% $OpenXM: OpenXM/doc/Papers/bfct.tex,v 1.1 2001/02/06 07:54:18 noro Exp $
\documentclass{jarticle}
\usepackage[theorem,useeps,FVerb]{jssac}
\title{Risa/Asir $B$K$*$1$k(B Weyl Algebra $B>e$N%0%l%V%J4pDl7W;;$*$h$S$=$N1~MQ(B}
\author{$BLnO$(B $B@59T(B\affil{$B?@8MBg3X(B}\mail{noro@math.kobe-u.ac.jp}}
\art{$BO@J8(B}

\begin{document}
\maketitle

\section{Weyl Algebra}

\subsection{Leibnitz rule}

$BBN(B $K$ $B>e$N(B $n$ $B<!85(B Weyl Algebra 
$$D_n = K\langle x_1,\cdots,x_n,\partial_1,\cdots,\partial_n \rangle$$
$B$K$*$1$k@Q$N7W;;$O(B, $B$h$/CN$i$l$F$$$k$h$&$K(B Leibnitz rule $B$G7W;;$G$-$k(B. 
$B$9$J$o$A(B, $2n$ $BJQ?tB?9`<04D(B 
$K[x,\xi]=K[x_1,\cdots,x_n,\xi_1,\cdots,\xi_n]$ $B$+$i(B
$D_n$ $B$X$N(B $K$ $B@~7A<LA|(B $\Psi$ $B$r(B 
$\Psi(x^\alpha \xi^\beta) = x^\alpha \partial^\beta$
$B$GDj5A$9$k$H$-(B, 
$$\Psi(f)\Psi(g) = \Psi(\sum_{k_1,\ldots,k_n \ge 0} {1 \over {k_1!\cdots k_n!}}
{{\partial^k f} \over {\partial \xi^k}} {{\partial^k g} \over {\partial x^k}})$$
($f,g\in K[x,\xi]$) $B$H$J$k(B. Weyl Algebra $B$K$*$1$k7W;;$N8zN(2=$N$?$a$K$O(B 
$B@Q$N8zN(2=$,=EMW$G$"$j(B, $B$=$N$?$a$K$O(B Leibnitz rule $B$r$I$&;H$&$+$,80$H$J$k(B. 
$BFC$K(B, $B%0%l%V%J4pDl7W;;$r9M$($k>l9g(B, monomial $\times$ polynomial $B$N9bB.(B
$B2=$,I,MW$G$"$k(B. 

\subsection{Asir $B$K$*$1$k<BAu(B}

Asir $B$K$*$1$k=i4|$N<BAu$G$O(B, $B$3$l$r(B

$m$ : monomial, $f= \sum f_j$ ($f_j$ : monomial) $B$KBP$7(B
$mf = \sum mf_j$

$B$H$7$F7W;;$7$F$$$?(B. $B$3$3$G(B monomial $B$I$&$7$N@Q$O(B
$$x^k D^l\cdot x^a D^b = (x_1^{k_1}(D_1^{l_1}x_1^{a_1})D_1^{b_1})\cdots
(x_n^{k_n}(D_n^{l_n}x_n^{a_n})D_n^{b_n})$$
$B$H$7$F(B, $B1&JU$N(B $D_i^{k_i}x_i^{a_i}$ $B$r(B Leibnitz rule $B$K$h$j(B
$$D_i^{l_i}x_i^{a_i} = \sum_{j=0}^{Min(l_i,a_i)}j!{l_i\choose j}
{a_i\choose j}x_i^{a_i-j}D_i^{l_i-j}$$
$B$H$7(B, $B$3$l$i$N@Q$r(B, $BDL>o$N2D49$JB?9`<0$H$7$FE83+$9$k(B, $B$H$$$&J}K!(B
$B$r:N$C$F$$$?(B. $B$7$+$7(B, $B$3$N(B monomial $B$N@Q<+BN$,B?9`<0$H$J$k$?$a(B, 
$B$3$l$i$rB-$79g$o$;$k:]$K$K9`$NHf3S$*$h$S%j%9%H$N$D$J$.49$($,B??t(B
$B@8$8(B, $BFC$K(B $f$ $B$N9`?t$,B?$$>l9g$K8zN($,0-$$$3$H$,J,$+$C$?(B. 

