\documentstyle{jarticle}
\title{$B7W;;Be?tF~Lg(B I}
\author{$B9b;3?.5#(B}
\date{1999$BG/(B10$B7n(B19$BF|(B, ``$BBe?t3X$H7W;;(B'' $B$K$F(B}
\begin{document}
\maketitle

$BK\F|$O7W;;Be?tF~Lg$H$$$&%?%$%H%k$G$*OC$5$;$F$$$?$@$/$N$G$9$,(B,
$B$^$:<+J,$O(B ''$B7W;;Be?t(B'' $B$r$I$&$$$C$?;X?K$G$d$C$F$$$k$N$+(B
3 $B$D$"$2$F$*$-$?$$$H;W$$$^$9(B.
\begin{enumerate}
\item $B?t3X$K$*$$$F(B
      $B7W;;$G$-$k$b$N$O$J$K$+$O$C$-$j$5$;$k(B. $B?t3X$NK\$dO@J8$r$_$?$H$-(B
      $B$3$l$O7W;;5!$G7W;;$G$-$k(B, $B$3$l$OB?J,$G$-$J$$(B, $B$G$-$k$H$7$F$b(B,
      $B;~4V$,$+$+$k(B, $B$H$$$&$3$H$rEz$($i$l$k$h$&$K$9$k(B.
\item $B%"%k%4%j%:%`(B, $B%7%9%F%`$N3+H/$r%P%i%s%9NI$/H/E8$5$;$k(B.
      $B@$3&$N$$$m$$$m$J8&5f%0%k!<%W$r$_$k$K(B, $B$3$N%P%i%s%9$,$H$l$F$$$k(B
      $B$H$3$m$G$h$$;E;v$,$*$3$J$o$l$F$$$k$3$H$,B?$$$h$&$K8+<u$1$i$l$^$9(B.
\item $B8zN(E*$K7W;;$9$kJ}K!$r9M$($k(B. 
\end{enumerate}
$B$?$H$($P(B $BKYEDNIG7;a$NL>Cx(B ``$BBe?tF~Lg(B'' $B$r3+$$$F$_$^$7$g$&(B.
$B$3$NK\$NFbMF$K$D$$$F(B, $B$J$K$,7W;;$G$-$F(B, $B$J$K$,7W;;$G$-$J$$$+$,(B
$B$"$kDxEY$O$C$-$j$7$F$-$F(B, $B4XO"$9$k%7%9%F%`$,3+H/$5$l$?$N$O(B
$BHf3SE*:G6a$N$3$H$G$9(B.
1 $B$O7W;;$NN)>l$+$i?t3XA4BN$r8+D>$9$H$$$&$3$H$G$"$j(B,
$B>CLW@o$G$O$"$j$^$9$,(B, $B$d$C$F$_$k$H$-$o$a$F3Z$7$$$7(B, $B?7$7$$?t3X$NE83+(B,
$B?7$7$$1~MQ?t3X$X$NE83+$b$3$N$h$&$JCOF;$J>CLW@o$+$i$&$^$l$F(B
$B$/$k$N$G$O$J$$$+$H;W$C$F$$$^$9(B.
$B$^$?(B, $B7W;;5!$,$$$m$$$m$J$b$N$r9=@.$9$k$N$r7P83$9$k$N$O(B
$B$o$1$b$J$/BgJQ6=J3$7$^$9(B.

