Annotation of OpenXM/doc/Papers/rims2005-noro.tex, Revision 1.4
1.1 takayama 1: \documentclass{jarticle}
2: %\usepackage[FVerb,theorem]{rims02}
3: \topmargin -0.5in
1.4 ! noro 4: \oddsidemargin -0.1in
! 5: \evensidemargin -0.1in
1.1 takayama 6: \textheight 9.5in
1.4 ! noro 7: \textwidth 6.4in
1.1 takayama 8: \IfFileExists{my.sty}{\usepackage{my}}{}
9: \IfFileExists{graphicx.sty}{\usepackage{graphicx}}{}
10: \IfFileExists{epsfig.sty}{\usepackage{epsfig}}{}
11: \newtheorem{definition}{$BDj5A(B}
12: \newtheorem{example}{$BNc(B}
13: \newtheorem{proposition}{$BL?Bj(B}
14: \newtheorem{remark}{$BCm0U(B}
15: \title{Risa/Asir $B$K$*$1$k?7$7$$7A<0$N?t<0$N<h$j07$$$K$D$$$F(B}
16: \author{$BLnO$(B $B@59T(B, $B9b;3?.5#(B \\ ($B?@8MBgM}(B)}
17: \date{}
18: \begin{document}
19: \maketitle
20: \def\gr{Gr\"obner $B4pDl(B}
21: \def\st{\, s.t. \,}
22: \def\noi{\noindent}
23: \def\Q{{\bf Q}}
24: \def\Z{{\bf Z}}
25: \def\NF{{\rm NF}}
26: \def\ini{{\rm in}}
27: \def\FN{{\tt FNODE}}
28: \def\QT{{\tt QUOTE}}
29: \def\ve{\vfill\eject}
30: \newcommand{\tmred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}\limits^{\scriptstyle *}}}
31:
32: \begin{abstract}
33: Risa/Asir $B$K$*$1$k=@Fp$J?t<0$N<h$j07$$$r<B8=$9$k$?$a(B,
34: $BLZ9=B$$GI=8=$5$l$??t<0$G$"$k(B \FN $B9=B$BN$rJ];}$9$k(B \QT $B7?$r07$($k$h$&(B
35: $B$K$7$?(B. $BI8=`2=$r$O$8$a$H$9$k(B \QT $B$KBP$9$k<o!9$NA`:n(B, $B$*$h$S(B
36: $B%Q%?!<%s%^%C%A%s%0$K$h$k=q$-49$($r<BAu$7$?(B. $B$3$l$i$rMQ$$$?(B
37: $B$$$/$D$+$NNc$r<($9(B. $B$^$?(B, weight $B$rMQ$$$?(B
38: $BHs2D49Be?t$K$*$1$k=q$-49$($N2DG=@-$K$D$$$F=R$Y$k(B.
39: \end{abstract}
40:
41: \section{Risa/Asir $B$K$*$1$k?t<0$N<h$j07$$(B}
42: Risa/Asir $B$K$*$$$F$O(B, $B%f!<%6$K$h$jF~NO$5$l$??t<0$O(B, $B$$$C$?$s(B \FN $B$H8F(B
43: $B$P$l$kLZ9=B$$KJQ49$5$l$?$N$A(B, {\tt eval()} $B$K$h$j:F5"E*$K@55,FbItI=8=(B
44: (Risa $B%*%V%8%'%/%H(B) $B$KJQ49$5$l$k(B. Risa $B%*%V%8%'%/%H$H$O(B, $B@hF,$K6&DL$N(B
45: $B<1JL;R%U%#!<%k%I$r;}$D0l72$N9=B$BN$G$"$j(B, {\tt arf\_add()} $B$J$I$N%H%C(B