\subsection{$B2~NI(B}

Leibnitz rule $B$K8=$l$kHyJ,$r(B $f\in K[x,\xi]$$B$K:nMQ$5$;$F$b(B monomial
$B$N=g=x$OJQ$o$i$J$$$3$H$KCm0U$9$l$P(B, $B:8$+$i$+$1$k(B monomial $m$ $B$r8GDj(B
$B$7$?$H$-(B, $BA09`$N(B monomial $B$I$&$7$N@Q$K$*$$$F(B

\begin{enumerate}
\item $D_i^{l_i}x_i^{a_i}$ $B$rJQ7A$7$FF@$i$l$kOB$r>o$K(B $\sum_{j=0}^{l_i}$ 
$B$H$9$k(B. $l_i>Min(l_i,a_i)$ $B$N>l9g(B, $B78?t$O(B 0 $B$H$7$F$*$/(B. 

\item $BE,Ev$J=g=x$rDj$a$F(B, $B@Q$r9=@.$9$k(B monomial $B$r@0Ns$7(B, $BBP1~$9$k0LCV$N(B
monomial $B$I$&$7$NOB$r$^$::n$k(B. $B$3$l$O4{$K@0Ns$5$l$F$$$k(B $f_j$ $B$HF1=g$G$h$$(B. 

\item $B:G8e$K$=$l$i$rB-$79g$o$;$k(B. 
\end{enumerate}
$BMW$9$k$K(B, Leibnitz rule $B$K$*$$$F(B, $BHyJ,$K4X$9$kOB$r@h$K7W;;$9$k$H$$$&$3(B
$B$H$@$,(B, $B$3$l$G(B $m$ $B$N(B $\partial$ $B$K4X$9$k<!?t$,$h$[$I9b$/$J$$8B$j8zN((B
$B2=$9$k(B. ($B$3$l$O(B, Kan/sm1 $B$G4{$K9T$o$l$F$$$?(B. ) $B$5$i$K(B, $B<!$N2~NI$,9M$((B
$B$i$l$k(B.

\begin{itemize}
\item 
$(x_1^{k_1}(D_1^{l_1}x_1^{a_1})D_1^{b_1})\cdots
(x_n^{k_n}(D_n^{l_n}x_n^{a_n})D_n^{b_n})$
$B$N(B $B7W;;$r(B incremental $B$K9T$&(B. 

$B$3$l$O(B ${1 \over {k_1!\cdots k_n!}}$ $B$K8=$l$k6&DL$NItJ,@Q$r=EJ#$7$F7W(B
$B;;$7$J$$$H$$$&$3$H$G$"$j(B, $B<!?t$,9b$$>l9g$KM-8z$+$b$7$l$J$$(B. 

\item $l_i, a_i$ $B$,(B 0 $B$N>l9g$N(B shortcut.

$B0lHL$K<!?t$,(B 0 $B$K$J$kItJ,$,B?$$$G$3$l$O=EMW$G$"$k(B.

\item exponent vector, monomial $B$J$I$rI=$99=B$BN$N%a%b%j4IM}$r<+A0$G9T$&(B. 

$B$3$l$O(B, Weyl $B$K8B$i$:(B, $BM-8BBN>e$N>l9g$K$O0lHL$KMQ$$$F$$$k(B. $B$3$l$b(B
$B8zN($KBg$-$/1F6A$9$k(B. 
\end{itemize}
$B0J>e$N2~NI$r2C$($k$3$H$G(B, $B=i4|<BAu$KHf3S$7$F==J,8zN(2=$9$k$3$H$,$G$-$?(B. 

\subsection{Weyl Algebra $B$K$*$1$k(B Buchberger $B%"%k%4%j%:%`(B}

$BB?9`<04D>e$N(B Buchberger $B%"%k%4%j%:%`<BAu$K$*$$$F$O(B, $B<o!9$N(B criteria $B$*(B
$B$h$S(Bselection strategy $B$,9M0F$5$l<BMQ2=$5$l$F$$$k$,(B, Weyl Algebra $B$K$*(B
$B$$$F$b(BBuchberger's criterion (S-polynomial $B$r9=@.$9$kB?9`<0$NF,9`$N(B 
GCD $B$,(B 1 $B$N>l9g$K$O$=$N(B S-polynomial $B$O(B 0 $B$K@55,2=$5$l$k(B) $B0J30$O$=$N$^$^(B
$B;H$($k(B. $B$h$C$F(B, $B$"$H$O@Q$rA0@a$N$b$N$K49$($l$P(B, $BB?9`<04DMQ$N%I%i%$%P(B, 
$B%5%V%k!<%A%s$,;H$((B, $BMF0W$K(B Weyl Algebra $B>e$N(B Buchberger $B%"%k%4%j%:%`$N(B
$B<BAu$,F@$i$l$k(B. 