$B$5$F(B, $B$3$&$$$C$?OC$7$O$3$N$/$i$$$K$7$F$*$$$F(B, $B<B:](B
$B?t3XE*$J:`NA$r$D$+$C$F(B, $B<+J,$,>e$N$h$&$J;X?K$r$b$D$h$&$K$J$C$?Lu$r(B
$B$9$3$7$G$b$D$?$($k$3$H$,$G$-$?$i$H;W$$$^$9(B.
$B<+J,$,??7u$K9M$($F$-$?$3$H$r$*OC$9$k$H$J$k$H(B, $B$I$&$7$F$b(B
$B%0%l%V%J4pDl$NOCBj$K$J$C$F$7$^$$$^$9(B.
$B%0%l%V%J4pDl$O69$$0UL#$N7W;;Be?t(B($B$D$^$j7W;;4DO@$H$G$b$$$&$Y$-$b$N(B)
$B$NCf?4$G$"$k;;K!$G$"$j(B, $BF~Lg$H$7$F$3$NOCBj$r$H$j$"$2$k$N$O(B
$BE,@Z$@$H$*$b$o$l$^$9(B.


\bigbreak
${\bf N} := \{ 0, 1, 2, \ldots \}$,
${\bf k}$ $B$rBN(B,
${\bf k} [x] := {\bf k}[x_1, \ldots, x_n]$
$B$r(B $n$ $BJQ?t$NB?9`<04D$H$7$^$7$g$&(B.
$a x^\alpha := a x_1^{\alpha_1} \cdots x_n^{\alpha_n}$,
$a \in {\bf k}$, $ \alpha \in {\bf N}^n$
$B$r%b%N%_%"%k(B(monomial)$B$H$h$S$^$9(B.
$w \in {\bf R}^n$ $B$KBP$7$F(B,
$$  w \cdot \alpha = \sum_{i=1}^n w_i \alpha_i $$
$B$r(B, $B%b%N%_%"%k(B $a x^\alpha$ $B$N(B
$w$-weight $B$H$h$S$^$9(B.

$B%0%l%V%J4pDl$NM}O@$G0lHVBg@Z$J35G0$rF3F~$7$^$7$g$&(B.
$f$ $B$rB?9`<0$H$9$k$H$-(B,
$${\rm in}_w(f) = \mbox{ $f$ $B$N$J$+$N%b%N%_%"%k$G:G9b<!$N(B $w$-weight $B$r(B
                        $B$b$D$b$N$NOB(B}
$$
$B$H$*$-$^$9(B.  
$B$3$NB?9`<0$r(B, $f$ $B$N(B $w$ $B$K4X$9$k(B, $B%$%K%7%"%kB?9`<0$H$$$$$^$9(B.

$BNc(B:
  $w = (10,0)$,
  $f = x_1^2 + x_2^2 -1 $ $B$H$*$/$H(B,
  ${\rm in}_w(f) = x_1^2$ $B$G$9(B.

$B$5$F$3$l$H4XO"$7$?35G0$H$7$F%$%K%7%c%k%b%N%_%"%k$rF3F~$7$^$7$g$&(B.
$\prec$ $B$r(B $x^\alpha$ $BC#$N4V$N(B term order $B$H$7$^$7$g$&(B.
term order $B$H$O<!$N>r7o$r$_$?$9A4=g=x$G$9(B.
\begin{enumerate}
\item $1 \preceq x^\alpha$
\item $x^\alpha \prec x^\beta$ $B$J$i$P(B 
$x^{\alpha+\gamma} \prec x^{\beta+\gamma}$
\end{enumerate}
weight $w$ $B$*$h$S(B term order $\prec$ $B$rMQ$$$F<!$N$h$&$K?7$7$$=g=x$r(B
$BF3F~$7$^$9(B.
\begin{eqnarray*}
& & x^\alpha \prec_w x^\beta \\
&\Leftrightarrow&
  w \cdot \alpha < w \cdot \beta \ \mbox{or }
  (w \cdot \alpha = w \cdot \beta) \ \mbox{ and } x^\alpha \prec x^\beta
\end{eqnarray*}
$B$3$N=g=x$O(B $w$ $B$,$-$a$kItJ,=g=x$r(B $\prec$
$B$G$h$j$3$^$+$/$7$?$b$N$G$"$j$^$9(B.
$B$5$F$3$N=g=x$rMQ$$$F(B, $BB?9`<0(B $f$ $B$KBP$7$F(B,
$${\rm in}_{\prec_w}(f) = \mbox{ $f$ $B$N$J$+$N%b%N%_%"%k$G(B
$B=g=x(B $\prec_w$ $B$G:GBg$N%b%N%_%"%k(B}
$$
$B$H$*$-$^$9(B.  
$B$3$l$O$+$J$i$:%b%N%_%"%k$H$J$j$^$9(B.

