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Diff for /OpenXM/src/k097/lib/minimal/example-ja.tex between version 1.2 and 1.3

version 1.2, 2000/08/02 05:14:30 version 1.3, 2000/08/09 03:45:27
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 % $OpenXM: OpenXM/src/k097/lib/minimal/example-ja.tex,v 1.1 2000/08/02 03:23:36 takayama Exp $  % $OpenXM: OpenXM/src/k097/lib/minimal/example-ja.tex,v 1.2 2000/08/02 05:14:30 takayama Exp $
 \documentclass[12pt]{jarticle}  \documentclass[12pt]{jarticle}
 \newtheorem{example}{Example}  \newtheorem{example}{Example}
 \def\pd#1{ \partial_{#1} }  \def\pd#1{ \partial_{#1} }
Line 50  complement of an affine variety via D-module computati
Line 50  complement of an affine variety via D-module computati
 Journal of pure and applied algebra, 139 (1999), 201--233. $B$r;2>H(B)  Journal of pure and applied algebra, 139 (1999), 201--233. $B$r;2>H(B)
 \end{enumerate}  \end{enumerate}
   
 \begin{example} \rm  \begin{example} \rm  \label{example:cusp}
 %Prog: minimal-test.k    test18()  %Prog: minimal-test.k    test18()
 $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2} \right) \right]$  $I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2} \right) \right]$
 $B$N>l9g(B.  $B$N>l9g(B.
Line 262  $(-{\bf 1},{\bf 1})$-minimal resolution
Line 262  $(-{\bf 1},{\bf 1})$-minimal resolution
 Degree shifts  Degree shifts
 [    [    0 ]  , [    0 , 2 , 2 , 2 ]  , [    2 , 2 , 2 , 3 , 3 ]  ]  [    [    0 ]  , [    0 , 2 , 2 , 2 ]  , [    2 , 2 , 2 , 3 , 3 ]  ]
 \end{verbatim}}  \end{verbatim}}
 %% $B$3$N(B resolution $B$O<B$O(B, toric $B$N(B resolution $B$N(B Koszul complex $B$K$J$C$F$k(B  %%Prog:test23() of minimal-test.k
 %% $B$O$:(B.  $B$3$N6K>.<+M3J,2r$O<B$O(B $B9TNs(B $(1,2,3)$ $B$G$-$^$k(B affine toric ideal
   $B$N6K>.<+M3J,2r$N(B Koszul complex $B$K$J$C$F$k(B.
   Gel'fand, Kapranov, Zelevinsky $B$K$h$C$FF3F~$5$l$?(B $D/I$
   $B$N(B resolution $B$r<+A3$K1dD9$7$?<!$N(B 2 $B=EJ#BN$r9M$($h$&(B.
   $$
   \begin{array}{ccccccccc}
   0 & \longrightarrow & D^2 & \stackrel{d^1}{\longrightarrow}
                       & D^3 & \stackrel{d^2}{\longrightarrow}
                       & D & \longrightarrow & 0 \\
     &                 & u^1 \downarrow      &
                       & u^2 \downarrow      &
                       & u^3 \downarrow      &   \\
   0 & \longrightarrow & D^2 & \stackrel{d^1}{\longrightarrow}
                       & D^3 & \stackrel{d^2}{\longrightarrow}
                       & D & \longrightarrow & 0
   \end{array}
   $$
   $B$3$3$G$O(B, ($B8m2r$b$J$$$H;W$&$N$G(B) $D$ $B$GF1<!2=%o%$%kBe?t(B,
   $d^i$ $B$G(B affine toric ideal $B$NF1<!2=$NB?9`<04D$G$N6K>.<+M3J,2r(B
   $$ d^2 = \pmatrix{ \pd{1}^2 - \pd{2}^2 h \cr
                      -\pd{1} \pd{2} + \pd{3} h \cr
                      \pd{2}^2 - \pd{1} \pd{3} \cr }, \
      d^1 = \pmatrix{ -\pd{2} & -\pd{1} & -h \cr
                      \pd{3}  & \pd{2}  & \pd{1} \cr }
   $$
   $B$r$"$i$o$9$b$N$H$9$k(B.
   $B$^$?(B $\ell = x_1 \pd{1} + 2 x_2 \pd{2} + 3 x_3 \pd{3}$ $B$H$*$/$H$-(B $u^i$ $B$r(B
   $B<!$N$h$&$K$-$a$k(B.
   $$ u^1=\pmatrix{ \ell + 4 h^2 & 0 \cr
                    0 & \ell+5 h^2 \cr}, \quad
      u^2=\pmatrix{\ell+2 h^2 & 0 & 0 \cr
                   0 & \ell + 3 h^2 & 0 \cr
                   0 & 0 & \ell+ 4 h^2 \cr}, \quad
      u^3 = \pmatrix{ \ell \cr}.
   $$
   