\section{b-function $B$N7W;;(B}
\label{orig}

\subsection{b-function}
$D=D_n=K\langle x_1,\cdots,x_n,\partial_1,\cdots,\partial_n\rangle$
($\partial_i = \partial/\partial x_i$), $f \in K[x_1,\cdots,x_n]$ $B$H$9(B
$B$k$H$-(B$P(s)f^{s+1}=b_f(s)f^s$ $B$J$k(B $P(s) \in D[s]$ $B$,B8:_$9$k$h$&$J:G>.(B
$B<!?t$N(B $b_f(s) \in K[s]$ $B$r(B $f$ $B$N(B (global) $b$-function $B$H8F$V(B 
\footnote{$BK\Mh$O(B $\tilde b(s)$ $B$H=q$/$Y$-$@$,(B, $BK\9F$G$O(B $b(s)$ $B$H(B
$B=q$/$3$H$K$9$k(B.}.
$BBg0$5W(B\cite{oaku-bfct} $B$K$h$j(B $b$-function $B$O0J2<$N$h$&$K7W;;$G$-$k(B. $B$^$:(B,
$n+1$ $B<!85$N(B Weyl Algebra $D_{n+1}=K\langle
t,x_1,\cdots,x_n,\partial_t,\partial_1,\cdots,\partial_n\rangle$ $B$r9M(B
$B$($k(B.

\begin{Def}
$D_{n+1}$ $B$N85(B
$P = \sum c_{\mu\nu\alpha\beta}t^\mu x^\alpha \partial_t^\nu \partial^\beta$
$B$KBP$7(B, $${\rm ord}_F(P) := \max\{\nu-\mu | $B$"$k(B \alpha, \beta $B$KBP$7(B
c_{\mu\nu\alpha\beta} \neq 0 \}$$
$$\hat{\sigma}(P) := 
\sum_{\nu-\mu={\rm ord}_F(P)} c_{\mu\nu\alpha\beta}t^\mu x^\alpha \partial_t^\nu \partial^\beta$$
\end{Def}

\begin{Def}
${\rm ord}_F(P) = m$ $B$J$k(B $P \in D_{n+1}$ $B$KBP$7(B, $\psi(P) \in D[s]$ $B$r(B
$$
\psi(P)(t\partial_t) = 
\left\{
\parbox[c]{2in}{
$\hat{\sigma}(t^mP)$ \quad\quad  $(m\ge 0)$\\
$\hat{\sigma}(\partial_t^{-m}P)$ \quad $(m< 0)$
}
\right.
$$
$B$GDj5A$9$k(B. 
\end{Def}

\begin{Th} 
$$I=Id(t-y_1f,\partial_1+y_1 (\partial f/\partial x_1) \partial_t, \cdots, 
\partial_n+ y_1 (\partial f/\partial x_n) f_n \partial_t)$$
$B$KBP$7(B, $G_1$ $B$r(B $I_1 = I \cap D$ $B$N%0%l%V%J4pDl$H$9$k(B. $B$3$N;~(B, 
$$Id(\psi(G_1)) \cap K[s] = Id(b(-s-1))$$
\end{Th}
$BBg0$5W(B \cite{oaku-bfct} $B$G$O(B 
$$(Id(\psi(G_1)) \cap K[x,s]) \cap K[s]$$$B$J$kFsCJ3,$N(B elimination $B$K$h(B
$B$k7W;;$rDs0F$7$F$$$k(B. $B$3$l$O(B, local $B$J(B b-function$B$N7W;;$K(B 
$(Id(\psi(G_1)) \cap K[x,s])$ $B$,I,MW$H$J$k$?$a$H9M$($i$l$k$,(B, global
b-function $B$N$_$r5a$a$k>l9g$K$O(B, $BD>@\(B $K[s]$ $B$H$N8r$o$j$r5a$a$F9=$o$J(B
$B$$(B. $B$h$C$F(B, $BG$0U$N=g=x$K4X$9$k(B $Id(\psi(G_1))$ $B$N%0%l%V%J4pDl$,5a$^$C$F(B
$B$$$l$P(B, $b(s)$ $B$O2D49B?9`<04D$N>l9g$HF1MM$K(B, $BL$Dj78?tK!$h$j5a$a$k(B
$B$3$H$,$G$-$k(B. 

