===================================================================
RCS file: /home/cvs/OpenXM/doc/Papers/bfct.tex,v
retrieving revision 1.1
retrieving revision 1.3
diff -u -p -r1.1 -r1.3
--- OpenXM/doc/Papers/bfct.tex	2001/02/06 07:54:18	1.1
+++ OpenXM/doc/Papers/bfct.tex	2001/02/07 09:29:45	1.3
@@ -1,4 +1,4 @@
-% $OpenXM$
+% $OpenXM: OpenXM/doc/Papers/bfct.tex,v 1.2 2001/02/06 08:38:31 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}
@@ -10,6 +10,14 @@
 
 \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}
 
 $BBN(B $K$ $B>e$N(B $n$ $B<!85(B Weyl Algebra 
@@ -135,7 +143,7 @@ $$
 
 \begin{Th} 
 $$I=Id(t-y_1f,\partial_1+y_1 (\partial f/\partial x_1) \partial_t, \cdots, 
-\partial_n+ y_1 (\partial f/\partial x_n) f_n \partial_t)$$
+\partial_n+ y_1 (\partial f/\partial x_n) \partial_t)$$
 $B$KBP$7(B, $G_1$ $B$r(B $I_1 = I \cap D$ $B$N%0%l%V%J4pDl$H$9$k(B. $B$3$N;~(B, 
 $$Id(\psi(G_1)) \cap K[s] = Id(b(-s-1))$$
 \end{Th}
@@ -151,14 +159,14 @@ 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}
 \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, $J\cap \Q[P] =
 Id(b(P))$ $B$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 $\Z_{(p)} = \{a/b | a\in \Z, b \notin
+$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 $GF(p)$ 
 $B$X$NI8=`E*<M1F(B ($B$*$h$S$=$N(B $D$ $B$X$N3HD%(B) $B$H$9$k(B.
 
@@ -245,7 +253,7 @@ $Id(\psi(G_1))\cap K[s]$$B$r(B Section \ref{mod1} $
 $B$H$7$F5a$a$kJ}K!(B ($BJ}K!(B 2), $B$*$h$S(B Section \ref{mod2} $B$G=R$Y$?J}K!(B ($BJ}(B
 $BK!(B 3)$B$K$h$k7W;;;~4V$r$5$^$6$^$JB?9`<0$KBP$7$FHf3S$9$k(B. $B$$$:$l$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
+$B$?(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
 $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
@@ -303,15 +311,36 @@ $(x_1x_2)^2+(x_3x_4)^2+(x_5x_6)^2+(x_7x_8)^2$ &16 & --
 
 \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. 
+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$$$F=R$Y$?(B. $b$-function $B7W;;$O(B 
+Kan/sm1, Macaulay 2 $B$K$b<BAu$5$l$$$F$k$,(B, $BK\9F$G=R$Y$?$h$&$J(B, $B:G>.B?9`(B
+$B<0$rL$Dj78?tK!$G5a$a$kJ}K!$rMQ$$$?Nc$O$J$$$h$&$G$"$k(B. $B0lJ}$G(B 
+$b$-function $B$O(B$f$ $B$N6I=j%b%N%I%m%_!<$H4X78$9$k$3$H$,CN$i$l$F$$$k$,(B,
+Singular $B$K$*$$$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}
 
+\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.:
 A Modular Method to Compute the Rational Univariate
 Representation of Zero-Dimensional Ideals.
@@ -336,6 +365,10 @@ J. Pure Appl.\ Algebra (in press).
 Saito, M., Sturmfels, B., Takayama, N.:
 Gr\"obner Deformations of Hypergeometric Differential Equations.
 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.:
 On the theory of $b$-functions.