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version 1.1, 2001/02/06 07:54:18 version 1.3, 2001/02/07 09:29:45
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 % $OpenXM$  % $OpenXM: OpenXM/doc/Papers/bfct.tex,v 1.2 2001/02/06 08:38:31 noro Exp $
 \documentclass{jarticle}  \documentclass{jarticle}
 \usepackage[theorem,useeps,FVerb]{jssac}  \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}  \title{Risa/Asir $B$K$*$1$k(B Weyl Algebra $B>e$N%0%l%V%J4pDl7W;;$*$h$S$=$N1~MQ(B}
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 \section{Weyl Algebra}  \section{Weyl Algebra}
   
   $B$5$^$6$^$J7W;;5!Be?t%7%9%F%`>e$G(B Weyl Algebra $B$K4X$9$k1i;;$,<BAu$5$l$F(B
   $B$$$k(B. $BBeI=E*$J$b$N$H$7$F(B, Kan/sm1 \cite{Kan}, Macaulay2
   \cite{Mac2}\cite{Tsai}, Maple Ore algebra package,
   Singular \cite{Singular}$B$J$I$,$"$k(B. $B0J2<$G$O(B Risa/Asir $B$K$*$1$k(B Weyl
   Algebra $B4XO"5!G=$N<BAu$K$D$$$F=R$Y$k$,(B, $B$3$3$G=R$Y$i$l$F$$$k2~NI$=$NB>(B
   $B$O(B, $BJ88%$H$7$F;2>H$9$k$3$H$O$G$-$J$$$b$N$N(B, $B>e5-%7%9%F%`$=$l$>$l$K$*$$(B
   $B$F:N$jF~$l$i$l$F$$$k$H9M$($i$l$k(B.
   
 \subsection{Leibnitz rule}  \subsection{Leibnitz rule}
   
 $BBN(B $K$ $B>e$N(B $n$ $B<!85(B Weyl Algebra  $BBN(B $K$ $B>e$N(B $n$ $B<!85(B Weyl Algebra
Line 135  $$
Line 143  $$
   
 \begin{Th}  \begin{Th}
 $$I=Id(t-y_1f,\partial_1+y_1 (\partial f/\partial x_1) \partial_t, \cdots,  $$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)$$  \partial_n+ y_1 (\partial f/\partial x_n) \partial_t)$$
 $B$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,  $B$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,
 $$Id(\psi(G_1)) \cap K[s] = Id(b(-s-1))$$  $$Id(\psi(G_1)) \cap K[s] = Id(b(-s-1))$$
 \end{Th}  \end{Th}
Line 151  b-function $B$N$_$r5a$a$k>l9g$K$O(B, $BD>@\(B $K[s
Line 159  b-function $B$N$_$r5a$a$k>l9g$K$O(B, $BD>@\(B $K[s
 \subsection{\Q $B>e$N(B Weyl Algebra $B$K$*$1$k:G>.B?9`<0$N(B modular $B7W;;(B}  \subsection{\Q $B>e$N(B Weyl Algebra $B$K$*$1$k:G>.B?9`<0$N(B modular $B7W;;(B}
 \label{mod1}  \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  $D$ $B$r(B $\Q$ $B>e$N(B Weyl Algebra, $J$ $B$r(B $D$ $B$N(B ideal, $P \in D$ $B$+$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] =  $B?t78?t$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$P(B  Id(b(P))$ $B$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.  $b(s)$ $B$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.  $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 $G$  $J$ $B$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  $B$H$7(B, $G$ $B$N3F85$N78?t$,(B $\Z_{(p)} = \{a/b | a\in \Z, b \notin
 p\Z\}$ $B$KB0$9$k$h$&$J(B $p$$B$rA*$V(B. $\phi_p$ $B$r(B $\Z_{(p)}$ $B$+$i(B $GF(p)$  p\Z\}$ $B$KB0$9$k$h$&$J(B $p$$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.  $B$X$NI8=`E*<M1F(B ($B$*$h$S$=$N(B $D$ $B$X$N3HD%(B) $B$H$9$k(B.
   
Line 245  $Id(\psi(G_1))\cap K[s]$$B$r(B Section \ref{mod1} $
Line 253  $Id(\psi(G_1))\cap K[s]$$B$r(B Section \ref{mod1} $
 $B$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  $B$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$$$:$l$b(B,  $BK!(B 3)$B$K$h$k7W;;;~4V$r$5$^$6$^$JB?9`<0$KBP$7$FHf3S$9$k(B. $B$$$:$l$b(B,
 $b$-function $B$N7W;;$O(Bmodular $B7W;;$G:G>.B?9`<05a$a$kJ}K!$K$h$j9T$C(B  $b$-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  $B$?(B. $BNcBj$O(B\cite{oaku-bfct}, \cite{yano-bfct} $B$+$i:N$C$?(B. $BI=(B 2 $B$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  $x^a+xy^{b-1}+y^b$ $B$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;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=|$$$F$"$k(B. ``--'' $B$O(B, $BB>$NJ}K!$HHf3S$7$F;~4V$,$+$+$j2a$.$k$?(B  $B%s;~4V$O=|$$$F$"$k(B. ``--'' $B$O(B, $BB>$NJ}K!$HHf3S$7$F;~4V$,$+$+$j2a$.$k$?(B
Line 303  $(x_1x_2)^2+(x_3x_4)^2+(x_5x_6)^2+(x_7x_8)^2$ &16 & --
Line 311  $(x_1x_2)^2+(x_3x_4)^2+(x_5x_6)^2+(x_7x_8)^2$ &16 & --
   