$I \subset {\bf k}[x]$ $B$r%$%G%"%k$H$7$^$7$g$&(B.
\begin{eqnarray*}
{\rm in}_{\prec_w}(I) &=& \langle {\rm in}_{\prec_w}(f)\,|\, f \in I \rangle \\
{\rm in}_{w}(I) &=& \langle {\rm in}_{w}(f)\,|\, f \in I \rangle 
\end{eqnarray*} 
$B$H?7$7$$%$%G%"%k$rDj5A$7$^$9(B.
${\rm in}_{\prec_w}(I)$ $B$O%b%N%_%"%k$G@8@.$5$l$k%$%G%"%k(B,
$B$9$J$o$A%b%N%_%"%k%$%G%"%k$H$J$j$^$9(B.

$BNc(B:
$n=1$, $I = \langle g \rangle$, $g = 1-x^2$.
$w>0$ $B$J$i(B ${\rm in}_w(I) = \langle x^2 \rangle$,
$w<0$ $B$J$i(B ${\rm in}_w(I) = \langle 1 \rangle$
$B$H$J$j$^$9(B.

$BNc(B:
$n=2$, $I = \langle x^2+y^2-4, xy-1 \rangle$,
$w = (100,1)$ $B$N$H$-(B,
${\rm in}_w(I) = \langle y^4, x \rangle$
$B$H$J$j$^$9(B.

\setlength{\unitlength}{1cm}
\begin{picture}(7,7)(-1,-1)
\put(1,0){\vector(1,0){4}}
\put(0,1){\vector(0,1){4}}
\put(-1,0){\line(1,0){2}}
\put(0,-1){\line(0,1){2}}
\put(1,0){\line(0,1){5}}
\put(0,4){\line(1,0){5}}
\put(1,-0.5){$x$}
\put(-0.5,4){$y^4$}
\end{picture}


$B%b%N%_%"%k%$%G%"%k$O(B
$B?^$N$h$&$K?^<($9$k$3$H$b$G$-$k(B.
$B%b%N%_%"%k%$%G%"%k$OAH9g$;O@E*$H4DO@$,8r:x$9$kIqBf$G$"$j(B,
$B%9%?%s%l!<$N8&5f$,M-L>$G$9$,(B, 
$B:G6a$O%0%l%V%J4pDl$N;kE@$,2C$o$j(B
$BBgJQ3hH/$K8&5f$,$*$3$J$o$l$F$$$^$9(B.
$B$?$H$($P(B, $B%b%N%_%"%k%$%G%"%k$N<+M3J,2r$r(B,
$B?^$N$h$&$J%b%N%_%"%k$N?^7A$r$b$H$KM}2r$9$k(B
Bayer, Sturmfels $B$N(B Scarf complex $B$NM}O@$J$I$O(B,
$B:G6a$N=EMW$J7k2L$G$7$g$&(B.
$B$^$?F|K\$G$O(B, $B:eBg$NF|Hf$5$s$dBg?y7/$,(B,
$B%$%K%7%c%k%$%G%"%k$H%b%N%_%"%k$NAH9g$;O@$H$$$&N)>l$G(B
$B8&5f$r$9$9$a$F$$$^$9(B.


$B$5$F(B, $\prec_w$ $B$K4X$9$k%$%G%"%k(B $I$ $B$N%0%l%V%J4pDl(B $G$ $B$H$O(B,
$B<!$N>r7o$r$_$?$9B?9`<0$NM-8B=89g$G$9(B.
\begin{enumerate}
\item $\langle G \rangle = I$,
\item $\langle {\rm in}_{\prec_w}(G) \rangle = {\rm in}_{\prec_w}(I)$.
\end{enumerate}

$B%0%l%V%J4pDl$,$b$H$^$l$P(B,
$B%$%K%7%c%k%$%G%"%k$,7W;;$G$-$^$9(B.

