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version 1.2, 2005/12/19 01:19:47

version 1.4, 2006/03/28 01:03:23

Line 1

\documentclass{jarticle}

%\usepackage[FVerb,theorem]{rims02}

\topmargin -0.5in

\oddsidemargin -0in

\oddsidemargin -0.1in

\evensidemargin -0in

\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}}{}

Line 151 $formula$ \>: 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.

Line 283 $Condition$ $B$*$h$S(B $Action$ $B$K$O%f!<%6Dj5A4X?

Line 284 $Condition$ $B$*$h$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.

Line 313 $Condition$ $B$*$h$S(B $Action$ $B$K$O%f!<%6Dj5A4X?

Line 314 $Condition$ $B$*$h$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

$B$G@8@.$5$l$k<+M37k9gBe?t(B $A$ $B$r(B $ K \langle z_1, \ldots, z_n, h \rangle $

$$ K \langle z_1, \ldots, z_n, h \rangle $$

$B$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.

Line 349 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}

Line 375 $ \{ 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$3$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.

%%<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.

$B%0%l%V%J!<4pDl$r7W;;$G$-$k$h$&$K$J$k$HM=A[$5$l$k(B.

\end{proposition}

$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.

Line 512 Risa/Asir $B$K$*$1$k?t<0$NCf4VE*I=8=$G$"$k(B \FN $B

Line 509 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

$G$-algebra $B$N35G0$rF3F~$7$F(B, Singular $B$K<BAu$7$?(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$rH/E8$5$;(B,

$B$o$l$o$l$N%"%W%m!<%A$K$h$j(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.

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