===================================================================
RCS file: /home/cvs/OpenXM/doc/Papers/rims2005-noro.tex,v
retrieving revision 1.2
retrieving revision 1.4
diff -u -p -r1.2 -r1.4
--- OpenXM/doc/Papers/rims2005-noro.tex	2005/12/19 01:19:47	1.2
+++ OpenXM/doc/Papers/rims2005-noro.tex	2006/03/28 01:03:23	1.4
@@ -1,10 +1,10 @@
 \documentclass{jarticle}
 %\usepackage[FVerb,theorem]{rims02}
 \topmargin -0.5in
-\oddsidemargin -0in
-\evensidemargin -0in
+\oddsidemargin -0.1in
+\evensidemargin -0.1in
 \textheight 9.5in
-\textwidth 6in
+\textwidth 6.4in
 \IfFileExists{my.sty}{\usepackage{my}}{}
 \IfFileExists{graphicx.sty}{\usepackage{graphicx}}{}
 \IfFileExists{epsfig.sty}{\usepackage{epsfig}}{}
@@ -151,7 +151,8 @@ $formula$ \>: Risa object
 $B4D$r$h$j0lHLE*$K$7$F$"$kE@$G0[$J$C$F$$$k(B 
 \footnote{Mathematica $B$K$*$$$F@Q$N(B {\tt Orderless} $BB0@-$r30$9$3$H$G(B,
 $B@Q$rHs2D49$K$G$-$k$,(B, $B4JC12=$K$*$$$F0[>o$J5sF0$r<($9$h$&$K$J$k(B (Ver. 4).
-Ver. 5 $B$G$O(B, $B78?t$^$GHs2D49$K$J$k(B.}.
+Ver. 5 $B$N=i4|$NHG$G$O(B, $B78?t$^$GHs2D49$K$J$C$?$,(B, $B:G6a$N$b$N$G$OD>$C$F$$$k(B
+$B$h$&$G$"$k(B.}.
 $B$5$i$K(B, $BI8=`7A$X$NJQ49;~$K(B, $B@Q$K4X$9$kJ,G[B'$rMxMQ$7$FE83+$5$l$?I8=`7A(B
 $B$rF@$k$3$H$b$G$-$k(B. 
 
@@ -283,7 +284,7 @@ $Condition$ $B$*$h$S(B $Action$ $B$K$O%f!<%6Dj5A4X?
 \FN $B4V$KA4=g=x$rF~$l$F(B, $B=q$-49$($?>l9g$K=g=x$,Bg$-$/(B($B>.$5$/(B)$B$J$k(B
 $B>l9g$K$N$_=q$-49$($r9T$&$H$$$&J}K!$G$"$k(B. $B$3$N>l9g(B, $B@Q$r9=@.$9$kM-8B8D(B
 $B$N(B \FN $B$NJB$YJQ$($NCf$G:G$b=g=x$,>e(B($B2<(B)$B$N$b$N$KE~C#$9$k$HDd;_$9$k(B. 
-$B$b$A$m$s(B, $Action$ $B$,J#;($J>l9g$K$O$3$N$h$&$K4JC1$K$O9T$+$J$$$,(B, 
+$Action$ $B$,J#;($J>l9g$K$O$3$N$h$&$K4JC1$K$O9T$+$J$$$,(B, 
 $B=q$-49$($NJ}8~@-$r<($9$b$N$H$7$FA4=g=x$rM?$($k$3$H$OM-8z$G$"$m$&(B.
 $B$h$C$F(B, $B=q$-49$(5,B'$K1~$8$F(B, $BA4=g=x$r$I$&A*$V$+$,LdBj$G$"$k(B.
 
@@ -313,29 +314,28 @@ $Condition$ $B$*$h$S(B $Action$ $B$K$O%f!<%6Dj5A4X?
 $B$K1~MQ$9$k$3$H$r9M$($k(B.
 