46: $B%W%l%Y%k1i;;4X?t$O(B, $B<u$1<h$C$?9=B$BN$N<1JL;R$r8+$F(B, $BE,@Z$J4X?t$K?6$jJ,(B
47: $B$1$k$H$$$&A`:n$r9T$&(B. Risa $B%*%V%8%'%/%H$H$7$F$O(B, $B?t(B, $BB?9`<0(B, $BM-M}<0(B,
48: $B%j%9%H(B, $BG[Ns$J$I(B30 $B<oN`<e$,Dj5A$5$l$F$$$k(B. $B$5$i$K(B, $B?t$O(B, $BM-M}?t(B, $BIbF0(B
49: $B>.?t(B, $BM-8BBN$J$I$5$i$K:Y$+$/J,N`$5$l$k(B. $B$$$C$?$s(B Risa $B%*%V%8%'%/%H$KJQ(B
50: $B49$5$l$F$7$^$($P(B, $B$=$l$>$l8GM-$NJ}K!$K$h$j(B, $B8zN($h$$1i;;$,E,MQ$G$-$k$,(B,
51: $B0lJ}$G(B, $B$?$H$($PB?9`<0$,6/@)E*$KE83+$5$l$F$7$^$&$J$I(B, $BK\Mh$NF~NO$,;}$C(B
52: $B$F$$$?>pJs$,<:$o$l$k$3$H$b$"$k(B. $B$^$?(B, $B86B'$H$7$FB?9`<0$N@Q$O2D49$H2>Dj(B
53: $B$5$l$F$$$k$?$a(B, $BHyJ,:nMQAG$J$I(B, $BHs2D49$JBP>]$r07$&>l9g$KIT<+A3$JA`:n$r(B
54: $B6/$$$i$l$FMh$?(B.
55:
56: \begin{example}
57:
58: {\tt dx} $B$r(B $\partial/\partial x$ $B$N0UL#$K;H$*$&$H;W$C$F$b(B
59: \begin{verbatim}
60: [0] x*dx;
61: dx*x;
62: [1] dx*x;
63: dx*x;
64: \end{verbatim}
65: $B$N$h$&$K(B, $B>!<j$K=g=x$,JQ$($i$l$F$7$^$&(B.
66: \end{example}
67: $B$^$?(B, $B0JA0$+$i;XE&$5$l$F$$$k(B, Risa/Asir $B$K<0$N4JC12=5!G=$,7gG!$7$F$$$kE@(B
68: $B$K$D$$$F$b(B, $B$"$i$f$k$b$N$rB?9`<0$KJQ49$7$F$+$i4JC12=$9$k$N$OIT<+A3$G$"$k(B.
69:
70: $B$3$N$h$&$J(B, $B?t<0$N=@Fp$J<h$j07$$$O(B, Maxima, Maple, Mathematica $B$J$I(B
71: $B$NF@0U$H$9$k$H$3$m$G$"$j(B, $B$3$l$^$G$O(B, Risa/Asir $B$NL\;X$9$H$3$m$O(B, $BB?9`<0(B
72: $B1i;;$N9bB.=hM}$G$"$k$H$7$F(B, $BFC$K$3$N$h$&$JJ}8~$N3+H/$O?J$a$F$3$J$+$C$?(B.
73: $B$7$+$7(B, $B;H$o$lJ}$,B?MM2=$7$?7k2L(B, $B$h$jB?MM$J?t<0$N<h$j07$$$,I,MW$H$J$k(B
74: $B>lLL$,B?$/$J$C$F$-$?$?$a(B, $B$h$j0lHL$N?t<0$N1i;;$*$h$S4JC12=(B, $B=q$-49$(5,B'$K$h$k(B
75: $B=q$-49$($N<BAu$KCe<j$7$?(B.
76:
77: \section{\QT $B7?(B}
78:
79: $BA0@a$G=R$Y$?$h$&$K(B, Risa/Asir $B$K$*$$$F$O(B, $BF~NO$5$l$??t<0$O(B,
80: Risa $B%*%V%8%'%/%H$KJQ49$5$l$kA0$K(B, \FN $B$H8F$P$l$kLZ9=B$$GJ];}$5$l$F$$$k(B.
81: $B$3$N(B \FN $B$r%\%G%#It$K;}$D(B Risa $B%*%V%8%'%/%H$G$"$k(B \QT $B7?$r(B
82: $BDj5A$7$?(B. $B$3$l$K$h$j(B, $BI>2AA0$NLZ9=B$$rJ];}$G$-$k(B.