\subsection{\Q $B>e$N(B Weyl Algebra $B$K$*$1$k:G>.B?9`<0$N(B modular $B7W;;(B}
\label{mod1}

$D$ $B$r(B $\Q$ $B>e$N(B Weyl Algebra, $J \subset D$, $P \in D$ $B$+$D(B $P$ $B$O@0(B
$B?t78?t$H$7(B, $J\cap \Q[P] \neq \{0\}$ $B$H$9$k(B. $B$3$N;~(B, $J\cap \Q[P] =
Id(b(P))$ $B$H$9$l$P(B
$b(s)$ $B$O(B$D/J$ $B$K$*$1$k(B $P$ $B$N(B \Q $B>e$N:G>.B?9`<0$H$J$k(B. 
$B$3$3$G(B, $b(s) \in \Z[s]$ $B$+$D(B \Z $B>e86;OE*$H<h$l$k(B. 

$J$ $B$N(B, $B=g=x(B $<$ $B$K4X$9$k%0%l%V%J4pDl$G(B, $B3F85$NF,78?t$,(B 1 $B$G$"$k$b$N$r(B $G$ 
$B$H$7(B, $G$ $B$N3F85$N(B $B$N78?t$,(B $\Z_{(p)} = \{a/b | a\in \Z, b \notin
p\Z\}$ $B$KB0$9$k$h$&$J(B $p$$B$rA*$V(B. $\phi_p$ $B$r(B $\Z_{(p)}$ $B$+$i(B $GF(p)$ 
$B$X$NI8=`E*<M1F(B ($B$*$h$S$=$N(B $D$ $B$X$N3HD%(B) $B$H$9$k(B.

\begin{Lem}
$\phi_p(G)$ $B$O(B $Id(\phi_p(G))$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J4pDl$G(B, $\phi_p(b(P)) \in
Id(\phi_p(G))$.
\end{Lem}
\begin{Proof}
$G$ $B$+$i:n$i$l$k(B S-polynomial $B$N(B $G$ $B$K$h$k(B 0 $B$X$N(B reduction $B$,(B $Z_{(p)}$ $B>e(B
$B$G9T$($k(B. $B$h$C$F(B, $B$=$N(B $\phi_p$ $B$K$h$kA|$,$=$N$^$^(B $\phi_p(G)$ $B$K$h$k(B
reduction $B$K$J$k$+$i(B, $\phi_p(G)$ $B$O(B $Id(\phi_p(G))$ $B$N%0%l%V%J4pDl$H(B
$B$J$k(B. $B8eH>$bF1MM$G$"$k(B. 
\end{Proof}

$B2>Dj$K$h$j(B $\phi_p(b(s))$ $B$O(B 0$B$G$J$$$+$i(B, $\phi_p(P)$
$B$N(B $\phi_p(D)/Id(\phi_p(G))$ $B$K$*$1$k:G>.B?9`<0(B $b_p(s)$ $B$,B8:_$7$F(B
$b_p(s) | \phi_p(b(s))$ $B$,@.$jN)$D(B. $BFC$K(B $\deg(b_p(s)) \le \deg(b(s))$.

\begin{Th}
\label{invimg}
$b_p(s)$ $B$r(B $\phi_p(D)/Id(\phi_p(G))$ $B$K$*$1$k(B $\phi_p(P)$ $B$N:G>.B?9`(B
$B<0$H$9$k$H$-(B, $\deg(f(s)) = \deg(b_p(s))$, $f(P)
\in Id(G)$$B$J$k(B $f \in \Z[s]$ $B$,B8:_$9$l$P(B, $f(s) = b(s)$.
\end{Th}
\begin{Proof}
$B2>Dj$h$j(B $b(s)|f(s)$. $B0lJ}$G(B $\deg(f(s)) = \deg(b_p(s)) \le \deg(b(s))$ $B$@$+$i(B
$b(s)=f(s)$.
\end{Proof}