 \section{$B$*$o$j$K(B}  \section{$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  Risa/Asir $B$K$*$1$k(B, Weyl Algebra $B4XO"5!G=$N<BAu$*$h$S(B, $B$=$N1~MQ$H$7$F(B
 $B$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  $b$-function$B$N7W;;J}K!$N2~NI$K$D$$$F=R$Y$?(B. $b$-function $B7W;;$O(B
 $B%$%G%"%k$KBP$7$F(B $b$-function $B$,7W;;$G$-$k$h$&$K$J$C$?$3$H$O3N$+$G$"$k(B. $B$7$+$7(B,  Kan/sm1, Macaulay 2 $B$K$b<BAu$5$l$$$F$k$,(B, $BK\9F$G=R$Y$?$h$&$J(B, $B:G>.B?9`(B
 $B4{$KB>$NJ}K!$G7k2L$,CN$i$l$F$$$k$b$N$G$b7W;;IT2DG=$JLdBj$OB8:_$7(B, $B$^$?(B  $B<0$rL$Dj78?tK!$G5a$a$kJ}K!$rMQ$$$?Nc$O$J$$$h$&$G$"$k(B. $B0lJ}$G(B
 $B$$$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.  $b$-function $B$O(B$f$ $B$N6I=j%b%N%I%m%_!<$H4X78$9$k$3$H$,CN$i$l$F$$$k$,(B,
 $B$3$l$i$KBP=h$9$k$?$a$K$O$5$i$J$k2~NI(B, $B$"$k$$$O?7$7$$J}K!$,I,MW$G$"$m$&(B.  Singular $B$K$*$$$F$O(B, $BA4$/0[$J$kN)>l$+$i(B isolated singularity $B$G$N%b%N(B
   $B%I%m%_!<9TNs$r5a$a$k5!G=$rDs6!$7$F$$$k(B. $B$3$l$K$D$$$F(B, $B8zN($NLL$+$i(B
   $B$NHf3S$bI,MW$H9M$($i$l$k$,(B, $BF@$i$l$k7k2L$,0[$J$k$3$H$b$"$j$^$@>\:Y(B
   $B$JHf3S$O9T$C$F$$$J$$(B.
   
   $BK\9F$G=R$Y$?J}K!$K$h$j(B, $B$h$j9-$$HO0O$NB?9`<0$*$h$S%$%G%"%k$KBP$7$F(B
   $b$-function $B$,7W;;$G$-$k$h$&$K$J$C$?$3$H$O3N$+$G$"$k(B. $B$7$+$7(B, $B4{$KB>(B
   $B$NJ}K!$G7k2L$,CN$i$l$F$$$k$b$N$G$b7W;;IT2DG=$JLdBj$OB8:_$7(B, $B$^$?$$$o$f(B
   $B$kB?=E(B $b$-function $B$KBP$7$F$O(B, $B:G>.B?9`<0$K$h$kJ}K!$OL5NO$G$"$k(B. $B$3$l(B
   $B$i$KBP=h$9$k$?$a$K$O$5$i$J$k2~NI(B, $B$"$k$$$O?7$7$$J}K!$,I,MW$G$"$m$&(B.
   
 \begin{thebibliography}{99}  \begin{thebibliography}{99}
   
   \bibitem{Mac2} Grayson, D., Stillman, M.:
   Macaulay 2, a software system for research in algebraic geometry.
   {\tt http://www.math.ucuc.edu/Macaulay2}.
   
   \bibitem{Singular} Greuel, G.-M., Pfister, G., Sch\"onemann, H.:
   SINGULAR, A Computer Algebra System for Polynomial Computations.
   {\tt http://www.singular.uni-kl.de/}.
   
   \bibitem{Tsai} Leykin, A., Tsai, H.:
   D-module package for Macaulay 2.
   {\tt http://www.math.cornell.edu/\verb+~+tsai}.
   
 \bibitem{RUR} Noro, M., Yokoyama, K.:  \bibitem{RUR} Noro, M., Yokoyama, K.:
 A Modular Method to Compute the Rational Univariate  A Modular Method to Compute the Rational Univariate
 Representation of Zero-Dimensional Ideals.  Representation of Zero-Dimensional Ideals.
Line 336  J. Pure Appl.\ Algebra (in press).
Line 365  J. Pure Appl.\ Algebra (in press).
 Saito, M., Sturmfels, B., Takayama, N.:  Saito, M., Sturmfels, B., Takayama, N.:
 Gr\"obner Deformations of Hypergeometric Differential Equations.  Gr\"obner Deformations of Hypergeometric Differential Equations.
 Algorithms and Computation in Mathematics {\bf 6}, Springer (2000).  Algorithms and Computation in Mathematics {\bf 6}, Springer (2000).
   
   \bibitem{Kan} Takayama, N.:
   Kan --- A system for doing algebraic analysis by computer.
   {\tt http://www.math.kobe-u.ac.jp/KAN}.
   
 \bibitem{yano-bfct} Yano, T.:  \bibitem{yano-bfct} Yano, T.:
 On the theory of $b$-functions.  On the theory of $b$-functions.
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