83:
84: \subsection{\QT $B$NF~NO(B, $B4pK\A`:n(B}
85:
86: \QT $B7?$KBP$9$k;MB'1i;;$J$I$N4pK\1i;;$O(B, $BLZ$KBP$9$kA`:n$H$7$FDj5A$9$k(B. $B$5$i$K(B,
87: $BLZ$KBP$9$k0lHLE*$JA`:n(B ($BB0@-(B, $B;R$N<h$j=P$7(B, $BLZ$N:F9=@.$J$I(B) $B$r(B Asir
88: $B$N4X?t$H$7$FM?$($k$3$H$G(B, $B%f!<%6$K$h$k?t<0$NA`:n$,2DG=$H$J$k(B.
89: \QT $B7?$KBP$9$kA`:n$O(B, $B<B:]$K$O(B \FN $B$KBP$9$kA`:n$G$"$k(B.
90: \FN $B$O(B
91: \begin{center}
92: ($id$ $arg_0$ $arg_1$ $\ldots$)
93: \end{center}
94: $B$H$$$&%j%9%H$GI=8=$5$l(B, $arg_i$ $B$N8D?t(B, $B7?$O(B $id$ $B$K$h$j$5$^$6$^$G$"$k(B.
95: \QT $B$NF~NO(B, $BJQ49$J$I$N4pK\A`:n$O<!$NDL$j$G$"$k(B.
96:
97: \begin{itemize}
98: \item \QT $B$NF~NO(B
99:
100: \QT $B$O(B {\tt quote}($Expr$) $B$^$?$O(B {\tt `}$Expr$ ($B%P%C%/%/%)!<%H$D$-(B)
101: $B$K$h$jF~NO$G$-$k(B.
102:
103: \item \QT $B$H(B Risa $B%*%V%8%'%/%H$NAj8_JQ49(B
104:
105: Risa $B%*%V%8%'%/%H$+$i(B \QT $B$r@8@.$9$k$N$O(B {\tt objtoquote}($Obj$),
106: $B5U$K(B \QT $B$rI>2A$7$F(B Risa $B%*%V%8%'%/%H$r@8@.$9$k$N$O(B {\tt eval\_quote}($Expr$)
107: $B$G9T$&(B.
108:
109: \item \QT $B$NJ,2r(B, $B9g@.(B
110:
111: {\tt quote\_to\_funargs}($Expr$) $B$O(B
112: \QT $Expr$ $B$N(B \FN $B$N<1JL;R(B, $B0z?t$r%j%9%H$H$7$FJV$9(B.
113:
114: {\tt funargs\_to\_quote}($List$) $B$O(B, $B$=$N5U$G$"$k(B.
115: \end{itemize}
116:
117: \subsection{\FN $B$NI8=`7A(B}
118: \FN $B$KBP$9$k%Q%?!<%s%^%C%A%s%0(B, $B=q$-49$($rMF0W$K9T$&$?$a$K(B,
119: \FN $B$KBP$9$kI8=`7A$rDj5A$7$?(B. $BI8=`7A$N7W;;$O(B {\tt qt\_normalize}($Expr$[,$Mode$])
120: $B$G9T$&(B. $Mode$ $B$O8e=R$9$kE83+%b!<%I;XDj$G$"$k(B.
121:
122: \begin{tabbing}
123: AAAAAAAAAAA \= \kill
124: $nf$ \> : $formula$ $|$ $functor$ ($nf$ [, $\ldots$]) $|$ $sum\_of\_monom$\\
125: $sum\_of\_monom$ \>: $monom$ [$+$ $\cdots$]\\
126: $monom$ \>: [$formula$ $*$ ] $nfpow$ [$*$ $\cdots$]\\
127: $nfpow$ \>: $nf$ $|$ $nf^{nf}$\\
128: $formula$ \>: Risa object
129: \end{tabbing}
130: $B$*$*$6$C$Q$K$$$($P(B, $BI8=`7A(B $nf$ $B$H$O(B, $BI8=`7A$N%Y%-@Q$N(B Risa $B%*%V%8%'%/%H(B
131: $B78?t$D$-$NOB$G$"$k(B.
