% $OpenXM: OpenXM/src/k097/lib/minimal/example-ja.tex,v 1.3 2000/08/09 03:45:27 takayama Exp $
\documentclass[12pt]{jarticle}
\newtheorem{example}{Example}
\def\pd#1{ \partial_{#1} }
%% [2] should be replaced by \cite{....}

\begin{document}
\section{$BNc(B}
$B2f!9$,(B $(u,v)$-$B6K>.<+M3J,2r$N9=@.$K6=L#$r$b$C$?F05!$N(B
$B0l$D$O(B, $D$ $B2C72(B $M$ $B$N@)8B%3%[%b%m%8$N7W;;$N8zN(2=$G$"$k(B.
[2] $B$G$O(B, $M$ $B$N(B Schreyer resolution $B$,(B $(-{\bf 1},{\bf 1})$
$B$KE,9g$7$?(B $M$ $B$N<+M3J,2r$G$"$k$3$H$r>ZL@$7(B,
$B$3$l$rMQ$$$?@)8B%3%[%b%m%8$N7W;;K!$rM?$($?(B.
$B$3$NJ}K!$rE,MQ$9$k$K$O(B
$(-w,w)$ $B$KE,9g$7$?(B $M$ $B$N<+M3J,2r$J$i$J$s$G$b$h$/(B,
Schreyer resolution $B$r$H$kI,A3@-$O$J$$(B.
[2] $B$NJ}K!$G$O(B, 1 $BE@$X$N@)8B$r7W;;$9$k$N$K<!85(B
$$O\left( (\mbox{ $B<+M3J,2r$N(B betti $B?t(B}) \times 
  \left(\mbox{$b$$B4X?t$N:GBg@0?t:,(B}\right)^n\right)$$
$B$N%Y%/%H%k6u4V$NJ#BN$N(B ${\rm Ker}/{\rm Im}$ $B$r(B
$B7W;;$9$kI,MW$,@8$8$k(B.
( $BItJ,B?MMBN$X$N@)8B$K$O(B $D_m$, $(m < n)$ $B<+M32C72$NJ#BN(B
$B$N%3%[%b%m%8$r7W;;$9$kI,MW$,$"$k(B.)
$B$7$?$,$C$F(B, betti $B?t$,Bg$-$/$J$k$H(B, $B7W;;$9$Y$-(B $B%Y%/%H%k6u4V$N<!85$,(B
$BBg$-$/$J$j(B, $B%a%b%jITB-$r$^$M$$$F$$$?(B.
$(-w,w)$-$B6K>.<+M3J,2r$N(B betti $B?t$O(B Schreyer resolution $B$N(B betti $B?t$K(B
$BHf3S$7$F$+$J$j>.$5$/$$$^$^$G@)8B$,7W;;$G$-$J$+$C$?Nc$b7W;;$G$-$k$h$&$K(B
$B$J$C$?(B.

Schreyer resolution $B$N(B betti $B?t$H(B $(-w,w)$-$B6K>.<+M3J,2r$N(B betti $B?t$r(B
$B$$$/$D$+$NNc$K$D$$$FHf3S$7$F$_$h$&(B.

$BHf3S$NA0$K$$$/$D$+5-9f$HM=HwCN<1$rF3F~$9$k(B.
\begin{enumerate}
\item Schreyer resolution $B$N(B betti $B?t$O(B $(-w,w)$ $B$@$1$G$J$/(B
tie-breaking order $B$K$b0MB8$9$k(B.
$B0J2<(B tie-breaking order $B$H$7$F(B, graded reverse lexicographic order
$B$rMQ$$$k(B.
\item $BB?9`<0(B $f$ $B$N(B $b$-$B4X?t$N:G>.@0?t:,$r(B $-r$ $B$H$9$k$H$-(B
${\rm Ann}(D f^{-1})$ $B$G(B
$1/f^r$ $B$rNm2=$9$k(B $D$ $B$N%$%G%"%k$N$"$k@8@.85$N=89g$r$"$i$o$9(B.
$B2<$N<BNc$N>l9g$G$O4X?t(B {\tt Sannfs(f,v)} $B$N=PNO$r$"$i$o$9(B.
\item $F(G)$ $B$G(B $G$ $B$N(B formal Laplace $BJQ49$r$"$i$o$9(B.
\item $F^h(G)$ $B$G(B $G$ $B$N(B formal Laplace $BJQ49$r(B homogenize $B$7$?$b$N$r$"$i$o$9(B.
\item Grothendieck $B$NHf3SDjM}$K$h$l$P(B 
$I = F({\rm Ann} D f^{-1})$ $B$H$*$/$H$-(B $D/I$ $B$N86E@$X$N@)8B%3%[%b%m%8(B
$B$,6u4V(B $ {\bf C}^n \setminus V(f)$ $B$N(B ${\bf C}$-$B78?t%3%[%b%m%872$K0lCW$9$k(B.
( "An algorithm for de Rham cohomology groups of the
complement of an affine variety via D-module computation", 
Journal of pure and applied algebra, 139 (1999), 201--233. $B$r;2>H(B)
\end{enumerate}

\begin{example} \rm  \label{example:cusp}
%Prog: minimal-test.k    test18()
$I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2} \right) \right]$
$B$N>l9g(B.
$B%$%G%"%k(B $I$ $B$O(B     
$$ -2x\pd{x}-3y\pd{y}+h^2 ,  -3y\pd{x}^2+2x\pd{y}h $$
$B$G@8@.$5$l$k(B.