$BDjM}(B:
$\prec_w$ $B$K4X$9$k%$%G%"%k(B $I$ $B$N%0%l%V%J4pDl$r(B $G$ $B$H$*$/$H(B,
$${\rm in}_{w}(I) = \langle {\rm in}_w(G) \rangle. $$


\medbreak
$B%0%l%V%J4pDl$NNc$r$"$2$F$*$-$^$7$g$&(B.
$B:#8e(B, $\prec$ $B$r(B lexicographic order $B$H$7$^$9(B
($B$D$^$j(B, $B$^$:(B $x_1$ $B$N6R$r$/$i$Y(B, $B>!Ii$,$D$J$J$$$J$i(B,
$x_2$ $B$N6R$r$/$i$Y$k(B, $B>!Ii$,$D$+$J$$$J$i(B, ....
$B$H$9$k=g=x(B.)

$$I = \langle x^2 + y^2 - 4, xy -1 \rangle, $$
$ w = (100,0)$ $B$H$9$k$H(B,
$$ G = \{ \underline{-y^4}+4y-1, \underline{x}+y^3-4y \} $$
$B$,(B $\prec_w$ $B$K4X$9$k%0%l%V%J4pDl$H$J$j$^$9(B.
$B%0%l%V%J4pDl$O(B, weight $w$ $B$rJQ2=$5$;$k$H(B, $BJQ2=$7$^$9(B.
$B$?$H$($P(B,
$w = (1,1)$ $B$N$H$-$O(B,
$$ G = \{ \underline{xy}-1, \underline{x^2}+y^2-4,
          \underline{y^3}+x-4y \} $$
$B$H$J$j$^$9(B.

$B%0%l%V%J4pDl$r7W;;$9$k%"%k%4%j%:%`$r(B Buchberger $B%"%k%4%j%:%`$H$$$$$^$9(B.
$w \geq 0$ ($B$D$^$j(B $w_i \geq 0$) $B$N$H$-$O(B,
Buchberger $B$,$=$N7W;;K!$rM?$($^$7$?(B.
$B4pK\E*$K%f!<%/%j%C%I$N8_=|K!$N3HD%$G$9(B.
$B8_=|K!$G$O(B, Knuth $B$N4pK\;;K!$G$*$J$8$_$N$h$&$K(B
($BF|K\8lLu(B, 4$B4,(B, p.251,
$ u(x) = x^8 + x^6 - 3 x^4 - 3 x^3 + 8 x^2 + 2 x -5$,
$ v(x) = 3 x^6+5 x^4-4x^2-9x+21$
$B$N(B gcd $B$r8_=|K!$G7W;;$9$k$H(B,
$ -1288744821/543589225 $ $B$H$J$k(B),
$B78?t$,Hs>o$K$*$*$-$/$J$kLdBjE@$,$"$j$^$9(B.
$BF1$8$3$H$,(B, Buchberger $B%"%k%4%j%:%`$K$D$$$F$bH/@8$7$F(B,
$B7W;;$N%\%H%k%M%C%/$K$J$j$^$9(B.
$B$3$NLdBj$K$D$$$F$O(B,
$B?tM}2J3X(B 1998$BG/(B11$B7n9f$K(B $BLnO$(B, $B2#;3N>;a$,2r@b$r=q$$$F$$$k$N$G(B
$B8fMw2<$5$$(B.