\begin{Alg}
\label{modbfct}
\begin{tabbing}
Input: \= $B9`=g=x(B $<$, $<$ $B$K4X$9$k%0%l%V%J4pDl(B $G \subset D$ ($BF,78?t$,(B 1),\\
      \> $Id(G) \cap \Q[P] \neq \{0\}$ $B$J$k(B $P\in D$ ($B@0?t78?t(B)\\
Output: $Id(G) \cap \Q[P] = Id(b(P))$ $B$J$k(B $b(s) \in \Q[s]$\\
do \= \{\\
   \> $p \leftarrow G \in \Z_{(p)}$ $B$J$kL$;HMQ$NAG?t(B\\
   \> $b_p \leftarrow$ $\phi_p(P)$ $B$N(B $\phi_p(D)/Id(\phi_p(G))$ $B$K$*$1$k:G>.B?9`<0(B\\
   \> $d \leftarrow deg(b_p)$\\
   \> $c \leftarrow NF(P^d,G)+\sum_{i=0}^{d-1}c_i NF(P^i,G)$ ($c_i$ $B$OL$Dj78?t(B)\\
   \> $C \leftarrow \{C_t(c_0,\cdots,c_{d-1})=0 | C_t$ $B$O(B $c$ $B$N3F(B {\rm monomial} $B$N78?t(B\}\\
   \> if \= $C$ $B$,2r(B $\{c_0=a_0,\cdots,c_{d-1}=a_{d-1}\}$ $B$r;}$D(B \{\\
   \>    \> return $s^d+\sum_{i=0}^{d-1}a_is^i$\\
   \> \}\\
\}
\end{tabbing}
\end{Alg}
$BK!(B $p$ $B$G$N:G>.B?9`<0$O(B, $NF(\phi_p(P^k),Id(\phi_p(G)))$ $B$r=g$K7W;;$7$F(B
$B@~7A4X78$rC5$9$3$H$GF@$i$l$k(B. 
$b(s) \in \Z[s]$ $B$H$7$?;~$N<g78?t$r3d$i$J$$(B $p$ $B$KBP$7$F$ODjM}(B \ref{invimg}
$B$N(B $f$$B$OB8:_$9$k$+$i(B, $B%"%k%4%j%:%`(B \ref{modbfct} $B$ODd;_$9$k(B. $B$^$?(B, $C$ $B$,(B
$B2r$r;}$D$J$i(B, $BK!(B $p$ $B$G$N0l0U@-$h$j$=$l$O0l0UE*$K7h$^$k(B. 
$C$ $B$N5a2r$O(B, $BK!(B $p$ $B$G$N0l0U@-$rMxMQ$7$F(B, Hensel $B9=@.$K$h$j8zN($h$/5a$a$k(B
$B$3$H$,$G$-$k(B. $B>\:Y$O(B \cite{RUR} $B$r;2>H(B.

\subsection{$B%[%m%N%_%C%/$J(B ideal $B$KBP$9$k(B $b$-function}
\label{mod2}

ideal $I \subset D_n$,$w \in \R^n \setminus \{0\}$ $B$KBP$7(B,
weight $(-w,w)$ $B$K4X$9$k(B initial ideal $in_{(-w,w)}(I)$ $B$,(B, Weyl Algebra
$B$K$*$1$k%0%l%V%J4pDl7W;;$K$h$j5a$^$k(B (\cite{SST} Theorem 1.1.6).
$I$ $B$N(B characteristic variety $B$N<!85$,(B $n$ $B$G$"$k(B
$B$H$-(B $I$ $B$O%[%m%N%_%C%/$H8F$P$l$k$,(B, $B$3$N$H$-(B \cite{SST} Theorem 5.1.2
$B$K$h$j(B
$$in_{(-w,w)}(I) \cap K[s] = Id(b(s))$$ ($s =
w_1\theta_1+\cdots+w_n\theta_n, \theta_i = x_i\partial_i$) $B$J$k(B 0 $B$G$J(B
$B$$B?9`<0(B $b(s)$ $B$,B8:_$9$k(B. $B$3$N(B $b(s)$ $B$r(B, $I$ $B$N(B (global) $b$-function
$B$H8F$V(B.
$BFC$K(B, $BB?9`<0(B $f$ $B$KBP$7(B, 
$$I_f=Id(t-f,\partial_1+(\partial f/\partial x_1) \partial_t, \cdots,
\partial_n+ (\partial f/\partial x_n) \partial_t)$$$B$H$*$/;~(B, $I_f$ 
$B$O%[%m%N%_%C%/$G(B, $t$ $B$r@hF,$NJQ?t$H$9$k;~(B $I_f$ $B$N(B 
$w=(1,0,\cdots,0)$ $B$K4X$9$k(B global $b$-function $B$r(B $B(s)$$B$H$9$l$P(B,
$b_f(s) = B(-s-1)$ $B$,@.$jN)$D(B. $B$h$C$F(B, $D/in_{(-w,w)}(I_f)$ $B$K$*$1$k(B 
$t\partial_t$ $B$N:G>.B?9`<0$+$i(B $b_f(s)$ $B$r5a$a$k$3$H$b$G$-$k(B.
$B$3$3$G(B, $in_{(-w,w)}(I_f)$ $B$N7W;;$K$O(B non-term order $B$K$h$k%0%l%V%J4pDl(B
$B7W;;$,I,MW$H$J$k$,(B, \cite{SST} $B$G=R$Y$i$l$F$$$k@F<!2=$K$h$j(B
$B$"$k(B term order $B$N$b$H$G$N%0%l%V%J4pDl7W;;$K5"Ce$G$-$k(B. 

