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% $OpenXM: OpenXM/doc/Papers/bfct.tex,v 1.1 2001/02/06 07:54:18 noro Exp $

% $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}

Line 10

\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

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}

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}

\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.

Line 336 J. Pure Appl.\ Algebra (in press).

Line 365 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.

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