   $B$3$N$H$-IU?o$9$k(B 1 $B=EJ#BN$O(B
   $$L^1 \ni f \mapsto (-d^1(f), u^1(f)) \in L^2 \oplus L^1, $$
   $$  L^2\oplus L^1 \ni (f,g)\mapsto (-d^2(f), u^2(f)+d^1(g)) \in L^3\oplus L^2,
   $$
   $$
     L^3\oplus L^2 \ni (f,g)\mapsto u^3(f)+d^2(g) \in L^3.
   $$
   $B$G$"$?$($i$l$k(B.
   $B$3$3$G(B $L^1 = D^2$, $L^2 = D^3$, $L^3 = D$ $B$G$"$k(B.
   $B$3$N(B 1 $B=EJ#BN$N6qBN7A$O0J2<$N$H$&$j(B.
   \footnotesize{
   \begin{verbatim}
    [
     [
       [    x1*Dx1+2*x2*Dx2+3*x3*Dx3 ]
       [    Dx1^2-Dx2*h ]
       [    -Dx1*Dx2+Dx3*h ]
       [    Dx2^2-Dx1*Dx3 ]
     ]
     [
       [    -Dx1^2+Dx2*h , x1*Dx1+2*x2*Dx2+3*x3*Dx3+2*h^2 , 0 , 0 ]
       [    Dx1*Dx2-Dx3*h , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+3*h^2 , 0 ]
       [    -Dx2^2+Dx1*Dx3 , 0 , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+4*h^2 ]
       [    0 , -Dx2 , -Dx1 , -h ]
       [    0 , Dx3 , Dx2 , Dx1 ]
     ]
     [
       [    Dx2 , Dx1 , h , x1*Dx1+2*x2*Dx2+3*x3*Dx3+4*h^2 , 0 ]
       [    -Dx3 , -Dx2 , -Dx1 , 0 , x1*Dx1+2*x2*Dx2+3*x3*Dx3+5*h^2 ]
     ]
    ]
   \end{verbatim}
   }
 \end{example}  \end{example}
   
   
   
   
 $(-w,w)$-$B6K>.<+M3J,2r$H(B $B6K>.<+M3J,2r$,$3$H$J$kNc$r$5$,$7$F$$$k$,(B  
 $B$3$l$O$^$@8+$D$+$C$F$$$J$$(B.  
   
 \section{$B<BAu(B}  \section{$B<BAu(B}
 $B$3$3$G$O(B  $B$3$3$G$O(B
 \begin{verbatim}  \begin{verbatim}
 /* OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v  /* OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 1.25
    1.23 2000/08/01 08:51:03 takayama Exp  */     2000/08/02 05:14:31 takayama Exp  */
 \end{verbatim}  \end{verbatim}
 $BHG$N(B {\tt minimal.k} $B$K=`5r$7$F<BAu$N35N,$r2r@b$9$k(B.  $BHG$N(B {\tt minimal.k} $B$K=`5r$7$F<BAu$N35N,$r2r@b$9$k(B.
   