\begin{tabular}{|l|l|}
\hline
Resolution type &  Betti numbers          \\ \hline
Schreyer &                        1, 4, 4, 1    \\ \hline
$(-{\bf 1},{\bf 1})$-minimal &    1, 2, 1 \\ \hline
minimal &                         1, 2, 1    \\
\hline
\end{tabular}

\noindent
$(-{\bf 1},{\bf 1})$-minimal resolution
{\footnotesize \begin{verbatim}
 [ 
  [ 
    [    -2*x*Dx-3*y*Dy+h^2 ] 
    [    -3*y*Dx^2+2*x*Dy*h ] 
  ]
  [ 
    [    -3*y*Dx^2+2*x*Dy*h , 2*x*Dx+3*y*Dy ] 
  ]
 ]
Degree shifts 
[    [    0 ]  , [    0 , 1 ]  ] 
\end{verbatim}}
Schreyer Resolution  %%Prog: a=test18();  sm1_pmat(a[3]);
{\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}}
\end{example}

\begin{example} \rm
%Prog: minimal-test.k    test17b()
$I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2z^2} \right) \right]$
$B$N>l9g(B.

\begin{tabular}{|l|l|}
\hline
Resolution type &  Betti numbers          \\ \hline
Schreyer &                        1, 8, 16, 11, 2    \\ \hline
$(-{\bf 1},{\bf 1})$-minimal &    1, 4, 5, 2 \\ \hline
minimal &                         1, 4, 5, 2    \\
\hline
\end{tabular}

\noindent
$(-{\bf 1},{\bf 1})$-minimal resolution
{\footnotesize \begin{verbatim}
 [ 
  [ 
    [    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 ] 
  ]
  [ 
    [    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 ] 
    [    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 ] 
    [    2*x*Dy*Dz , 0 , z , -y ] 
  ]
  [ 
    [    y , -2*x*Dy*Dz , 3*y*z , z , 2*x*Dx ] 
    [    Dz , -3*Dx^2*h , 2*x*Dx+3*y*Dy+3*z*Dz+6*h^2 , Dy , -3*Dy*Dz ] 
  ]
 ]
Degree shifts 
[    [    0 ]  , [    0 , 0 , 2 , 2 ]  , [    2 , 0 , 3 , 2 , 1 ]  ] 
\end{verbatim}}
\end{example}

\begin{example} \rm
%Prog: minimal-test.k    test22();
$I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3+y^3+z^3} \right) \right]$
$B$N>l9g(B.

%% Uli Walther $B$N(B $BO@J8(B (MEGA 2000) $B$NNc$H(B betti $B?t$rHf3S$;$h(B.
\begin{tabular}{|l|l|}
\hline
Resolution type &  Betti numbers          \\ \hline
Schreyer &    1, 12, 44, 75, 70, 39, 13, 2     \\ \hline
$(-1,-2,-3,1,2,3)$-minimal & 1, 4, 5, 2  \\ \hline
minimal & 1, 4, 5, 2                      \\
\hline
\end{tabular}

\noindent
$(-1,-2,-3,1,2,3)$-minimal resolution
{\footnotesize \begin{verbatim}
 [
  [
    [    x*Dx+y*Dy+z*Dz-3*h^2 ]
    [    y*Dz^2-z*Dy^2 ]
    [    x*Dz^2-z*Dx^2 ]
    [    x*Dy^2-y*Dx^2 ]
  ]
  [
    [    0 , -x , y , -z ]
    [    -x*Dz^2+z*Dx^2 , x*Dy , x*Dx+z*Dz-3*h^2 , z*Dy ]
    [    -x*Dy^2+y*Dx^2 , -x*Dz , y*Dz , x*Dx+y*Dy-3*h^2 ]
    [    -y*Dz^2+z*Dy^2 , x*Dx+y*Dy+z*Dz-2*h^2 , 0 , 0 ]
    [    0 , Dx^2 , -Dy^2 , Dz^2 ]
  ]
  [
    [    -x*Dx+3*h^2 , y , -z , -x , 0 ]
    [    -Dz^3-Dy^3 , -Dy^2 , Dz^2 , Dx^2 , -x*Dx-y*Dy-z*Dz ]
  ]
 ]
Degree shifts
[    [    0 ]  , [    0 , 4 , 5 , 3 ]  , [    3 , 5 , 6 , 4 , 9 ]  ]
\end{verbatim}}
\end{example}


\begin{example} \rm
%Prog: minimal-test.k    test21();
$I = F^h\left[{\rm Ann}\left( D \frac{1}{x^3-y^2z^2+y^2+z^2} \right) \right]$
$B$N>l9g(B.