$w \geq 0$ $B$G$J$$$H$-$K(B, $B%0%l%V%J4pDl$r7W;;$9$k;;K!$r(B
tangent cone $B%"%k%4%j%:%`$H$$$$$^$9(B.
$B$3$l$K$D$$$F$O(B, 1980$BG/Be$N$J$+$P$h$j(B,
90$BG/Be$N$O$8$a$K$+$1$F(B, $B%h!<%m%C%Q$rCf?4$K$5$+$s$K8&5f$,(B
$B$*$3$J$o$l$^$7$?(B.
$B$=$N@.2L$O(B Mora $B$N(B tangent cone $B%"%k%4%j%:%`$H$7$F7k<B$7$F$$$^$9(B.

\medbreak

$B$5$FI,MW$J35G0$,$G$=$m$C$?$N$G(B,
$BOC$r3K?4ItJ,$X$9$9$a$^$7$g$&(B.

$B7W;;4DO@$N?t3XE*4pK\86M}(B, $B?t3XE*4pK\LdBj$H$G$b$$$&$Y$-$b$N$N$&$A(B,
$B<!$NFs$D$r$3$3$G$O6/D4$7$?$$$H;W$$$^$9(B.
($B8zN($N4QE@$+$i$_$k$HB>$N6/D4$9$Y$-MWAG$,$"$j$^$9$,(B,
$B;d$O$3$N4QE@$OITF@0U$J$N$G;DG0$J$,$i$U$l$k$3$H$,$G$-$^$;$s(B.)
\begin{enumerate}
\item ${\rm in}(\cdot)$ $B$N(B $BH>O"B3@-$r3hMQ$7$F%"%k%4%j%:%`$r:n$k(B.
\item $B%$%G%"%k(B $I$ $B$h$j@8$8$k%b%N%_%"%k%$%G%"%k(B
${\rm in}_{\prec_w}(I)$ $B$N@-<A$r$7$i$Y$k(B.
\end{enumerate}
$B2r@O$G$N%"%J%m%8$r$$$($P(B,
${\rm in}_w(I)$ $B$O(B $I$ $B$NBh0l6a;w$G$"$j(B,
$B%0%l%V%J4pDl$NM}O@$O%$%G%"%k$K$*$1$k@]F0O@(B
$B$G$"$k$o$1$G$9(B.

$BH>O"B3$H$$$&E@$+$i$_$k$H(B, 1930$BG/Be$N(B Macaulay $B$K$h$k<!$NDjM}$,(B
$B:G=i$N4pAC$H$$$($^$9(B.

$BDjM}(B:
$w \geq 0$ $B$N$H$-(B,
$$ \dim_{\bf k} {\bf k}[x]/I =
   \dim_{\bf k} {\bf k}[x]/{\rm in}_{\prec_w}(I).
$$
weight $w$ $B$,0lHL$@$H(B, $=$ $B$,(B $\geq$ $B$H$J$k(B.

\medbreak
${\bf k} = {\bf C}$ $B$N$H$-(B,
$ \dim_{\bf k} {\bf k}[x]/I $ $B$O(B $V(I)$ $B$NE@$N8D?t$G$"$j(B,
$B$D$^$j(B, $I$ $B$KBP1~$9$kBe?tJ}Dx<07O$N:,$N8D?t$K(B
$BEy$7$$$o$1$G$9$,(B,
$B$3$N?t$O(B, $B%b%N%_%"%k%$%G%"%k(B
${\rm in}_{\prec_w}(I)$ $B$K$?$$$7$F(B,
$   \dim_{\bf k} {\bf k}[x]/{\rm in}_{\prec_w}(I) $
$B$r7W;;$9$l$P$$$$$o$1$G$9(B.
$B%b%N%_%"%k%$%G%"%k$KBP$7$F$O(B, $B=c?h$KAH9g$;E*$JLdBj$H$J$k(B.

