% $OpenXM: OpenXM/src/k097/lib/minimal/example-ja.tex,v 1.2 2000/08/02 05:14:30 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
%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}}
%% $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
%% $B$O$:(B.
\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}
$B$3$3$G$O(B
\begin{verbatim}
/* OpenXM: OpenXM/src/k097/lib/minimal/minimal.k,v 
   1.23 2000/08/01 08:51:03 takayama Exp  */
\end{verbatim}
$BHG$N(B {\tt minimal.k} $B$K=`5r$7$F<BAu$N35N,$r2r@b$9$k(B.

$B$^$@=q$$$F$J$$(B.
	
\end{document}