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Diff for /OpenXM/src/k097/lib/minimal/minimal-note-ja.txt between version 1.3 and 1.4

version 1.3, 2000/06/08 08:37:53 version 1.4, 2000/06/09 08:04:54
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 $OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.2 2000/05/24 15:24:54 takayama Exp $  $OpenXM: OpenXM/src/k097/lib/minimal/minimal-note-ja.txt,v 1.3 2000/06/08 08:37:53 takayama Exp $
   
 SpairAndReduction() :  SpairAndReduction() :
    $BM?$($i$l$?(B pair $B$r(B reduction $B$9$k(B.     $BM?$($i$l$?(B pair $B$r(B reduction $B$9$k(B.
Line 77  test8() $B$G(B sm1 $B$G=q$$$?J}$N(B Schreyer $B$r
Line 77  test8() $B$G(B sm1 $B$G=q$$$?J}$N(B Schreyer $B$r
 kernel = image  kernel = image
 $B$H$J$C$F$$$k$N$G0J8e$3$N(B option $B$O(B 1 $B$N$^$^;H$&$3$H$H$9$k(B.  $B$H$J$C$F$$$k$N$G0J8e$3$N(B option $B$O(B 1 $B$N$^$^;H$&$3$H$H$9$k(B.
 $BMW$9$k$K(B k0 $B$N%3!<%I$,$I$&$d$i$*$+$7$$$i$7$$(B.  $BMW$9$k$K(B k0 $B$N%3!<%I$,$I$&$d$i$*$+$7$$$i$7$$(B.
   ==>
   6/8 $B$N%N!<%H$h$j(B.
   syzygy $B$r(B homogenization $B$r2p$7$F7W;;$9$k$N$OLdBj$"$j(B.
   --> usage of isExact
   $BMW$9$k$K(B kernel = image $B$N%3!<%I$bJQ(B.  Homogenized $B$N$^$^$d$kI,MW$"$j(B.
   
 -----------------------------------  -----------------------------------
 June 8, 2000 (Thu), 9:10 (Spain local time)  June 8, 2000 (Thu), 9:10 (Spain local time)
Line 100  test10();
Line 105  test10();
   LaScala-Stillman $B$NJ}K!$G$D$/$C$?(B, schreyer resol $B$,(B exact $B$+(B    LaScala-Stillman $B$NJ}K!$G$D$/$C$?(B, schreyer resol $B$,(B exact $B$+(B
   $BD4$Y$k(B.    $BD4$Y$k(B.
   $BNcBj$O(B, ann(1/(x^3-y^2 z^2)) $B$N(B Laplace $BJQ49(B.    $BNcBj$O(B, ann(1/(x^3-y^2 z^2)) $B$N(B Laplace $BJQ49(B.
     ==> OK.  IsExact_h $B$G$7$i$Y$k(B.  (IsExact $B$O$@$a$h(B)
   
   
   
   June 8, 2000 (Thu), 19:35
   load["minimal-test.k"];;
   test11();
     LaScala-Stillman $B$NJ}K!$G$D$/$C$?(B, minimal resol $B$,(B exact $B$+(B
     $BD4$Y$k(B.
     $BNcBj$O(B, ann(1/(x^3-y^2 z^2)) $B$N(B Laplace $BJQ49(B.
   
   SwhereInTower $B$r;H$&$H$-$O(B,
   SsetTower() $B$G(B gbList $B$rJQ99$7$J$$$H$$$1$J$$(B.
   $B$b$A$m$s;HMQ$7$?$i(B, $B$=$l$rLa$9$3$H(B.
   SpairAndReduction, SpairAndReduction2 $B$G(B,
     SsetTower(StowerOf(tower,level));
     pos = SwhereInTower(syzHead,tower[level]);
   
     SsetTower(StowerOf(tower,level-1));
     pos2 = SwhereInTower(tmp[0],tower[level-1]);
   $B$H(B, SwhereInTower $B$NA0$K(B setTower $B$r$/$o$($?(B.
   ( $B0c$&%l%Y%k$G$NHf3S$N$?$a(B.)
   
   IsExact_h $B$O(B, 0 $B%Y%/%H%k$r4^$`>l9g(B, $B$?$@$7$/F0:n$7$J$$$h$&$@(B.
   test11().
   test11a() $B$G(B, 0 $B%Y%/%H%k$r<j$G=|$$$?9TNs$N(B exactness $B$r%A%'%C%/(B. ==> OK.
   