$B$?$H$($P(B,
$x^2+y^2-4=0, xy-1=0$ $B$,2?8D2r$r$b$D$+9M$($F$_$^$7$g$&(B.
$B%b%N%_%"%k%$%G%"%k(B
${\rm in}_{\prec_w}(I)$
$B$O(B, $I$ $B$N%0%l%V%J4pDl$r7W;;$9$k$3$H$K$h$j7W;;$G$-$^$9(B.
$B$5$-$[$I$N(B ${\rm in}$ $B$N7W;;Nc$r$_$l$P(B,
$B:,$N?t$O(B, 
$\dim_{\bf C} {\bf C}[x,y]/\langle y^4, x \rangle$ $B$KEy$7$$(B.
$B$7$?$,$C$F(B, $4$ $B$,:,$N8D?t$H$J$k(B.

$B$A$J$_$K(B, $B$3$N(B Macaulay $B$NDjM}$H%0%l%V%J4pDl$r7W;;$9$k(B
Buchberger $B%"%k%4%j%:%`$N$/$_$"$o$;$O(B
$B:,$N?t$r7W;;$9$k$b$C$H$b4pK\E*$J%"%k%4%j%:%`$H$$$($k$H;W$$$^$9$,(B,
$B8zN($NE@$+$i$OITK~B-$J$b$N$G$9(B.
$B8zN(@-$N4QE@$+$i:,$N?t$N7W;;K!$O8=:_$G$b@9$s$K8&5f$5$l$F$$$^$9(B.

$BH>O"B3@-$K$+$s$9$kDjM}$r$b$&0l$D>R2p$9$k$^$($K(B,
$B$3$3$G(B, $B%0%l%V%J4pDl$K4X$9$k;29M=q$r$$$/$D$+$"$2$F$*$-$^$7$g$&(B.
TeX $B$N$*$+$2$@$H;W$$$^$9$,(B, $B8&5f$,$5$+$s$K$*$3$J$o$l$F$+$i(B, 
$BK\$H$7$F$^$H$a$i$l$k;~4V%5%$%/%k$,$+$J$jC;$/$J$C$F$$$^$9(B.
$B$+$J$j:G?7$N8&5f@.2L$bK\$H$7$F=PHG$5$l$F$$$^$9(B.

$BF~Lg=q$H$7$F$O(B, $B<!$NK\$,$"$j$^$9(B.
\begin{enumerate}
\item Cox, O'Shea, Little,
{\it Ideals, Varieties and Algorithms}, Springer
\item Adams, Loustannan,
{\it An Introduction to Gr\"obner Bases}, AMS
\item Becker, Weispfenning,
{\it Gr\"obner Bases}, Springer
\end{enumerate}
1 $B$O(B, $B@~7ABe?tDxEY$rM=HwCN<1$H$7$F$+$+$l$?%0%l%V%J4pDl(B,
Buchberger $B%"%k%4%j%:%`$*$h$SBe?t4v2?$NF~Lg=q$G(B, 
$B3XIt3X@8$N%;%_%J!<$K$b:GE,$G$9(B.
2 $B$O(B, $BBe?t$N4pAC$rJY6/$7$?$3$H$N$"$k?M8~$1$K(B, $B<j$C<h$jAa$/(B
$BMWE@$r@bL@$7$F$"$kK\$G$9(B.
3 $B$O(B, $B2C72$J$I$NBe?t$N4pAC35G0$b@bL@$7$J$,$i(B,
$B$,$C$A$j=q$$$F$"$kK\$G$9(B.
$B%"%k%4%j%:%`$N@bL@$OF~Lg=qCf$$$A$P$s=<<B$7$F$$$F(B,
$B@bL@$I$&$j$K<BAu$9$k$H$^$"$^$"$N%W%m%0%i%`$r=q$1$^$9(B.
3$B$N8eH>$O$h$j9b5i$JOCBj$H$7$F=`AG%$%G%"%kJ,2r$N%"%k%4%j%:%`$,@bL@(B
$B$7$F$"$j$^$9(B.
$B$3$NK\$N=PHG8e(B,
Eisenbud, Huneke, Vasconcelos $B$d(B Shimoyama, Yokoyama $B$J$I$N(B
$B?7$7$$J}K!$,$G$F$-$^$7$?$,(B, ($B$b$A$m$s(B)$B$U$l$i$l$F$$$^$;$s(B.

