v) {  def sm1_gb(f,v) {
   f =ToString_array(f);    f =ToString_array(f);
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