   
   ---------------------------------
   June 9, 6:20
   SpairAndReduction
   $B$H(B
   SpairAndReduction2
   $B$N0c$$(B.
   SpairAndReduction  :  SlaScala  (LaScala-Stillman's algorithm $B$G;H$&(B)
   SpairAndReduction2 :  Sschreyer (schreyer  algorithm $B$G;H$&(B, laScala $B$O$J$7(B.)
   
   0 $B$r<+F0$G=|$/%3!<%I$r=q$3$&(B.
   
   SpruneZeroRow() $B$r(B Sminimal() $B$K2C$($?(B.
   test11() $B$b@5$7$/F0:n$9$k$O$:(B.
   IsExact_h $B$O(B schreyer $B$r(B off $B$7$F(B, ReParse $B$7$F$+$i(B,
   $B8F$S=P$9$3$H(B.
   
   
   #ifdef TOTAL_STRATEGY
     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));
   TOTAL_STRATEGY $B$rMQ$$$kI,MW$,$"$k$N$G$O(B??
   Example 1:  Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
             v=[[2*x*Dx + 3*y*Dy+6, 0],
                [3*x^2*Dy + 2*y*Dx, 0],
                [0,  x^2+y^2],
                [0,  x*y]];
            a=Sminimal(v);
   strategy $B$,$*$+$7$$$H$$$C$F$H$^$k(B. $BM}M3$O(B?
   
   a=test_ann3("x^3+y^3+z^3); $B$O;~4V$,$+$+$j$=$&(B.
   a=test_ann3("x^3+y^3"); OK.
   a=test_ann3("x^2+y^2+z"); OK.
   
   
   $B>e$N(B example 1 $B$N%(%i!<(B $B$N8+J}(B:
   Processing [    1 , 3 ]    Strategy = 2
        1 $B$N(B 3 $BHVL\$N(B spair $B$N(B reduction $B$r=hM}Cf(B.
        In(7)=reductionTable:
       [[ 1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
                                                      -- $B$3$l(B.
   SpairAndReduction:
   [    p and bases  , [    [    0 , 3 ]  , [    y*h , -x ]  ]  , [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ]  ]
   0 $B$N(B 0 $BHVL\$H(B 3 $BHVL\(B $B$N(B spair $B$r7W;;$7$F(B, 0 $B%l%Y%k$N(B gb $B$G(B reduction.
   [ 1 , 1 , 1 , 2 , 2 , 3 ] $B$K$"$k$h$&$K(B, strategy 3 $B0J30$O7W;;$:$_(B.
   ( $B7W;;$7$F$J$$$b$N$O(B %[null] $B$H$J$C$F$k(B. )
   [    level= , 1 ]
   [    tower2= , [    [   ]  ]  ]   ( $B0lHV2<$J$N$G(B, tower $B$O$J$7$h(B. )
   [    y*h , -es^3*x ]
   [gi, gj] = [    2*x*Dx+3*y*Dy+6*h^2 , 2*y*Dx*h+3*x^2*Dy ]
   1
   Reduce the element 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy
   by  [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ]
   result is [    3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , 1 , [    0 , 0 , 0 , 0 , 0 , 0 ]  ]
   vdegree of the original = -1
   vdegree of the remainder = -1
   [    3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , [    y*h , 0 , 0 , -x , 0 , 0 ]  , 3 , 5 , -1 , -1 ]
   
   In(11)=freeRes:
   [    [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ]  , [    %[null] , [    0 , 0 , y^2 , 0 , -x , 0 ]  , [    0 , -y , x , 0 , 1 , 0 ]  , [    -y*h , 0 , 0 , x , 0 , 1 ]  , %[null] ]  , [    %[null] ]  ]
   $B$r$_$l$P$o$+$k$h$&$K(B, SlaScala $B$G(B, freeRes $B$K$3$N85$,(B [0,5] $B$K2C$((B
   $B$i$l$?(B.
   