 $B$^$@=q$$$F$J$$(B.  $B<BAu$N@bL@$N$?$a$NNc$H$7$F%$%G%"%k(B
   $$ I = D \cdot \{  -2x\pd{x}-3y\pd{y}+h^2,  -3y\pd{x}^2+2x\pd{y}h \} $$
   $B$N(B $(u,v) = (-1,-1,1,1)$-$B6K>.J,2r$N9=@.(B
   $B$r9M$($h$&(B.
   %%Prog: minimal-note-ja.txt  6/9 (Fri) $B$*$h$S0J8e$N(B bug fix $B$N5-O?$r;2>H(B.
   %%$BNc$H$7$F(B, $B%$%G%"%k(B
   %%$$ I = D \cdot \{ x^2 + y^2, x y \} $$
   %%$B$N(B $(u,v) = (-1,-1,1,1)$-$B6K>.J,2r$N9=@.(B
   %%$B$r9M$($h$&(B.
   %%($B$3$N>l9g$OB?9`<04D$NF1<!<0$G@8@.$5$l$k$N$G(B, $BB?9`<04D$G$N(B
   %% $B6K>.<+M3J,2r$N7W;;$HF1$8$3$H$K$J$k(B.)
   $B$3$N>l9g(B,
   $I$ $B$N%0%l%V%J4pDl(B $G$ $B$O(B
   {\footnotesize
   \begin{verbatim}
    [
      [    -2*x*Dx-3*y*Dy+h^2 ]
      [    -3*y*Dx^2+2*x*Dy*h ]
      [    9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
      [    27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
    ]
   \end{verbatim}
   }  \noindent
   $B$H$J$C$F$*$j(B,
   Schreyer resolution $B$O(B
   {\footnotesize
   \begin{verbatim}
     [
      [
        [    -2*x*Dx-3*y*Dy+h^2 ]
        [    -3*y*Dx^2+2*x*Dy*h ]
        [    9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ]
        [    27*y^3*Dy^2+27*y^2*Dy*h^2-3*y*h^4-8*x^3*Dy*h ]
      ]
      [
        [    9*y^2*Dy+3*y*h^2 , 0 , 2*x , 1 ]
        [    -4*x^2*Dy*h , 0 , -3*y*Dy+4*h^2 , Dx ]
        [    2*x*Dy*h , 3*y*Dy-2*h^2 , Dx , 0 ]
        [    3*y*Dx , -2*x , 1 , 0 ]
      ]
      [
        [    -Dx , 1 , 2*x , 3*y*Dy-2*h^2 ]
      ]
     ]
   \end{verbatim}
   }  \noindent
   $B$G$"$k(B.  $1$ $B$,$?$/$5$s(B Schreyer resolution $B$NCf$K$O$"$k$3$H$K(B
   $BCm0U(B. $1$ $B$O6K>.<+M3J,2r$K$OI,MW$J$$85$G$"$k$3$H$r0UL#$9$k(B.
   $B6K>.<+M3J,2r$O(B, $BNc(B \ref{example:cusp} $B$K6qBN7A$r=q$$$F$*$$$?(B.
   
   \medbreak
   
   
   $B$3$N<BAu$G$O6K>.<+M3J,2r$r(B LaScala $B$N%"%k%4%j%:%`$r$b$H$K$7$F(B
   $B9=@.$9$k(B (LaScala and Stillman [??] $B$*$h$S?tM}2J3X$N5-;v(B ??? $B$r;2>H(B).
   
   $B$3$N%"%k%4%j%:%`$O4{CN$H$7$F(B, $B0c$$$N$_$r@bL@$7$h$&(B.
   LaScala $B$N%"%k%4%j%:%`$O(B,
   reduction $B$7$?$H$-$K(B $0$ $B$K$J$C$?>l9g(B, $B$=$N(B reduction $B$KIU?o$7$?(B
   syzygy $B$r(B $B6K>.<+M3J,2r$N85$H$7(B,
   reduction $B$7$?$H$-$K(B $0$ $B$K$J$i$J$+$C$?>l9g(B, $B$=$N85$r(B
   $B%0%l%V%J4pDl$N85$H$7$F2C$((B, $BIU?o$7$?(B syzygy $B$O6K>.<+M3J,2r$G$OM>7W$J$b$N$H(B
   $B$_$J$9(B.
   $B$o$l$o$l$O(B $(u,v)$-$B6K>.$J<+M3J,2r$r$b$H$a$?$$(B.
   $B$=$3$G>e$N<jB3$-$r<!$N$h$&$KJQ$($k(B.
   \begin{center}
   \begin{minipage}{10cm}
   Reduction $B$7$?$H$-$K(B modulo $(u,v)$-$B%U%#%k%?!<$G(B $0$ $B$K$J$C$?>l9g(B,
   $B$=$N(B reduction $B$KIU?o$7$?(B syzygy $B$r(B $B6K>.<+M3J,2r$N85$H$7(B, $B$5$i$K(B
   $B$=$N85$,(B $0$ $B$G$J$1$l$P%0%l%V%J4pDl$K2C$($k(B.
   reduction $B$7$?$H$-$K(B modulo $(u,v)$-$B%U%#%k%?!<$G(B $0$ $B$K$J$i$J$+$C$?>l9g(B,
   $B$=$N85$r%0%l%V%J4pDl$N85$H$7$F2C$((B,
   $BIU?o$7$?(B syzygy $B$O6K>.<+M3J,2r$G$OM>7W$J$b$N$H$_$J$9(B.
   \end{minipage}
   \end{center}
   