132: $B$3$3$G(B, $BOB$O(B \FN $B$H$7$F$O(B, n$B9`OB$H$7$FI=8=$5$l(B, $BOB$r9=@.$9$kC19`<0(B
133: $B$O(B, $B$"$kA4=g=x$K$h$j@0Ns$5$l$k(B. $B$^$?(B, $B@Q$b(B n$B9`@Q$H$7$FI=8=$5$l$k(B.
134: $B$9$J$o$A(B, $BI8=`7A$O(B, $BF~NO$5$l$??t<0$,(B, Risa $B%*%V%8%'%/%H$r78?t4D$H$9$k(B
135: $B7k9gBe?t$N85$G$"$k$H8+$J$7(B, $BOB$N2D49@-(B, $B@Q$N7k9g@-$K$h$j%U%i%C%H$K(B
136: $B@0M}$7$J$*$7$?$b$N$G$"$k(B.
137: \begin{example}
138: \begin{verbatim}
139:
140: [278] ctrl("print_quote",1)$ /* FNODE $B$r%j%9%H$GI=<((B */
141: [279] `(x+y+z);
142: [u_op,(),[b_op,+,[b_op,+,[internal,x],[internal,y]],[internal,z]]]
143: [280] qt_normalize(`(x+y+z));
144: [n_op,+,[internal,x],[internal,y],[internal,z]]
145: \end{verbatim}
146: 2 $B9`1i;;$GI=8=$5$l$?<0$,(B, $BI8=`7A$G$O(B n $B9`OB$GI=8=$5$l$F$$$k$3$H$,J,$+$k(B.
147: \end{example}
148:
149: $B$3$l$O(B, Mathematica $B$K$*$1$kI8=`7A(B \cite{MMA}
150: $B$H4pK\E*$KF1$8$G$"$k$,(B, $B@Q$N2D49@-$r2>Dj$7$F$$$J$$$3$H(B, $B$*$h$S(B, $B78?t(B
151: $B4D$r$h$j0lHLE*$K$7$F$"$kE@$G0[$J$C$F$$$k(B
152: \footnote{Mathematica $B$K$*$$$F@Q$N(B {\tt Orderless} $BB0@-$r30$9$3$H$G(B,
153: $B@Q$rHs2D49$K$G$-$k$,(B, $B4JC12=$K$*$$$F0[>o$J5sF0$r<($9$h$&$K$J$k(B (Ver. 4).
1.3 noro 154: Ver. 5 $B$N=i4|$NHG$G$O(B, $B78?t$^$GHs2D49$K$J$C$?$,(B, $B:G6a$N$b$N$G$OD>$C$F$$$k(B
155: $B$h$&$G$"$k(B.}.
1.1 takayama 156: $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
157: $B$rF@$k$3$H$b$G$-$k(B.
158:
159: \begin{example}
160: \begin{verbatim}
161:
162: [289] ctrl("print_quote",2)$ /* FNODE $B$r<0$GI=<((B */
163: ctrl(``print_quote'',2)$
164: [290] qt_normalize(`(x+y)^2);
165: ((x)+(y))^(2)
166: [291] qt_normalize(`(x+y)^2,1);
167: ((x)^(2))+((x)*(y))+((y)*(x))+((y)^(2))
168: \end{verbatim}
169: \end{example}
170:
171: \subsection{$B9`=g=x$*$h$S78?t4D$N@_Dj(B}
172:
173: $BC19`<0=g=x$*$h$S78?t4D$O2DJQ$G$"$j(B, $B$=$l$>$l<!$N$h$&$J4X?t$,MQ0U$5$l$F$$$k(B.