   $B<!$K(B SnextI $B$,(B SlaScala $B$h$j8F$P$l$F$3$N%(%i!<(B.
           i = SnextI(reductionTable_tmp,strategy,redundantTable,
                      skel,level,freeRes);
   In(22)=reductionTable:
   [    [    1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
   $B$J$N$G(B, $B:G8e(B $B$N(B 2 $B$,=hM}$5$l$k$O$:$@$,(B,
   In(25)=skel[2]:
   [    [    [    0 , 2 ]  , [    1 , -y^2 ]  ]  ]
   $B$N$h$&$K(B, 0 $BHVL\$H(B, 2 $BHVL\$N(B spair.
   $B$7$+$7(B,
   In(26)=bases:
   [    %[null] , [    0 , 0 , y^2 , 0 , -x , 0 ]  , [    0 , -y , x , 0 , 1 , 0 ]  , [    -y*h , 0 , 0 , x , 0 , 1 ]  , %[null] ]
   $B$N$h$&$K(B, 0 $BHVL\$O(B strategy 3 $B$J$N$G(B, $B$^$@$b$H$^$C$F$$$J$$(B.
   
   reductionTable_tmp=[    2 ]
   See also reductionTable, strategy, level,i
   ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations.
   --- Engine error or interrupt : In function : Error of class PrimitiveObject
   
   Type in Cleards() to exit the debug mode and Where() to see the stack trace.
   In(7)=reductionTable:
   [    [    1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
   In(8)=strategy:
   2
   In(9)=level:
   2
   
      RemoveRedundantInSchreyerSkelton = 0
   $B$H$7$F$bF1$8%(%i!<(B.
   
   -------------------------------------------------
   test_ann3("x*y+y*z+z*x");    OK.
   
   6/9 (Fri)
   Sminimal $B$N<BAu$KAjJQ$o$i$:6lO+$7$F$^$9(B.
   Sevilla $B$G$$$m$$$m$HD>$7$?7k2L(B,
   Sminimal $B$O$&$^$/$&$4$1$P@5$7$$Ez$($r$@$7$F$k$_$?$$$G$9$,(B
   (D<h> : homogenized Weyl $B$G(B ker = im $B$r(B check $B$7$F$k(B,
    V-adapted (strict) $B$+$I$&$+$N(B check routing $B$O$^$@=q$$$F$J$$(B),
   strategy $B$,$&$^$/$&$4$+$J$/$F$H$^$k>l9g$b$"$j$^$9(B
   ( strategy = 2 $B$N(B sp $B$r7W;;$9$k$N$K(B, strategy 3 $B$N(B $B85$rI,MW$H(B
     $B$7$?$j$9$k>l9g$"$j(B).
   
   
   strategy $B$O(B
   def Sdegree(f,tower,level) {
     local i,ww, wd;
     /* extern WeightOfSweyl; */
     ww = WeightOfSweyl;
     f = Init(f);
     if (level <= 1) return(StotalDegree(f));
     i = Degree(f,es);
     return(StotalDegree(f)+Sdegree(tower[level-2,i],tower,level-1));
   }
   $B$rMQ$$$F(B,
         ans_at_each_floor[j] = Sdegree(tower[i,j],tower,i+1)-(i+1)
   $B$G7W;;$7$F$^$9(B.
   
   $B$$$/$D$+=PNO$r$D$1$F$*$-$^$9$N$G(B, $B8!F$(B!!!
   
   $BNc(B 1:
   load["minimal-test.k"];;
   a=test_ann3("x^3-y^2*z^2"); $B0z?t$N(B annihilating ideal $B$N(B laplace $BJQ49$N(B
                               homogenization $B$N(B resolution.
         weight vector $B$O(B (-1,-1,-1,1,1,1)
   
   In(4)=sm1_pmat(a[1]);
    [
     [   0 $B<!(B
       [    y*Dy-z*Dz ]
       [    -2*x*Dx-3*z*Dz+h^2 ]
       [    2*x*Dy*Dz^2-3*y*Dx^2*h ]
       [    2*x*Dy^2*Dz-3*z*Dx^2*h ]
     ]
     [   1 $B<!(B
       [    3*Dx^2*h , 0 , Dy , -Dz ]
       [    6*x*Dy*Dz^2-9*y*Dx^2*h , -2*x*Dy*Dz^2+3*y*Dx^2*h , -2*x*Dx-3*y*Dy , 0 ]
       [    0 , 2*x*Dy^2*Dz-3*z*Dx^2*h , 0 , 2*x*Dx+3*z*Dz ]
       [    2*x*Dx+3*z*Dz-h^2 , y*Dy-z*Dz , 0 , 0 ]
       [    2*x*Dy*Dz , 0 , z , -y ]
     ]
     [  2 $B<!(B
       [    -2*x*Dx-3*y*Dy-3*z*Dz-6*h^2 , -Dy , -Dz , 3*Dx^2*h , 3*Dy*Dz ]
       [    3*y*z , z , y , -2*x*Dy*Dz , 2*x*Dx ]
     ]
    ]
   In(5)=
   
