OpenXM/src/k097/lib/minimal/minimal.k - diff

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Diff for /OpenXM/src/k097/lib/minimal/minimal.k between version 1.6 and 1.10

-version 1.6, 2000/05/06 07:58:37
+version 1.10, 2000/05/07 02:10:44
 Line 1
 Line 1
 Line 1
- /* $OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 1.5 2000/05/05 08:13:49 takayama Exp $ */
+ /* $OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 1.9 2000/05/06 13:41:12 takayama Exp $ */
  #define DEBUG 1
  /* #define ORDINARY 1 */
  /* If you run this program on openxm version 1.1.2 (FreeBSD),
 Line 37  def load_tower() {
 Line 37  def load_tower() {
 Line 37  def load_tower() {
      sm1(" [(parse) (k0-tower.sm1) pushfile ] extension ");
      sm1(" /k0-tower.sm1.loaded 1 def ");
    }
+   sm1(" oxNoX ");
  }
  load_tower();
  SonAutoReduce = true;
-Line 1043  def Sannfs2(f) {
+Line 1044  def Sannfs2(f) {
 Line 1043  def Sannfs2(f) {
 Line 1044  def Sannfs2(f) {
    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]]);
    pp = Map(p,"Spoly");
    return(Sminimal_v(pp));
    /* return(Sminimal(pp)); */
  }
+ HelpAdd(["Sannfs2",
+ ["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.",
+  "See also Sminimal_v, Sannfs3.",
+  "Example: a=Sannfs2(\"x^3-y^2\");",
+  "         b=a[0]; sm1_pmat(b);",
+  "         b[1]*b[0]:",
+  "Example: a=Sannfs2(\"x*y*(x-y)*(x+y)\");",
+  "         b=a[0]; sm1_pmat(b);",
+  "         b[1]*b[0]:"
+ ]]);
  /* Do not forget to turn on TOTAL_STRATEGY */
  def Sannfs2_laScala(f) {
    local p,pp;
-Line 1071  def Sannfs3(f) {
+Line 1085  def Sannfs3(f) {
 Line 1071  def Sannfs3(f) {
 Line 1085  def Sannfs3(f) {
    return(Sminimal_v(pp));
  }
+ HelpAdd(["Sannfs3",
+ ["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.",
+  "See also Sminimal_v, Sannfs2.",
+  "Example: a=Sannfs3(\"x^3-y^2*z^2\");",
+  "         b=a[0]; sm1_pmat(b);",
+  "         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.
-Line 1152  def Sschreyer(g) {
+Line 1174  def Sschreyer(g) {
 Line 1152  def Sschreyer(g) {
 Line 1174  def Sschreyer(g) {
                    /* i must be equal to f[2], I think. Double check. */
                    /* Correction Of Constant */
-                   c2 = -f[6];  /* or f[6]?  Double check. */
+                   /* 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 != place) && (! IsNull(bases[ii]))) {
+                      if ((ii != i) && (! IsNull(bases[ii]))) {
-                        bases[ii] = bases[ii]*c2;
+                        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];
+                   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];
-Line 1182  def Sschreyer(g) {
+Line 1231  def Sschreyer(g) {
 Line 1182  def Sschreyer(g) {
 Line 1231  def Sschreyer(g) {
      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[jj] = reducerBase[jj]-reducerBase[jj,ii]*reducerBase[ii];
+                 /* reducerBase[ii,ii] should be always constant. */
+                 reducerBase[jj] = reducerBase[ii,ii]*reducerBase[jj]-reducerBase[jj,ii]*reducerBase[ii];
                }
               }
             }
-Line 1310  def SpairAndReduction2(skel,level,ii,freeRes,tower,ww,
+Line 1362  def SpairAndReduction2(skel,level,ii,freeRes,tower,ww,
 Line 1310  def SpairAndReduction2(skel,level,ii,freeRes,tower,ww,
 Line 1362  def SpairAndReduction2(skel,level,ii,freeRes,tower,ww,
    return(ans);
  }
+ HelpAdd(["Sminimal_v",
+ ["It constructs the V-minimal free resolution from the Schreyer resolution",
+  "step by step.",
+  "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."]]);
  def Sminimal_v(g) {
    local r, freeRes, redundantTable, reducer, maxLevel,
          minRes, seq, maxSeq, level, betti, q, bases, dr,