   $B$A$J$_$K(B,
   $(-w,w)$-$B6K>.<+M3J,2r$H(B $B6K>.<+M3J,2r$,$3$H$J$kNc$r$5$,$7$F$$$k$,(B
   $B$3$l$O$^$@8+$D$+$C$F$$$J$$(B.
   $B$A$g$C$HIT;W5D$G$"$k(B.
   
   \bigbreak
   
   {\tt minimal.k} $B$N%=!<%9%3!<%I$G$O$3$NItJ,$O<!$N$h$&$K$J$C$F$$$k(B.
   {\footnotesize
   \begin{verbatim}
   def SlaScala(g,opt) {
       ...
       ...
                  f = SpairAndReduction(skel,level,i,freeRes,tower,ww);
                  if (f[0] != Poly("0")) {
                     place = f[3];
   if (Sordinary) {
                     redundantTable[level-1,place] = redundant_seq;
                     redundant_seq++;
   }else{
                     if (f[4] > f[5]) {                       ($B$$(B)
                       /* 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{                                   ($B$m(B)
                       redundantTable[level-1,place] = redundant_seq;
                       redundant_seq++;
                     }
   }
                     redundantTable_ordinary[level-1,place]
                        =redundant_seq_ordinary;
     ...
     ...
   }
   \end{verbatim}
   }
   
   $B>/!9D9$/$J$k$,(B, $B$3$NItJ,$K$"$i$o$l$kJQ?t$N@bL@$r$7$h$&(B.
   
   LaScala $B$N%"%k%4%j%:%`$G$O(B, $B:G=i$K7W;;$9$Y$-(B S-pair $B$N7W;;<j=g(B,
   $B$*$h$S(B Schreyer frame $B$r:n@.$9$k(B.
   Schreyer frame $B$O(B Schreyer resolution $B$N(B initial $B$G$"$k(B.
   $B$3$l$i$O$"$i$+$8$a(B
   {\tt SresolutionFrameWithTower(g,opt);}
   $B$G7W;;$5$l$F(B, {\tt tower} $B$*$h$S(B {\tt skel} $B$K3JG<$5$l$F$$$k(B.
   $B$3$l$i$NJQ?t$NCM$O(B, $B4X?t(B {\tt Sminimal()} $B$NLa$jCM$H$7$F(B
   $B8+$k$3$H$,$G$-$k(B.
   $B4X?t(B {\tt Sminimal()} $B$NLa$jCM$,JQ?t(B $a$ $B$K3JG<$5$l$F$$$k$H$9$k$H(B,
   {\tt a[0]} $B$,6K>.<+M3J,2r(B
   {\tt a[3]} $B$,(B Schreyer $B<+M3J,2r(B($B$H$/$K(B {\tt a[3,0]} $B$,(B
   $I$ $B$N%0%l%V%J4pDl(B),
   {\tt a[4]} $B$,(B,
   $B4X?t(B {\tt SlaScala()} $B$N(B
   $BJQ?t(B {\tt [rf[0], tower, skel, rf[3]]} $B$NCM$G$"$k(B.
   $B$7$?$,$C$F(B,  {\tt tower} $B$O(B {\tt a[4,1]} $B$K3JG<$5$l$F$$$k(B.
   $I$ $B$N>l9g$N(B {\tt tower} $B$O0J2<$N$H$&$j(B.
   {\footnotesize
   \begin{verbatim}
   In(25)=sm1_pmat(a[4,1]);
    [
      [    -2*x*Dx , -3*y*Dx^2 , -9*y^2*Dx*Dy , -27*y^3*Dy^2 ]
      [    -9*y^2*Dy , -3*es^2*y*Dy , -3*es*y*Dy , -3*y*Dx ]
      [    -Dx ]
    ]
   \end{verbatim}
   } \noindent
   $B$3$3$G(B ${\tt es}^i$ $B$O%Y%/%H%k$N(B $BBh(B $i$ $B@.J,$G$"$k$3$H$r$7$a$7$F$$$k(B.
   $B$?$H$($P(B,
   \verb# -3*es^2*y*Dy # $B$O(B
   \verb# [0, 0, -3*y*Dy, 0] # $B$r0UL#$9$k(B.
   