   $BNc(B 2:
   load["minimal-test.k"];;
   a=test_ann3("x*y+y*z+z*x");
   In(6)=sm1_pmat(a[1]);
    [
     [  0 $B<!(B
       [    2*x*Dx+x*Dz-y*Dz+z*Dz+h^2 ]
       [    -2*y*Dy+x*Dz-y*Dz-z*Dz-h^2 ]
       [    -2*x*Dy+2*z*Dy+x*Dz-y*Dz+3*z*Dz+h^2 ]
       [    -2*y*Dx+2*z*Dx-x*Dz+y*Dz+3*z*Dz+h^2 ]
     ]
     [  1 $B<!(B
       [    y-z , x-z , -y , x ]
       [    2*Dy-2*Dz , 2*Dx-2*Dz , 2*Dx+2*Dz , -2*Dy-2*Dz ]
       [    2*y*Dx-2*z*Dx+x*Dz-y*Dz-3*z*Dz-2*h^2 , 0 , 0 , 2*x*Dx+x*Dz-y*Dz+z*Dz+2*h^2 ]
       [    2*y*Dy-2*z*Dy+y*Dz-z*Dz+h^2 , 2*x*Dz-y*Dz+2*z*Dz+h^2 , -x*Dz+z*Dz , 2*x*Dy+x*Dz ]
       [    -2*y*Dy+2*z*Dy+y*Dz-z*Dz , y*Dz-4*z*Dz , -2*y*Dx+2*z*Dx-y*Dz+2*z*Dz , -2*z*Dy+y*Dz-3*z*Dz ]
     ]
     [  2 $B<!(B
       [    -2*y*Dx+2*z*Dx-y*Dz+2*z*Dz , x*y-x*z-y*z+z^2 , y-z , y , x+y-z ]
       [    -6*Dx*Dz-2*Dz^2 , x*Dz+y*Dz-5*z*Dz-4*h^2 , -2*Dy+2*Dz , 2*Dx+2*Dz , 4*Dz ]
     ]
    ]
   In(7)=
   
   $BNc(B 3:  $B$&$^$/9T$+$J$$Nc(B:
   
   Example 1:  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);
   strategy $B$,$*$+$7$$$H$$$C$F$H$^$k(B. $BM}M3$O(B?
   Negative weight vector $B$r;H$o$J$$$H$-$A$s$HF0$-$^$9(B.
   
   
   DEBUG $B=PNO(B:
   rf= [
     [
      [   Schreyer frame.
        [    0 , y^3 , 0 , 0 , -x^2 , 0 ]
        [    0 , 0 , y^2 , 0 , -x , 0 ]
        [    0 , y , -x , 0 , 0 , 0 ]
        [    y*h , 0 , 0 , -x , 0 , 0 ]
        [    0 , 0 , 0 , 3*y*Dy , 0 , -2*Dx ]
      ]
      [
        [    1 , 0 , -y^2 , 0 , 0 ]
      ]
       [   ]
     ]
     [
       [    2*x*Dx , e_*x^2 , e_*x*y , 2*y*Dx*h , e_*y^3 , 3*y^2*Dy*h ]
       [    es*y^3 , es^2*y^2 , es*y , y*h , 3*es^3*y*Dy ]
       [    1 ]
     ]
     [
       [   ]
      [
       [
         [    1 , 4 ]
         [    y^3 , -x^2 ]
       ]
       [
         [    2 , 4 ]
         [    y^2 , -x ]
       ]
       [
         [    1 , 2 ]
         [    y , -x ]
       ]
       [
         [    0 , 3 ]
         [    y*h , -x ]
       ]
       [
         [    3 , 5 ]
         [    3*y*Dy , -2*Dx ]
       ]
      ]
      [
       [
         [    0 , 2 ]
         [    1 , -y^2 ]
       ]
      ]
       [   ]
     ]
     [   resolution $B$9$Y$-(B $BItJ,2C72(B e_ $B$O(B $B%Y%/%H%k@.J,$N%^!<%/(B.
       [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ]
     ]
    ]
   