\subsection{$B%?%$%_%s%0%G!<%?(B}

Section \ref{orig} $B$G>R2p$7$?(B, $BFsCJ3,$N>C5n$K$h$kJ}K!(B ($BJ}K!(B 1)$B$H(B,
$Id(\psi(G_1))\cap K[s]$$B$r(B Section \ref{mod1} $B$G=R$Y$?$h$&$K:G>.B?9`<0(B
$B$H$7$F5a$a$kJ}K!(B ($BJ}K!(B 2), $B$*$h$S(B Section \ref{mod2} $B$G=R$Y$?J}K!(B ($BJ}(B
$BK!(B 3)$B$K$h$k7W;;;~4V$r$5$^$6$^$JB?9`<0$KBP$7$FHf3S$9$k(B. $B$$$:$l$b(B,
$b$-function $B$N7W;;$O(Bmodular $B7W;;$G:G>.B?9`<05a$a$kJ}K!$K$h$j9T$C(B
$B$?(B. $BNcBj$O(B\cite{oaku-bfct}, \cite{yano-bfct} $B$+$i:N$C$?(B. $B8eH>$N(B
$x^a+xy^{b-1}+y^b$ $B$K4X$7$F$O(B, $BK\9V5fO?Cf$NBg0$5W(B, $B9b;3N>;a$K$h$k9F$r(B
$B;2>H(B. $B7W;;$O(B, PentiumIII 1GHz $B>e$G9T$C$?(B. $BC10L$OIC$G%,!<%Y%C%8%3%l%/%7%g(B
$B%s;~4V$O=|$$$F$"$k(B. ``--'' $B$O(B, $BB>$NJ}K!$HHf3S$7$F;~4V$,$+$+$j2a$.$k$?(B
$B$aCfCG$7$?$3$H$r0UL#$9$k(B.  $BI=(B 1$B$G$O7W;;$,Hf3SE*MF0W$J$b$N$,07$o$l$F$*(B
$B$j(B, $B3FJ}K!$GBg:9$O$J$$$,(B, $BI=(B 2 $B$G$O(B, $BJ}K!(B 1 $B$G$O7W;;$,:$Fq$J$b$N$,(B, $BJ}(B
$BK!(B 2, 3 $B$K$h$j7W;;$G$-$F$$$k(B. $B$^$?(B, $B<!?t$,>e$,$k$K$D$l(B, $BJ}K!(B 3$B$,M%0L$H(B
$B$J$k(B. $B$3$l$O(B, $BJ}K!(B 2 $B$K8=$l$k%0%l%V%J4pDl7W;;$,(B, $BJ}K!(B 3 $B$K$*$1$k(B 
$in_{(-w,w)}(I_f)$ $B$N7W;;$KHf$Y$F:$Fq$H$J$k$?$a$G$"$k(B. $B$3$l$OI=(B 3 $B$K$*(B
$B$$$F$5$i$K82Cx$H$J$j(B, $BJ}K!(B 2 $B$G$O7W;;$G$-$J$$Nc$b=P(B
$B$FMh$k(B. $B<B:]$K$O(B, $BJ}K!(B 3 $B$G$O(B, $B:G>.B?9`<0$N7W;;$,(B 
dominant $B$H$J$kNc$b$7$P$7$P8+$i$l$k(B.