   $BJQ?t(B
   {\tt skel} $B$K$O(B
   S-pair (sp) $B$N7W;;<j=g$,$O$$$C$F$$$k(B.
   $I$ $B$N>l9g$K$O0J2<$N$H$&$j(B.
   {\footnotesize
   \begin{verbatim}
   In(16)=sm1_pmat(a[4,2]);
    [
      [   ]
     [
      [
        [    0 , 2 ]          G'[0] $B$H(B G'[2] $B$N(B sp $B$r7W;;(B (0)
        [    -9*y^2*Dy , 2*x ]
      ]
      [
        [    2 , 3 ]          G'[2] $B$H(B G'[3] $B$N(B sp $B$r7W;;(B (1)
        [    -3*y*Dy , Dx ]
      ]
      [
        [    1 , 2 ]          G'[1] $B$H(B G'[2] $B$N(B sp $B$r7W;;(B (2)
        [    -3*y*Dy , Dx ]
      ]
      [
        [    0 , 1 ]          G'[0] $B$H(B G'[1] $B$N(B sp $B$r7W;;(B (3)
        [    -3*y*Dx , 2*x ]
      ]
     ]
     [
      [
        [    0 , 3 ]          G''[0] $B$H(B G''[3] $B$N(B sp $B$r7W;;(B
        [    -Dx , 3*y*Dy ]
      ]
     ]
      [   ]
    ]
   \end{verbatim}
   }  \noindent
   $B$3$3$G(B $G'$ $B$O(B Schreyer order $B$GF@$i$l$?(B $G$ $B$N(B syzygy $B$N@8@.85(B,
   $G''$ $B$O(B Schreyer order $B$GF@$i$l$?(B $G'$ $B$N(B syzygy $B$N@8@.85$r$"$i$o$9(B.
   $B$?$H$($P>e$NNc$G$O(B,
   $G'[0]$ $B$O(B
   $G[0]$ $B$H(B $G[2]$ $B$N(B sp $B$N7W;;$K$h$jF@$i$l$?(B syzygy,
   $G'[1]$ $B$O(B
   $G[2]$ $B$H(B $G[3]$ $B$N(B sp $B$N7W;;$K$h$jF@$i$l$?(B syzygy,
   ...
   $B$r0UL#$9$k(B.
   
   {\footnotesize
   \begin{verbatim}
        f = SpairAndReduction(skel,level,i,freeRes,tower,ww);
   \end{verbatim}
   } \noindent
   $B$G$O(B {\tt skel[level,i]} $B$K3JG<$5$l$?(B
   S-pair $B$r7W;;$7$F(B, {\tt freeRes[level-1]} $B$G(B reduction $B$r$*$3$J$&(B.
   Reduction $B$N$?$a$N(B Schreyer order $B$O(B \\
   {\tt StowerOf(tower,level-1)} $B$rMQ$$$k(B.
   $B$?$H$($P(B, ${\tt [level,i] = [1,3]}$ $B$N$H$-$K(B
   $B4X?t(B {\tt SpairAndReduction} $B$G(B
   $B$I$N$h$&$J7W;;$,$J$5$l$F$$$k$+(B $I$ $B$N>l9g$K$_$F$_$h$&(B.
   