   $BN,(B
   Processing [    1 , 3 ]    Strategy = 2
        1 $B$N(B 3 $BHVL\$N(B spair $B$N(B reduction $B$r=hM}Cf(B.
        In(7)=reductionTable:
       [[ 1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
                                                      -- $B$3$l(B.
   SpairAndReduction:
   [    p and bases  , [    [    0 , 3 ]  , [    y*h , -x ]  ]  , [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ]  ]
   0 $B$N(B 0 $BHVL\$H(B 3 $BHVL\(B $B$N(B spair $B$r7W;;$7$F(B, 0 $B%l%Y%k$N(B gb $B$G(B reduction.
   [ 1 , 1 , 1 , 2 , 2 , 3 ] $B$K$"$k$h$&$K(B, strategy 3 $B0J30$O7W;;$:$_(B.
   ( $B7W;;$7$F$J$$$b$N$O(B %[null] $B$H$J$C$F$k(B. )
   [    level= , 1 ]
   [    tower2= , [    [   ]  ]  ]   ( $B0lHV2<$J$N$G(B, tower $B$O$J$7$h(B. )
   [    y*h , -es^3*x ]
   [gi, gj] = [    2*x*Dx+3*y*Dy+6*h^2 , 2*y*Dx*h+3*x^2*Dy ]
   1
   Reduce the element 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy
   by  [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , %[null] ]
   result is [    3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , 1 , [    0 , 0 , 0 , 0 , 0 , 0 ]  ]
   vdegree of the original = -1
   vdegree of the remainder = -1
   [    3*y^2*Dy*h+6*y*h^3-3*x^3*Dy , [    y*h , 0 , 0 , -x , 0 , 0 ]  , 3 , 5 , -1 , -1 ]
   
   In(11)=freeRes:
   [    [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ]  , [    %[null] , [    0 , 0 , y^2 , 0 , -x , 0 ]  , [    0 , -y , x , 0 , 1 , 0 ]  , [    -y*h , 0 , 0 , x , 0 , 1 ]  , %[null] ]  , [    %[null] ]  ]
   $B$r$_$l$P$o$+$k$h$&$K(B, SlaScala $B$G(B, freeRes $B$K$3$N85$,(B [0,5] $B$K2C$((B
   $B$i$l$?(B.
   
   $B<!$K(B SnextI $B$,(B SlaScala $B$h$j8F$P$l$F$3$N%(%i!<(B.
           i = SnextI(reductionTable_tmp,strategy,redundantTable,
                      skel,level,freeRes);
   In(22)=reductionTable:
   [    [    1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
   $B$J$N$G(B, $B:G8e(B $B$N(B 2 $B$,=hM}$5$l$k$O$:$@$,(B,
   In(25)=skel[2]:
   [    [    [    0 , 2 ]  , [    1 , -y^2 ]  ]  ]
   $B$N$h$&$K(B, 0 $BHVL\$H(B, 2 $BHVL\$N(B spair.
   $B$7$+$7(B,
   In(26)=bases:
   [    %[null] , [    0 , 0 , y^2 , 0 , -x , 0 ]  , [    0 , -y , x , 0 , 1 , 0 ]  , [    -y*h , 0 , 0 , x , 0 , 1 ]  , %[null] ]
   $B$N$h$&$K(B, 0 $BHVL\$O(B strategy 3 $B$J$N$G(B, $B$^$@$b$H$^$C$F$$$J$$(B.
   
   reductionTable_tmp=[    2 ]
   See also reductionTable, strategy, level,i
   ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations.
   --- Engine error or interrupt : In function : Error of class PrimitiveObject
   
   Type in Cleards() to exit the debug mode and Where() to see the stack trace.
   In(7)=reductionTable:
   [    [    1 , 1 , 1 , 2 , 2 , 3 ]  , [    3 , 2 , 1 , 2 , 3 ]  , [    2 ]  ]
   In(8)=strategy:
   2
   In(9)=level:
   2
   $B$3$N;~E@$^$G$G$b$H$^$C$?(B basis
    [
      [    2*x*Dx+3*y*Dy+6*h^2 , e_*x^2+e_*y^2 , e_*x*y , 2*y*Dx*h+3*x^2*Dy , e_*y^3 , 3*y^2*Dy*h+6*y*h^3-3*x^3*Dy ]
      [    %[null] , [    0 , 0 , y^2 , 0 , -x , 0 ]  , [    0 , -y , x , 0 , 1 , 0 ]  , [    -y*h , 0 , 0 , x , 0 , 1 ]  , %[null] ]
      [    %[null] ]
    ]
   