174:
175: \begin{itemize}
176: \item $BC19`<0=g=x$N@_Dj(B
177:
178: $BI8=`7ACf$NC19`<0=g=x$O(B, $B;XDj$,$J$$>l9g$K$O(B, $B%7%9%F%`$,7h$a$kITDj85(B
179: $B$*$h$S4X?t;R$N=g=x$+$iM6F3$5$l$k<-=q<0=g=x$,E,MQ$5$l$k(B. $B$=$N:](B, $B4X?t8F$S=P$7$O(B,
180: $BC1$J$kITDj85$h$j=g=x$,>e$G(B, $B4X?t;R$,Ey$7$$>l9g$K$O0z?t$,<-=q<0$KHf3S$5$l$k(B.
181: {\tt qt\_set\_ord}($VarList$) $B$K$h$j(B, $VarList$ $B$K8=$l$kITDj85$r@hF,(B
182: $B$H$7(B, $B$N$3$j$r%7%9%F%`$,7h$a$k$H$$$&=g=x$,@_Dj$5$l$k(B.
183:
184: %$BNc(B
185:
186: \item $B78?t4D$N@_Dj(B
187:
188: $B%G%U%)%k%H$G$O78?t4D$O?t$N$_$+$i$J$k$,(B, $B$$$/$D$+$N%Q%i%a%?$r78?t4D$N85(B
189: $B$H$7$F07$$$?$$>l9g(B, {\tt qt\_set\_coef}($ParamList$) $B$K$h$j;XDj$G$-$k(B.
190: $ParamList$ $B$K;XDj$5$l$?%Q%i%a%?$O(B, $B78?t4D$G$"$k2D49$JM-M}4X?tBN$NITDj85(B
191: $B$H$7$F07$o$l$k(B.
192:
193: \end{itemize}
194: \begin{example}
195: \begin{verbatim}
196:
197: [304] qt_normalize(`(b*x+a*y)*b*y,1);
198: ((a)*(y)*(b)*(y))+((b)*(x)*(b)*(y))
199: [305] qt_set_coef([a,b])$
200: [306] qt_normalize(`(b*x+a*y)*b*y,1);
201: ((b^2)*(x)*(y))+((b*a)*((y)^(2))) /* a,b $B$,78?t4D$KF~$C$?(B; x*y > y^2 */
202: [307] qt_set_ord([y,x])$
203: [308] qt_normalize(`(b*x+a*y)*b*y,1);
204: ((b*a)*((y)^(2)))+((b^2)*(x)*(y)) /* y^2 > x*y */
205: \end{verbatim}
206: \end{example}
207:
208:
209: \section{$B%Q%?!<%s%^%C%A%s%0$K$h$k=q$-49$((B}
210:
211: Risa/Asir $B$K$*$$$F$O(B, $BITDj85$H%W%m%0%i%`JQ?t$OL@3N$K6hJL$5$l$F$$$k(B.
212: $B$=$3$G(B, $B%Q%?!<%sJQ?t$H$7$F%W%m%0%i%`JQ?t$rMQ$$$k$3$H$K$7$?(B.
213: $B$9$J$o$A%Q%?!<%s$H$O(B, $B%W%m%0%i%`JQ?t$r4^$s$G$b$h$$(B \QT $B$G$"$k(B. $B$3$l(B
214: $B$KBP$7(B, $B$$$/$D$+$N=q$-49$(4X?t$rMQ0U$7$?(B.
215: \begin{itemize}
216: \item {\tt nqt\_match}($Expr$,$Patten$[,$Mode$])
217:
218: \QT $B<0(B $Expr$ $B$H%Q%?!<%s(B $Pattern$ $B$,%^%C%A$7$?$i(B 1 $B$rJV$9(B. $B$5$i$K(B,
219: $Pattern$ $BCf$K4^$^$l$k%W%m%0%i%`JQ?t$K%^%C%A$7$?CM$,<B:]$KBeF~$5$l$k(B.
220:
221: \item {\tt nqt\_match\_rewrite}($Expr$,$Rule$,$Mode$)
222:
223: $Rule$ $B$O(B [$Pattern$,$Action$] $B$^$?$O(B [$Pattern$,$Condition$,$Action$] $B$G(B
224: $B$"$k(B. $B$3$N4X?t$O(B, $Expr$ $B$,(B $Pattern$ $B$K%^%C%A$7$?$i(B, $Action$ $B$,I>2A(B
225: $B$5$l(B, $B$=$NCM$,JV$5$l$k(B.