 $B78?t4D$r(B $K$ $B$N>e$G(B $ z_1, \ldots, z_n, h $
-$B$G@8@.$5$l$k<+M37k9gBe?t(B $A$ $B$r(B
-$$ K \langle z_1, \ldots, z_n, h \rangle  $$
+$B$G@8@.$5$l$k<+M37k9gBe?t(B $A$ $B$r(B $ K \langle z_1, \ldots, z_n, h \rangle  $
 $B$H=q$/(B.
 $h$ $B$rI,MW$K1~$8$F(B $z_{n+1}$ $B$H=q$/$3$H$b$"$k(B.
 \begin{definition}\rm
 $A$ $B$G$N=q$-49$(5,B'(B($B$^$?$O4X78<0(B, $B:8JU$OI,$:C19`<0(B)
-$$ L_1 \rightarrow R_1, \ldots,  L_m \rightarrow R_m  $$
+$ L_1 \rightarrow R_1, \ldots,  L_m \rightarrow R_m  $
 $B$,(B, $BF1<!2=(B weight $B%Y%/%H%k(B $H$ $B$K$D$$$F(B,
 $BF1<!E*=q$-49$(5,B'$G$"$k$H$O(B, 
 $R_i$ $B$,(B $0$ $B$G$"$k$+$^$?$O(B,
-$$ {\rm deg}_H(L_i) = {\rm deg}_H(R_i \mbox{$B$NG$0U$N9`(B}) $$
+$ {\rm deg}_H(L_i) = {\rm deg}_H(R_i \mbox{$B$NG$0U$N9`(B}) $
 $B$,@.N)$9$k$3$H$G$"$k(B.
 \end{definition}
 $B$3$3$G(B ${\rm deg}_H(\prod z_i^{e_i})$ $B$O(B
 $\prod z_i^{e_i}$ $B$N(B weight $H$ $B$K$D$$$F$N(B($BHs2D49@-$rL5;k$7$?(B)$B<!?t$G$"$k(B.
 $B$D$^$j(B
-$${\rm deg}_H(\prod z_i^{e_i}) = \sum e_i H_i $$
+${\rm deg}_H(\prod z_i^{e_i}) = \sum e_i H_i $
 $B$HDj5A$9$k(B ($i$ $B$O=EJ#$7$F$"$i$o$l$k$3$H$b$"$k(B).  
 
 \begin{example} \rm
-$$ z_2 z_1 \rightarrow z_1 z_2 + h^2 , 
+$ z_2 z_1 \rightarrow z_1 z_2 + h^2 , 
  h z_i \rightarrow z_i h
-$$
+$
 $B$O(B $H=(1,1,1)$ $B$K$D$$$F$NF1<!E*=q$-49$(5,B'$G$"$k(B. 
 $B$3$NNc$O(B $x=z_1, \partial = z_2$ $B$H$7$?(B
 1 $BJQ?t$NF1<!2=(B Weyl $BBe?t$K$[$+$J$i$J$$(B.
@@ -349,11 +349,11 @@ well order $\succ$ $B$r0J2<$R$H$D8GDj$9$k(B.
 
 \begin{example} \rm
 $BA0$NNc$N=q$-49$(5,B'(B
-$$ z_2 z_1 \rightarrow z_1 z_2 + h^2 , 
+$ z_2 z_1 \rightarrow z_1 z_2 + h^2 , 
  h z_i \rightarrow z_i h
-$$
+$
 $B$K$5$i$K(B
-$$ z_2^{p+1} \rightarrow 0, z_1 z_2 \rightarrow p h^2 $$
+$ z_2^{p+1} \rightarrow 0, z_1 z_2 \rightarrow p h^2 $
 $B$r2C$($?5,B'$N=89g$r(B $R_p$ $B$H=q$/(B. $B$3$3$G(B $p$ $B$O<+A3?t$G$"$k(B.
 $R_p$  $B$O(B $H=(1,1,1)$ $B$K$D$$$F$NF1<!E*=q$-49$(5,B'$G$"$k(B.
 \end{example}
@@ -375,19 +375,16 @@ $ \{ L_i \rightarrow R_i \} $
 \end{definition}
 %% z_2 z_1 --> z_1 z_2 + z_2 z_1  $BNc(B. z_1 > z_2 (lex) $B$H$9$k(B. $B$3$l$O$@$a(B.
 $B=q$-49$(5,B'$,$"$k@5?t%Y%/%H%k(B $H$ $B$K$D$$$FF1<!E*$G$"$k$3$H$+$i(B, 
-$B$3$l$i$N>r7o$K$h$j=q$-49$($,Dd;_@-$r$b$D$3$H$,J,$+$k(B. $B$5$i$K(B
-$B<!$NL?Bj$,@.$jN)$D(B.
+$B$3$l$i$N>r7o$K$h$j=q$-49$($,Dd;_@-$r$b$D$3$H$,J,$+$k(B. $B$5$i$K(B, 
 %%<hyperlink|G-algebra|http://www.singular.uni-kl.de/Manual/latest/sing_407.htm>
 %% z_j z_i \rightarrow c_{ij} z_i z_j + d_{ij}, i<j, c_{ij} \in K^*
 %% {\rm deg}_w(d_{ij}) \leq w_i + w_j
-\begin{proposition}\rm
 $G$-algebra \cite{LEV} $B$N>r7o$N$&$A(B, 
 well order $B$NB8:_>r7o$r2>Dj$7$J$/$F$b(B,
 $BE,Ev$JF1<!2=(Bweight$B%Y%/%H%k(B, $BM-8z(B weight $B%Y%/%H%k$,B8:_$9$k$J$i$P(B,
 $h$ $B$r2C$($k@F<!2=(B,
 $h$ $B$r(B $1$ $B$H$*$/$3$H$K$h$kHs@F2=$K$h$j(B,
-$B%0%l%V%J!<4pDl$r7W;;$G$-$k$h$&$K$J$k(B.
-\end{proposition}
+$B%0%l%V%J!<4pDl$r7W;;$G$-$k$h$&$K$J$k$HM=A[$5$l$k(B.
 