   -------------------------------------
   
   Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
   a=Sminimal([x^2+y^2,x*y]);
   $B$3$l$G$b;w$?$h$&$J%(%i!<$r$@$;$k(B.
   $B$3$NJ}$,(B debug $B$7$d$9$$(B:
   Sweyl("x,y",[["x",-1,"y",-1,"Dx",1,"Dy",1]]);
   a=Sminimal([x*y,x^2+y^2]);
   $B$G$O%(%i!<$,$G$J$$$N$,IT;W5D(B.
   pruneZero $B$,F0$$$F$J$$$N$,JQ(B.
   
   rf= [
     [
      [
        [    y^3 , 0 , -x^2 ]
        [    0 , y^2 , -x ]
        [    y , -x , 0 ]
      ]
      [
        [    1 , 0 , -y^2 ]
      ]
       [   ]
     ]
     [
       [    x^2 , x*y , y^3 ]
       [    y^3 , es*y^2 , y ]
       [    1 ]
     ]
     [
       [   ]
      [
       [
         [    0 , 2 ]
         [    y^3 , -x^2 ]
       ]
       [
         [    1 , 2 ]
         [    y^2 , -x ]
       ]
       [
         [    0 , 1 ]
         [    y , -x ]
       ]
      ]
      [
       [
         [    0 , 2 ]
         [    1 , -y^2 ]
       ]
      ]
       [   ]
     ]
     [
       [    x^2+y^2 , x*y , y^3 ]
     ]
    ]
   [    0 , 0 ]
   Processing [    0 , 0 ]    Strategy = 1
   [    0 , 1 ]
   Processing [    0 , 1 ]    Strategy = 1
   [    1 , 2 ]
   Processing [    1 , 2 ]    Strategy = 1
   SpairAndReduction:
   [    p and bases  , [    [    0 , 1 ]  , [    y , -x ]  ]  , [    x^2+y^2 , x*y , %[null] ]  ]
   [    level= , 1 ]
   [    tower2= , [    [   ]  ]  ]
   [    y , -es*x ]
   [gi, gj] = [    x^2+y^2 , x*y ]
   1
   Reduce the element y^3
   by  [    x^2+y^2 , x*y , %[null] ]
   result is [    y^3 , 1 , [    0 , 0 , 0 ]  ]
   vdegree of the original = -3
   vdegree of the remainder = -3
   [    y^3 , [    y , -x , 0 ]  , 2 , 2 , -3 , -3 ]
   [    0 , 2 ]
   Processing [    0 , 2 ]    Strategy = 2
   [    1 , 1 ]
   Processing [    1 , 1 ]    Strategy = 2
   SpairAndReduction:
   [    p and bases  , [    [    1 , 2 ]  , [    y^2 , -x ]  ]  , [    x^2+y^2 , x*y , y^3 ]  ]
   [    level= , 1 ]
   [    tower2= , [    [   ]  ]  ]
   [    es*y^2 , -es^2*x ]
   [gi, gj] = [    x*y , y^3 ]
   1
   Reduce the element 0
   by  [    x^2+y^2 , x*y , y^3 ]
   result is [    0 , 1 , [    0 , 0 , 0 ]  ]
   vdegree of the original = -4
   vdegree of the remainder = %[null]
   [    0 , [    0 , y^2 , -x ]  , 1 , -1 , -4 , %[null] ]
   reductionTable_tmp=[    2 ]
   See also reductionTable, strategy, level,i
   ERROR(sm): error operator : SnextI: bases[i] or bases[j] is null for all combinations.
   --- Engine error or interrupt : In function : Error of class PrimitiveObject
   
   Type in Cleards() to exit the debug mode and Where() to see the stack trace.
   In(10)=reductionTable :
   [    [    1 , 1 , 2 ]  , [    3 , 2 , 1 ]  , [    2 ]  ]
   In(11)=bases:
   [    %[null] , [    0 , y^2 , -x ]  , [    -y , x , 1 ]  ]
   In(12)=  $B$3$l$O(B, [3, 2, 1]  $B$N85$N$&$A(B, 2,1 $B$,$b$H$^$C$F$$$k(B.
   [ 2 ] $B$N7W;;$K(B 0 $BHVL\$,I,MW$G$3$l$,$^$@$J$$(B.
   
   
   
   
Legend:
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changed lines
  Added in v.1.4
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