$B$b$&$9$3$7>e5i$NK\$H$7$F<!$N(B4$B:}$r$"$2$F$*$-$^$9(B.
\begin{enumerate}
\item Cox, O'Shea, Little,
{\it Using Algebraic Geometry}, Springer
\item Sturmfels,
{\it Gr\"obner Basis and Convex Polytopes}, AMS
\item Eisenbud,
{\it Commutative Algebra --- with a View Toward Algebraic Geometry}, Springer
\item Saito, Sturmfels, Takayama,
{\it Gr\"obner Deformations of Hypergeometric Differential Equations},
Springer
\end{enumerate}
1 $BHVL\$NK\$O(B,
$BBe?tJ}Dx<0$N2rK!(B (FGLM, $B8GM-CM(B, Mixed Volume $B$H%[%b%H%T!<K!$J$I(B),
Tangent Cone Algorithm,
Resultant,
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2 $BHVL\$NK\$b(B, 90 $BG/BeA0H>$N(B,
toric variety $B$H$=$N(B $B%$%K%7%!%k%$%G%"%k$NAH9g$;O@$K4X$9$k8&5f$r$^$H$a$?(B
$BK\$G$9(B.
3 $BHVL\$NK\$O(B, $B2D494DO@$NF~Lg=q$G$9$,(B, $B%0%l%V%J4pDl$K4X$9$k2r@b(B,
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$BCx<T$O2D494DO@$NBg@h@8$H$$$&$@$1$G$J$/(B,
Macaulay $B$J$I$N%7%9%F%`3+H/(B,
Macaulay $B$rMQ$$$?2D494D$K$+$s$9$k7W;;5!<B83$bH`$N<~JU$G$*$3$J$o$l$?$@$1$"$C$F(B,
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4 $BHVL\$NK\$O(B, 90$BG/Be8eH>$N%0%l%V%J4pDl$N8&5f$N(B
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$B0J2<$=$NFbMF$N0lIt$r2r@b$9$kM=Dj$G$9(B.

$B%7%9%F%`$O(B Free $B$N$b$N$,5!G=E*$K=<<B$7$F$$$^$9(B.
{\tt Macaulay2}, {\tt Singular}, {\tt CoCoA}
$B$J$I$,Be?t4v2?MQ(B,
Gr\"obner Basis $B$GBe?tJ}Dx<07O$r2r$/$K$O(B,
{\tt Asir}, {\tt Posso GB}, {\tt $F_4$}($B%Q%j(B6$B$N%U%)%2!<%k$N$H$3$m$KLdBj$r(B
$BAw$kI,MW$"$j(B)
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\bigbreak
$B$5$F(B ${\rm in}(\cdot)$ $B$NH>O"B3@-DjM}$r$b$&$R$H$D>R2p$7$?$$$o$1$G$9$,(B,
$B$3$NDjM}$OHyJ,:nMQAG4D$K$+$s$9$k$b$N$G$9(B.
$$ D = {\bf C}\langle x_1, \ldots, x_n, \partial_1, \ldots, \partial_n
              \rangle $$
$B$rHyJ,:nMQAG4D(B($B$^$?$O(B Weyl algebra)$B$H$7$^$7$g$&(B.
$B$b$C$H$-$A$s$HDj5A$9$k$H(B,
$D$ $B$O(B $x_i$, $\partial_j$ $BC#$G@8@.$5$l$k(B
associative algebra $B$G(B,
$B@8@.85$O$[$H$s$I2D49$H$$$($k<!$N4X78<0$r$_$?$7$^$9(B:
$$ x_i x_j = x_j x_i, \ 
  \partial_i \partial_j = \partial_j \partial_i, \ 
  \partial_i x_j = x_j \partial_i \ (i \not= j),
$$
$$ \partial_i x_i = x_i \partial_i + 1. $$

$D$ $B$O4X?t$N6u4V$K:nMQ$7$^$9(B.
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$f$ $B$r(B $x_1, \ldots, x_n$ $B$N4X?t$H$9$k$H$-(B,
$$  \partial_i \bullet f = \frac{\partial f}{\partial x_i}, \ 
    x_i \bullet f = x_i f
$$
$B$G:nMQ$rDj5A$7$^$9(B.