-         betti_levelplus, newbases, i, j,qq;
+         betti_levelplus, newbases, i, j,qq,tminRes;
    r = Sschreyer(g);
    sm1_pmat(r);
    Debug_Sminimal_v = r;
-Line 1337  def Sminimal_v(g) {
+Line 1402  def Sminimal_v(g) {
 Line 1337  def Sminimal_v(g) {
 Line 1402  def Sminimal_v(g) {
            if (level < maxLevel-1) {
              bases = freeRes[level+1];
              dr = reducer[level,q];
-             dr[q] = -1;
+             /* 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] = bases[i] + bases[i,q]*dr;
+               newbases[i] = dr[q]*bases[i] - bases[i,q]*dr;
              }
              Println(["level, q =", level,q]);
              Println("bases="); sm1_pmat(bases);
-Line 1353  def Sminimal_v(g) {
+Line 1419  def Sminimal_v(g) {
 Line 1353  def Sminimal_v(g) {
 Line 1419  def Sminimal_v(g) {
              minRes[level+1] = newbases;
              freeRes = minRes;
  #ifdef DEBUG
- /*  Do it later.
+             for (qq=q; qq<betti; qq++) {
-             for (qq=0; qq<betti; qq++) {
                  for (i=0; i<betti_levelplus; i++) {
-                   if (!IsZero(newbases[i,qq])) {
+                   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
            }
          }
        }
     }
-    return([Stetris(minRes,redundantTable),
+    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 */
+ /* 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
+   contains 1.
+ */
+ def CopyArray(m) {
+   local ans,i,n;
+   if (IsArray(m)) {
+      n = Length(m);
+      ans = NewArray(n);
+      for (i=0; i<n; i++) {
+        ans[i] = CopyArray(m[i]);
+      }
+      return(ans);
+   }else{
+      return(m);
+   }
+ }
+ HelpAdd(["CopyArray",
+ ["It duplicates the argument array recursively.",
+  "Example: m=[1,[2,3]];",
+  "         a=CopyArray(m); a[1] = \"Hello\";",
+  "         Println(m); Println(a);"]]);
+ def IsZeroVector(m) {
+   local n,i;
+   n = Length(m);
+   for (i=0; i<n; i++) {
+     if (!IsZero(m[i])) {
+       return(false);
+     }
+   }
+   return(true);
+ }
+ def SpruneZeroRow(res) {
+   local minRes, n,i,j,m, base,base2,newbase,newbase2, newMinRes;
+   minRes = CopyArray(res);
+   n = Length(minRes);
+   for (i=0; i<n; i++) {
+     base = minRes[i];
+     m = Length(base);
+     if (i != n-1) {
+       base2 = minRes[i+1];
+       base2 = Transpose(base2);
+     }
+     newbase = [ ];
+     newbase2 = [ ];
+     for (j=0; j<m; j++) {
+       if (!IsZeroVector(base[j])) {
+         newbase = Append(newbase,base[j]);
+         if (i != n-1) {
+           newbase2 = Append(newbase2,base2[j]);
+         }
+       }
+     }
+     minRes[i] = newbase;
+     if (i != n-1) {
+       if (newbase2 == [ ]) {
+         minRes[i+1] = [ ];
+       }else{
+         minRes[i+1] = Transpose(newbase2);
+       }
+     }
+   }
+   newMinRes = [ ];
+   n = Length(minRes);
+   i = 0;
+   while (i < n ) {
+     base = minRes[i];
+     if (base == [ ]) {
+       i = n; /* break; */
+     }else{
+       newMinRes = Append(newMinRes,base);
+     }
+     i++;
+   }
+   return(newMinRes);
+ }
+ def testAnnfs2(f) {
+   local a,i,n;
+   a = Sannfs2(f);
+   b=a[0];
+   n = Length(b);
+   Println("------ V-minimal free resolution -----");
+   sm1_pmat(b);
+   Println("----- Is it complex?  ---------------");
+   for (i=0; i<n-1; i++) {
+     Println(b[i+1]*b[i]);
+   }
+   return(a);
+ }
+ def testAnnfs3(f) {
+   local a,i,n;
+   a = Sannfs3(f);
+   b=a[0];
+   n = Length(b);
+   Println("------ V-minimal free resolution -----");
+   sm1_pmat(b);
+   Println("----- Is it complex?  ---------------");
+   for (i=0; i<n-1; i++) {
+     Println(b[i+1]*b[i]);
+   }
+   return(a);
+ }

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