   {\tt SpairAndReduction} $B$N<B9T;~(B
   $B$K<!$N$h$&$J%a%C%;!<%8$,$G$F$/$k(B.
   {\footnotesize
   \begin{verbatim}
   reductionTable= [
      [    1 , 2 , 3 , 4 ]
      [    3 , 4 , 3 , 2 ]
      [    3 ]
    ]
   [    0 , 0 ]
   Processing [level,i]= [    0 , 0 ]    Strategy = 1
   [    0 , 1 ]
   Processing [level,i]= [    0 , 1 ]    Strategy = 2
   [    1 , 3 ]
   Processing [level,i]= [    1 , 3 ]    Strategy = 2
   SpairAndReduction:
   [    p and bases  , [    [    0 , 1 ]  , [    -3*y*Dx , 2*x ]  ]  ,
      [-2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ]  ]
   [    level= , 1 ]
   [    tower2= , [    [   ]  ]  ]
   [    -3*y*Dx , 2*es*x ]
   [gi, gj] = [    -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h ]
   1
   Reduce the element 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h
   by  [ -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ]
   result is [    9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h , 1 , [    0 , 0 , 0 , 0 ]  ]
   vdegree of the original = 0
   vdegree of the remainder = 0
   [  9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h ,
     [ -3*y*Dx , 2*x , 0 , 0 ]  , 3 , 2 , 0 , 0 ]
   \end{verbatim}
   }  \noindent
   $B:G=i$KI=<($5$l$k(B {\tt reductionTable} $B$N0UL#$O$"$H$G@bL@$9$k(B.
   $B<!$N9T$KCmL\$7$h$&(B.  $B$3$3$G$O(B {\tt skel[0,4]} $B$N(B S-pair
   $B$r7W;;$7$F(Breduction $B$7$F$$$k(B.
   {\footnotesize
   \begin{verbatim}
   SpairAndReduction:
   [    p and bases  , [    [    0 , 1 ]  , [    -3*y*Dx , 2*x ]  ]  ,
      [-2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h , %[null] , %[null] ]  ]
   \end{verbatim}
   }  \noindent
   {\tt [0, 1]} $B$O(B  $G'[0]$ $B$H(B $G'[1]$ $B$N(B sp $B$r7W;;(B
   $B$;$h$H$$$&0UL#$G$"$k(B.
   ${\tt level} = 0$ $B$G4{$K$b$H$^$C$F$$$k(B $B%V%l%V%J4pDl$O(B
   $G[0]$ $B$H(B $G[1]$ $B$N$_$G$"$j(B,
   $B$=$l$i$O$=$l$>$l(B,
   \verb# -2*x*Dx-3*y*Dy+h^2 , -3*y*Dx^2+2*x*Dy*h #
   $B$G$"$k(B.
   {\tt SpairAndReduction} $B$O(B $G[0]$, $G[1]$ $B$N$_$rMQ$$$F(B,
   S-pair  \\
   \verb# 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h #
   $B$r(B reduction $B$9$k(B.
   $B7k6I(B reduction $B$N7k2L$O(B 0 $B$G$O$J$/$F(B, \\
   \verb# 9*y^2*Dx*Dy+3*y*Dx*h^2+4*x^2*Dy*h #
   $B$H$J$k(B.
   LaScala $B$N%"%k%4%j%:%`$N(B 2 $BG\$*F@%7%9%F%`$G(B,
   $B$3$l$,?7$7$$%0%l%V%J4pDl$N85(B {\tt G[place]} $B$H$J$j(B,
   reduction $B$N2aDx$h$j(B syzygy $B$bF@$i$l$k(B.
   