$I$ $B$r(B $D$ $B$N:8%$%G%"%k$H$9$k$H$-(B
$I \bullet f = 0$ $B$H$J$k4X?t(B $f$ $B$r(B $I$ $B$N2r$H$$$$$^$9(B.

\noindent
$BNc(B:
$I = D \cdot \{ \partial_1^2 + \partial_2^2 -4,
  \partial_1 \partial_2 -1 \}
$
($\partial_1^2 + \partial_2^2 -4$,
 $\partial_1 \partial_2 -1 $ $B$G@8@.$5$l$k:8%$%G%"%k(B)
$B$H$9$k$H$-(B,
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$f = e^{\lambda_1 x_1 + \lambda_2 x_2}$
$B$H=q$1$^$9(B.
$B$3$3$G(B, $(\lambda_1, \lambda_2)$ $B$O(B
$B<!$NBe?tJ}Dx<07O(B
$$ \lambda_1^2 + \lambda_2^2 -4 = 0,\ \lambda_1 \lambda_2 -1=0 $$
$B$N:,$G$9(B.
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$B7k6I(B $I$ $B$N@5B'2r$N6u4V$O(B 4 $B<!85$N(B ${\bf C}$ $B%Y%/%H%k6u4V$H$J$j$^$9(B.
\medbreak

$B$5$F(B,
${\rm rank}(I)$ $B$r(B ${\bf C}^n$ $B$N(B generic point $p$ $B$N$^$o$j$G$N(B
$I$ $B$N@5B'2r$N<!85$H$7$^$9(B.
${\rm in}_{(u,v)}$ $B$O(B $2n$ $B<!85$N%Y%/%H%k(B $(u,v)$ $B$G>r7o(B
$u+v\geq 0$ $B$r$_$?$9$b$N$KBP$7$FB?9`<0$N>l9g$HF1$8$h$&$KDj5A(B
$B$9$k$3$H$,$G$-$^$9(B.

$BDjM}(B(Saito, Sturmfels, Takayama, 1998):
$I$ $B$r(B $D$ $B$N:8%$%G%"%k$H$9$k$H$-(B,
$$ {\rm rank}(I) \geq {\rm rank}({\rm in}_{(-w,w)}(I)) $$
$B$,G$0U$N(B $w \in {\bf R}^n$ $B$KBP$7$F$J$j$?$D(B.
$B$H$/$K(B regular holonomic $B$N$H$-Ey9f$,$J$j$?$D(B.
\medbreak

\noindent
$BNc(B:
$w=(100,0)$ $B$H$9$k$H(B,
${\rm in}_{(-w,w)}(I) = D \cdot \{ \partial_2^4, \partial_1 \}$
$B$H$J$k(B.
\medbreak
$B$J$*(B, ${\rm rank}$ $B$r7W;;$9$k$K$O(B, $B%0%l%V%J4pDl$rMQ$$$?(B
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$B$J$*(B, regular holonomic $B$G$"$k:8%$%G%"%k(B $I$ $B$KBP$7$F$O(B,
$B$3$NDjM}$,(B rank $B$N9bB.$J7W;;K!$r$"$?$($k>l9g$,$"$j$^$9(B.

\medbreak
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$B$3$l$K$D$$$F$OOC$9;~4V$,$J$+$C$?$N$G(B,
Eisenbud $B$NK\$r8fMw2<$5$$(B.

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\bigbreak
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$B$G$-$^$9(B.)




\end{document}