226: $B$=$N:](B, $Action$ $BCf$N%Q%?!<%sJQ?t$,(B, $B%^%C%A$7$?CM$KCV$-49$($i$l$k(B.
227: $Condition$ $B$,;XDj$5$l$F$$$k>l9g$K$O(B, $Condition$ $BCf$N%Q%?!<%sJQ?t$,F1MM(B
228: $B$KCV$-49$($i$lI>2A$5$l(B, 0 $B$G$J$$>l9g$K(B $Action$ $B$,I>2A$5$l$k(B. $B%^%C%A$7$J$$(B
229: $B>l9g$K$O(B $Expr$ $B$=$N$b$N$,JV$5$l$k(B.
230: \end{itemize}
231: \begin{example}
232: \begin{verbatim}
233:
234: [318] nqt_match(`x*y*z-3*u,`X*Y+Z);
235: 1
236: [319] [X,Y,Z];
237: [x,(y)*(z),(-3)*(u)]
238: [320] nqt_match_rewrite(`x*y*z,[`X*Y,`X+Y],1);
239: ((y)*(z))+(x)
240: \end{verbatim}
241: \end{example}
242:
243: $B$$$:$l$b<B9TA0$K0z?t$,I8=`7A$KJQ49$5$l$k$,(B, $Mode$ $B$O$=$N:]$KE83+$r9T$&$+(B
244: $B$I$&$+$r;X<($9$k(B. $B%^%C%A%s%0$K$*$$$F$O(B, $B:G=i$K%^%C%A$7$?;~E@$N>pJs$,JV$5$l$k(B
245: \footnote{$B8=>u$G$O<BAu$,IT40A4$G$"$j(B, $BF10l%Q%?!<%sJQ?t$,J#?t8=$l$k%Q%?!<%s(B
246: $B$KBP$7$F$O%^%C%A%s%0$K<:GT$9$k>l9g$,$"$k(B.}.
247: $Condition$ $B$*$h$S(B $Action$ $B$K$O%f!<%6Dj5A4X?t$r4^$a$k$3$H$,$G$-$k(B.
248: $B$3$l$K$h$j(B, $BJ#;($J=q$-49$(5,B'$r=q$/$3$H$,$G$-(B, $B$^$?=q$-49$(5,B'$N?t$r>/$J$/2!$($k(B
249: $B$3$H$,$G$-$k(B.
250: $B8=>u$G$O(B, Mathematica $B$G2DG=$J(B, $B%Q%?!<%sJQ?t$K%^%C%A$9$k7?$N;XDj$,$G$-(B
251: $B$J$$$?$a(B, $Condition$ $B$K$*$$$F7?H=Dj$r9T$&$3$H$K$J$k(B. $B$3$N$?$a(B, \QT $B$K(B
252: $BBP$9$k$$$/$D$+$N7?H=Dj4X?t$rMQ0U$7$?(B. $B$3$l$i$rMQ$$$F(B, $B=q$-49$(5,B'=89g$r(B
253: $BM?$($F(B, $B=q$-49$(5,B'$,E,MQ$G$-$J$k$J$k$^$G=q$-49$($rB3$1$k4X?t(B
254: {\tt qt\_rewrite}($Expr$,$Rules$,$Mode$) $B$r%f!<%64X?t$H$7$F5-=R$7$?(B.