   $B$5$F(B, $(u,v)$-$B6K>.J,2r$r:n$k$K$O(B, reduction $B$7$?M>$j$,(B
   $(u,v)$-$B%U%#%k%?!<$G(B modulo $B$7$F(B $0$ $B$+$I$&$+D4$Y$J$$$H$$$1$J$$(B.
   $B$3$N$?$a(B,
   $B4X?t(B {\tt Sdegree()} $B$rMQ$$$F(B, reduction $B$9$kA0$N85(B, $B$*$h$SM>$j$N(B
   $B%7%U%HIU$-(B $(u,v)$-order $B$r7W;;$9$k(B.
   $B$3$NNc$G$O(B, $BN>J}$H$b(B $0$ $B$G$"$k(B.
   {\footnotesize
   \begin{verbatim}
   vdegree of the original = 0
   vdegree of the remainder = 0
   \end{verbatim}
   }
   $B$7$?$,$C$F(B, modulo $(u,v)$-$B%U%#%k%?!<$G$b(B $0$ $B$G$J$$(B.
   
   $B=`Hw@bL@$,$*$o$C$?(B. $B:G=i$N%W%m%0%i%`(B {\tt SlaScala()} $B$N@bL@$KLa$k(B.
   {\tt SpairAndReduction()} $B$NLa$jCM(B
   {\tt f[0]} $B$K$O(B, reduction $B$7$?M>$j(B,
   {\tt f[4]}, {\tt f[5]} $B$K$O(B,
   reduction $B$9$kA0$N85(B, $B$*$h$SM>$j$N(B
   $B%7%U%HIU$-(B $(u,v)$-order $B$,3JG<$5$l$F$$$k(B.
   $B$3$NNc$N>l9g$K$O(B ($B$m(B) $B$N>l9g$,<B9T$5$l$F(B,
   $BIU?o$7$?(B syzygy $B$O(B $B6K>.<+M3J,2r$K$OITMW$J$b$N$H$7$F(B
   {\tt redundantTable} $B$KEPO?$5$l$k(B:
   {\footnotesize
   \begin{verbatim}
                       redundantTable[level-1,place] = redundant_seq;
   \end{verbatim}
   }  \noindent
   $BM>$j(B {\tt f[0]} $B$O(B, laScala $B$N%"%k%4%j%:%`$N(B 2 $BG\$*F@8=>]$GF@$i$l$?(B,
   $B?7$7$$%V%l%V%J4pDl$N85$G$"$k$,(B, $B$3$l$rJ]B8$9$Y$->l=j$N%$%s%G%C%/%9$O(B,
   $BLa$jCM(B {\tt f[3]}({\tt place}) $B$K3JG<$5$l$F$$$k(B:
   {\footnotesize
   \begin{verbatim}
                     bases[place] = f[0];
                     freeRes[level-1] = bases;
                     reducer[level-1,place] = f[1];
   \end{verbatim}
   } \noindent
   $B$3$N(B reduction $B$GF@$i$l$?(B syzygy ($B$NK\<AE*ItJ,(B)$B$O(B,
   $BJQ?t(B {\tt reducer} $B$KEPO?$5$l$k(B.
   $B0J>e$G(B $(u,v)$-$B6K>.<+M3J,2rFCM-$N=hM}$NItJ,$N2r@b$r=*$($k(B.
   
   
   \bigbreak
   $B0J2<$G$O(B, LaScala $B$N%"%k%4%j%:%`$N$o$l$o$l$N<BAu$N35N,$HLdBjE@$r(B
   $B=R$Y$k(B.
   
   $B$^$:(B, $BJQ?t(B
   {\tt reductionTable} $B$N0UL#$r@bL@$7$h$&(B.
   LaScala $B$N%"%k%4%j%:%`$G$O(B,
   {\tt level - Sdegree(s)}
   $B$N>.$5$$(B S-pair $B$+$i7W;;$7$F$$$/(B.
   $B4X?t(B {\tt Sdegree} $B$O<!$N$h$&$K:F5"E*$KDj5A$5$l$F$$$k(B.
   {\footnotesize
   \begin{verbatim}
   /* f is assumed to be a monomial with toes. */
   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));
   }
   \end{verbatim}
   }  \noindent
   $B$3$3$G(B {\tt StotalDegree(f)} $B$O(B $f$ $B$NA4<!?t$G$"$k(B.
   