\begin{table}[hbtp]
\begin{tabular}{c|c|c|c|c} \hline
 & $\deg(b(s))$ & $BJ}K!(B 1  & $BJ}K!(B 2 & $BJ}K!(B 3 \\ \hline
$x^5+x^3y^3+y^5$ & 7 &1.4 &2.4 &1.3 \\ \hline
$x^3+y^3+z^3+x^2y^2z^2+xyz$ & 5 &3.5 &3.9 & 10\\ \hline
$x^6+y^4+z^3$ &18 &0.4 &0.7 & 0.2 \\ \hline
$x^4+y^4+z^3+xyz$ &8 &0.3 &0.6 & 0.3\\ \hline
$(x^3-y^2z^2)^2$ &14 &0.7 &1.5 & 0.2\\ \hline
$y(x^5-y^2z^2)$ &18 &0.8 &0.5 & 0.1 \\ \hline
$y((y+1)x^3-y^2z^2)$ &11 &0.6 &1.1 & 0.7 \\ \hline
\end{tabular}
\caption{\cite{oaku-bfct} $B$+$i$NNcBj(B}
\end{table}


\begin{table}[hbtp]
\begin{tabular}{c|c|c|c|c} \hline
 & $\deg(b(s))$ & $BJ}K!(B 1  & $BJ}K!(B 2 & $BJ}K!(B 3 \\ \hline
$x^4+xy^4+y^5$ &13 &135 &5.0 & 3.4 \\ \hline
$x^4+xy^5+y^6$ &13 &15 &2.9 & 2.1 \\ \hline
$x^4+xy^6+y^7$ &19 &719 &5.1 & 6.9 \\ \hline
$x^4+xy^7+y^8$ &11 &19 &7.3 & 4.0 \\ \hline
$x^4+xy^8+y^9$ &25 &-- &14 & 11 \\ \hline
$x^4+xy^9+y^{10}$ &23 &-- &22 & 14\\ \hline
$x^5+xy^5+y^6$ &21 &-- &158 & 21\\ \hline
$x^5+xy^6+y^7$ &25 &-- &32 & 19\\ \hline
\end{tabular}

\caption{\cite{yano-bfct} $B$+$i$NNcBj(B}
\end{table}

\begin{table}[hbtp]
\begin{tabular}{c|c|c|c} \hline
 & $\deg(b(s))$ & $BJ}K!(B 2 & $BJ}K!(B 3 \\ \hline
$x^6+y^{12}z^8$ & 50 & 71 & 1.8 \\ \hline
$x_1x_2^3+x_3x_4^5+x_5x_6^7$ &106 & 404 & 14 \\ \hline
$(x_1x_2)^3+(x_3x_4)^3+(x_5x_6)^3$ &23 & -- & 27 \\ \hline
$(x_1x_2)^2+(x_3x_4)^2+(x_5x_6)^2+(x_7x_8)^2$ &16 & -- & 61 \\ \hline
\end{tabular}

\caption{\cite{yano-bfct} $B$+$i$NNcBj(B (non-isolated singularities)}
\end{table}

\section{$B$*$o$j$K(B}

Risa/Asir $B$K$*$1$k(B, Weyl Algebra $B4XO"5!G=$N<BAu$*$h$S(B, $B$=$N1~MQ$H$7$F(B $b$-function
$B$N7W;;J}K!$N2~NI$K$D$$$F=R$Y$?(B. $B$3$3$G=R$Y$?J}K!$K$h$j(B, $B$h$j9-$$HO0O$NB?9`<0$*$h$S(B
$B%$%G%"%k$KBP$7$F(B $b$-function $B$,7W;;$G$-$k$h$&$K$J$C$?$3$H$O3N$+$G$"$k(B. $B$7$+$7(B, 
$B4{$KB>$NJ}K!$G7k2L$,CN$i$l$F$$$k$b$N$G$b7W;;IT2DG=$JLdBj$OB8:_$7(B, $B$^$?(B
$B$$$o$f$kB?=E(B $b$-function $B$KBP$7$F$O(B, $B:G>.B?9`<0$K$h$kJ}K!$OL5NO$G$"$k(B. 
$B$3$l$i$KBP=h$9$k$?$a$K$O$5$i$J$k2~NI(B, $B$"$k$$$O?7$7$$J}K!$,I,MW$G$"$m$&(B. 

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\end{thebibliography}

\end{document}