255:
256: \begin{example}[$sl_2$$B$NE83+4D(B]
257: \begin{verbatim}
258:
259: [336] Rsl=[[`h*e,`e*h+2*e],[`h*f,`f*h-2*f],[`e*f,`f*e+h]]$
260: 0sec(7e-06sec)
261: [337] qt_rewrite(`e*f^2,Rsl,2);
262: ((f)*(f)*(e))+((2)*(f)*(h))+((-2)*(f))
263: 1.776e-15sec(0.008608sec)
264: [338] qt_rewrite(`h*e^3,Rsl,2);
265: ((e)*(e)*(e)*(h))+((6)*(e)*(e)*(e))
266: \end{verbatim}
267: \end{example}
268:
269:
270: \section{\FN $B$N=g=x$E$1(B}
271:
272: $B:#2s$N<BAu$NL\E*$O(B, $B%f!<%6$,5$7Z$K=q$-49$(5,B'$rM?$($F(B, $B0lHL$KHs2D49$JBe?t(B
273: $B$K$*$1$k7W;;$r5$7Z$K;n$;$k$h$&$J4D6-$r:n$k$3$H$G$"$k(B. $BM?$($i$l$?=q$-49$(5,B'(B
274: $B$NDd;_@-(B, $B$"$k$$$O9gN.@-$K4X$7$F$O(B, $B9`=q$-49$(7O$N8&5f<T$K$h$k8&5f$,KDBg(B
275: $B$K$"$k$,(B, $B$3$3$G$O?<F~$j$O$7$J$$(B. $B$3$3$G$O(B, $BL58B%k!<%W$K4Y$i$J$$$h$&$J(B
276: $B<BMQE*$J;X?K$H$7$F(B, \FN $B$KBP$9$k=g=x$E$1$*$h$S(B weight $B$N;HMQ$rDs0F$9$k(B.
277: $B$3$NJ}K!$O8e=R$9$k$h$&$KB?9`<04D$dHyJ,:nMQAG4D$GMQ$$$i$l$k(B weight
1.2 takayama 278: $B%Y%/%H%k$N9M$(J}$N<+A3$J0lHL2=$G$"$j(B, $BM}O@E*$K$b6=L#?<$$(B.
1.1 takayama 279:
280: $BNc$H$7$F(B, $B2D49@-$rDj5A$9$k>l9g$r9M$($k(B. $B?t3XE*$K$O(B, $BG$0U$N(B $X$, $Y$ $B$K(B
281: $BBP$7(B $XY=YX$ $B$G$h$$$,(B, $B$?$H$($P$3$N$^$^(B $[`X*Y,`Y*X]$ $B$H$$$&=q$-49$((B
282: $B5,B'$r(B
283: $B=q$/$H$b$A$m$sDd;_$7$J$$(B. $B$3$N>l9g(B, $B:G$b0BD>$J2r7hJ}K!$N0l$D$O(B,
284: \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
285: $B>l9g$K$N$_=q$-49$($r9T$&$H$$$&J}K!$G$"$k(B. $B$3$N>l9g(B, $B@Q$r9=@.$9$kM-8B8D(B
286: $B$N(B \FN $B$NJB$YJQ$($NCf$G:G$b=g=x$,>e(B($B2<(B)$B$N$b$N$KE~C#$9$k$HDd;_$9$k(B.
1.4 ! noro 287: $Action$ $B$,J#;($J>l9g$K$O$3$N$h$&$K4JC1$K$O9T$+$J$$$,(B,
1.1 takayama 288: $B=q$-49$($NJ}8~@-$r<($9$b$N$H$7$FA4=g=x$rM?$($k$3$H$OM-8z$G$"$m$&(B.
289: $B$h$C$F(B, $B=q$-49$(5,B'$K1~$8$F(B, $BA4=g=x$r$I$&A*$V$+$,LdBj$G$"$k(B.
290:
291: \subsection{\FN $B$N(B weight$B$H=q$-49$((B}
292:
293: $B0lHL$K(B \FN $f$ $B$N(B weight $w(f)$ $B$r(B
294: \begin{enumerate}
295: \item $f$ $B$,(B leaf $B$N>l9g(B, $BE,Ev$JCM$rM?$($k(B. $BFC$K78?t$N(B weight $B$O(B 0.
296: \item $f$ $B$,(B node $B$N>l9g(B, $f$ $B$N;R$N(B weight $BCM$r0z?t$H$7(B,
297: $B<1JL;R$G7h$a$i$l$?4X?t$r7W;;$7$F$=$NCM$r$H$k(B.