 $B$3$N1~MQ$K:]$7$F$O(B, $BM?$($i$l$?=q$-49$(5,B'$KBP$7(B, $BM-8z(B weight $B%Y%/%H%k(B
 $w$ $B$r8+$D$1$kI,MW$,$"$k(B. 
@@ -512,19 +509,19 @@ Risa/Asir $B$K$*$1$k?t<0$NCf4VE*I=8=$G$"$k(B \FN $B
 $BI,MW$G$"$k(B.
 
 $B$3$N=q$-49$($H(Bweight $B%Y%/%H%k$K$h$kC19`<0Hf3S$rAH$_9g$o$;$k$3$H$K$h$j(B, 
-$B<+M37k9gBe?t$K$*$1$k0lHLE*$J%0%l%V%J4pDl$N7W;;$rO@$8$?(B.
+$B<+M37k9gBe?t$K$*$1$k0lHLE*$J=q$-49$(7W;;$rO@$8$?(B.
 $B$3$3$GDs0F$7$?0lHL2=$O(B Weyl $BBe?t$NF1<!2=$NM}O@$r4^$`(B.  Risa/Asir $B$G?7(B
 $B$7$/F3F~$7$?(B, \QT $B$KBP$9$k0lHLE*$J(B weight $B%Y%/%H%k$N%a%+%K%:%`(B
 \verb@ qt_set_weight @ $B$K$h$j$o$l$o$l$NM}O@$H%"%k%4%j%:%`$N%W%m%H%?%$(B
 $B%W$rMF0W$K;n$9$3$H$,2DG=$G$"$k(B.  V. Levandovskyy \cite{LEV} $B$O(B 
-$G$-algebra $B$N35G0$rF3F~$7$F(B, Singular $B$K<BAu$7$?$,(B, $B$o$l$o$l$N35G0$O(B
-$BF1<!2=$r$H$*$7$F(B, well order $B$G$J$$>l9g$K$bE,MQ$G$-$k(B.  
-$B$o$l$o$l$N%"%W%m!<%A$K$h$j(B, 
+$G$-algebra $B$N35G0$rF3F~$7$F(B, Singular $B$K<BAu$7$?(B.
+$B$o$l$o$l$N%"%W%m!<%A$rH/E8$5$;(B, 
+$BF1<!2=$r$H$*$7$F(B, well order $B$G$J$$>l9g$K$bE,MQ$G$-$k(B 
+$B%0%l%V%J!<4pDl$NM}O@$r9=@.$9$l$P(B,  
 $G$-algebra $B$h$j9-$$HO0O$N(B algebra $B$r07$&$3$H$,2DG=$H$J$k(B.  
 $B0lHLE*$JOHAH$_$N1~MQ$H$7$F(B, $B>-MhE*$K$O(B $D$-$B2C72$N%"%k%4%j%:%`$r3HD%$7(B,
 Calderon-Moreno $BEy$NF3F~$7$?(B algebra $B$r6I=jE*$K07$&$J$I$N1~MQ$,8+9~$^(B
 $B$l$k(B.
-
 $B$^$?(B, \FN $B$r%f!<%68@8l$h$jA`:n$9$k4X?t$rMQ$$$k$3$H$K$h$j(B,
 $BF~NO(B, $B=PNO$N%f!<%6%$%s%?%U%'!<%9$rBgI}$K2~A1$G$-$k$3$H$K$b(B
 $BCm0U$7$F$*$-$?$$(B.