\begin{tabular}{|l|l|}
\hline
Resolution type &  Betti numbers          \\ \hline
Schreyer        & 1, 13, 43, 50, 21, 2                        \\ \hline
$(-{\bf 1},{\bf 1})$-minimal &  1, 7, 10, 4   \\ \hline
minimal &  1, 7, 10, 4                        \\
\hline
\end{tabular}

\noindent
$f=x^3-y^2z^2+y^2+z^2$ $B$H$*$$$?>l9g(B,
$B6u4V(B ${\bf C}^3 \setminus V(f)$ $B$N(B
$B%3%[%b%m%872$N<!85$O(B 
${\rm dim}\, H^i = 1$, $(i=0, 1)$,
${\rm dim}\, H^i = 0$, $(i=2, 3)$,
$B$H$J$k(B.
$B$3$N>l9g(B $D/I$ $B$N(B
$b$-$B4X?t$N:GBg@0?t:,$O(B $2$ $B$H$J$j(B,
$B%3%[%b%m%8$r7W;;$9$k$?$a$K(B
$B9M$($k@~7A6u4V$NJ#BN$N<!85$O(B, $10, 12, 9, 4$  $B$G$"$k(B. %%Prog: Srestall.sm1
$B0lJ}(B Schreyer resolution $B$+$i%9%?!<%H$7$F(B,
$B@~7A6u4V$NJ#BN$r9M$($k$H(B, $B$=$N<!85$O(B
130, 1078, 1667, 749, 40 $B$H$J$k(B. %%Prog: test21b()
\end{example}

\begin{example} \rm
%Prog: minimal-test.k    test20()
$I = D\cdot\{  x_1\pd{1}+2x_2\pd{2}+3x_3\pd{3} ,
    \pd{1}^2-\pd{2}h,
    -\pd{1}\pd{2}+\pd{3}h,
    \pd{2}^2-\pd{1}\pd{3} \}
$ $B$N>l9g(B.
$B$3$l$O(B $A=(1,2,3)$, $\beta=0$ $B$KIU?o$9$k(B GKZ $BD64v2?7O$N(B
homogenization.

\begin{tabular}{|l|l|}
\hline
Resolution type &  Betti numbers          \\ \hline
Schreyer &                  1, 10, 25, 23, 8, 1    \\ \hline
$(-{\bf 1},{\bf 1})$-minimal &    1, 4, 5, 2  \\ \hline
minimal &                         1, 4, 5,  2    \\
\hline
\end{tabular}

\noindent
$(-{\bf 1},{\bf 1})$-minimal resolution
{\footnotesize \begin{verbatim}
 [ 
  [ 
    [    x1*Dx1+2*x2*Dx2+3*x3*Dx3 ] 
    [    Dx1^2-Dx2*h ] 
    [    -Dx1*Dx2+Dx3*h ] 
    [    Dx2^2-Dx1*Dx3 ] 
  ]
  [ 
    [    Dx1*Dx2-Dx3*h , -x1*Dx2 , 2*x2*Dx2+3*x3*Dx3+3*h^2 , -x1*h ] 
    [    Dx1^2-Dx2*h , -x1*Dx1-3*x3*Dx3-2*h^2 , 2*x2*Dx1 , 2*x2*h ] 
    [    Dx2^2-Dx1*Dx3 , x1*Dx3 , x1*Dx2 , -2*x2*Dx2-3*x3*Dx3-4*h^2 ] 
    [    0 , Dx3 , Dx2 , Dx1 ] 
    [    0 , -Dx2 , -Dx1 , -h ] 
  ]
  [ 
    [    Dx2 , -Dx3 , -Dx1 , -2*x2*Dx2-3*x3*Dx3-4*h^2 , -x1*Dx2-2*x2*Dx3 ] 
    [    -Dx1 , Dx2 , h , -x1*h , -3*x3*Dx3-h^2 ] 
  ]
 ]
Degree shifts 
[    [    0 ]  , [    0 , 2 , 2 , 2 ]  , [    2 , 2 , 2 , 3 , 3 ]  ] 
\end{verbatim}}
%%Prog:test23() of minimal-test.k
$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}




\section{$B<BAu(B}
$B$3$3$G$O(B
\begin{verbatim}
/* OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 1.25 
   2000/08/02 05:14:31 takayama Exp  */
\end{verbatim}
$BHG$N(B {\tt minimal.k} $B$K=`5r$7$F<BAu$N35N,$r2r@b$9$k(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}