   \noindent
   $B$5$F(B, LaScala $B$N%"%k%4%j%:%`$G$O(B,
   Resolution $B$r2<$+$i=gHV$K7W;;$7$F$$$/$N$G$O$J$$(B.
   $B$3$l$,K\<AE*$JE@$G$"$k(B.
   $B$3$N=gHV$OJQ?t(B {\tt reductionTable} $B$K$O$C$F$$$k(B.
   $I$ $B$NNc$G$O(B
   {\footnotesize
   \begin{verbatim}
   reductionTable= [
      [    1 , 2 , 3 , 4 ]
      [    3 , 4 , 3 , 2 ]   skel[0] $B$KBP1~(B
      [    3 ]               skel[1] $B$KBP1~(B
    ]
   \end{verbatim}
   }  \noindent
   $B$H$J$k(B.
   
   $B8=:_$N<BAu$G$N7W;;B.EY(B, $B%a%b%j;HMQNL$N%\%H%k%M%C%/$r(B
   $B;XE&$7$F$*$/(B.
   LaScala $B$N%"%k%4%j%:%`$G$O(B, Schreyer Frame $B$r9=@.$7$F$+$i(B,
   $B6K>.<+M3J,2r$r9=@.$9$k(B.
   $B2<5-$N%W%m%0%i%`$NJQ?t(B {\tt redundantTable[level,q]} $B$K$O(B,
   $BBP1~$9$k(B syzygy $B$H(B $B%0%l%V%J4pDl$N85$,2?2sL\$N(B reduction $B$G@8@.(B
   $B$5$l$?$+$N?t$,$O$$$C$F$$$k(B.
   $B6K>.<+M3J,2r$N9=@.$G$O(B, $B:G8e$N(B reduction $B$N(B syzygy $B$+$i;O$a$F(B,
   Schreyer resolution $B$+$i6K>.<+M3J,2r$K$H$C$FM>J,$J85$r<h$j=|$$$F(B
   $B$$$/(B
   ({\tt seq} $B$r(B $1$ $B$E$D8:$i$7$F$$$/(B).
   {\footnotesize
   \begin{verbatim}
   def Sminimal(g,opt) {
   
     ....
   
     while (seq > 1) {
       seq--;
       for (level = 0; level < maxLevel; level++) {
         betti = Length(freeRes[level]);
         for (q = 0; q<betti; q++) {
           if (redundantTable[level,q] == seq) {
             Print("[seq,level,q]="); Println([seq,level,q]);
             if (level < maxLevel-1) {
               bases = freeRes[level+1];
               dr = reducer[level,q];
               dr[q] = -1;
               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;
               }
               ....
             }
             ....
           }
        }
      }
     }
     ....
   }
   \end{verbatim}
   } \noindent
   $BLdBj$O(B,
   $B6K>.<+M3J,2r<+BN$O$A$$$5$/$F$b(B, Schreyer Frame $B$,5pBg(B ($10000$ $BDxEY$N(B
   betti $B?t(B) $B$H$J$k$3$H$bB?$$>l9g$,$"$k$3$H$G$"$k(B.
   $B2<$NJQ?t(B {\tt bases} $B$K(B, Schreyer resolution $B$N(B {\tt level} $B<!$N(B
   syzygy $B$r$$$l$F$$$k(B. Schreyer Frame $B$K(B $10000$ $BDxEY$N(B betti
   $B?t$,$"$i$o$l$k$H$3$NJQ?t$O(B $B%5%$%:(B $10000$ $BDxEY$NG[Ns$H$J$k(B.
   $B$5$i$K(B, Schreyer $BJ,2r$+$i6K>.<+M3J,2r$N$?$a$KITMW$J85$r$H$j$N$>$$$?(B
   $BJ,2r$r:n$k$?$a$K(B\\
   \verb#              newbases[i] = bases[i] + bases[i,q]*dr;   # \\
   $B$J$k>C5n$r$*$3$J$$(B, $0$ $B$GKd$a$i$l$?Ns$^$?$O(B $0$ $B$GKd$a$i$l$?9T$r@8@.$7$F$$$k(B.
   $B$3$NItJ,$,(B, $B%a%b%j$N;HMQ$r05Gw$7$F$*$j(B, $B7W;;;~4V$b$D$+$C$F$$$k(B.
   
   
   
 \end{document}  \end{document}
   
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