298: \end{enumerate}
299: $B$K$h$j:F5"E*$K7h$a$k$3$H$,$G$-$k(B. $BOB$KBP$7$F$O(B $\max()$,
300: $B@Q$KBP$7$F$OOB(B, $B%Y%-$KBP$7$F$O@Q$rMQ$$$k$H(B, $B<!$N$h$&$K$J$k(B.
301: \begin{enumerate}
302: \item $w(f+g)=\max(w(f),w(g))$
303: \item $w(fg) = w(f)+w(g)$
304: \item $w(f^n)=nw(f)$
305: \end{enumerate}
306: % $B$h$C$F(B, $BM?$($i$l$?=q$-49$(5,B'=89g$KBP$7(B, $B$3$N$h(B
307: %$B$&$J(B weight $B$r8+$D$1$k(B, $B$9$J$o$A(B leaf $B$NCM$rE,@Z$K@_Dj$9$k$3$H$,=EMW$G(B
308: %$B$"$k(B. $B$3$N$h$&$J(B weight $B$,:n$l$k%/%i(B
309: %$B%9$rM?$($k$3$H(B, $B$*$h$S$=$N$h$&$J%/%i%9$KB0$9$k=q$-49$(5,B'=89g$KBP$7(B, $B>e$N(B
310: %$B@-<A$rK~$?$9(B weight $B$rA4$FM?$($k$3$H$O6=L#?<$$LdBj$G$"$k(B.
311: %\subsection{$B<+M37k9gBe?t$K$*$1$kF1<!=q$-49$(5,B'(B}
312:
313: $B0J2<$G$O(B, $B$3$N$h$&$J(B weight $B$rM-8B@8@.$N<+M37k9gBe?t$KBP$9$k=q$-49$((B
314: $B$K1~MQ$9$k$3$H$r9M$($k(B.
315:
316: $B78?t4D$r(B $K$ $B$N>e$G(B $ z_1, \ldots, z_n, h $
1.4 ! noro 317: $B$G@8@.$5$l$k<+M37k9gBe?t(B $A$ $B$r(B $ K \langle z_1, \ldots, z_n, h \rangle $
1.1 takayama 318: $B$H=q$/(B.
319: $h$ $B$rI,MW$K1~$8$F(B $z_{n+1}$ $B$H=q$/$3$H$b$"$k(B.
320: \begin{definition}\rm
321: $A$ $B$G$N=q$-49$(5,B'(B($B$^$?$O4X78<0(B, $B:8JU$OI,$:C19`<0(B)
1.4 ! noro 322: $ L_1 \rightarrow R_1, \ldots, L_m \rightarrow R_m $
1.1 takayama 323: $B$,(B, $BF1<!2=(B weight $B%Y%/%H%k(B $H$ $B$K$D$$$F(B,
324: $BF1<!E*=q$-49$(5,B'$G$"$k$H$O(B,
325: $R_i$ $B$,(B $0$ $B$G$"$k$+$^$?$O(B,
1.4 ! noro 326: $ {\rm deg}_H(L_i) = {\rm deg}_H(R_i \mbox{$B$NG$0U$N9`(B}) $
1.1 takayama 327: $B$,@.N)$9$k$3$H$G$"$k(B.
328: \end{definition}
329: $B$3$3$G(B ${\rm deg}_H(\prod z_i^{e_i})$ $B$O(B
330: $\prod z_i^{e_i}$ $B$N(B weight $H$ $B$K$D$$$F$N(B($BHs2D49@-$rL5;k$7$?(B)$B<!?t$G$"$k(B.
331: $B$D$^$j(B
1.4 ! noro 332: ${\rm deg}_H(\prod z_i^{e_i}) = \sum e_i H_i $
1.1 takayama 333: $B$HDj5A$9$k(B ($i$ $B$O=EJ#$7$F$"$i$o$l$k$3$H$b$"$k(B).
334:
335: \begin{example} \rm
1.4 ! noro 336: $ z_2 z_1 \rightarrow z_1 z_2 + h^2 ,
1.1 takayama 337: h z_i \rightarrow z_i h
1.4 ! noro 338: $
1.1 takayama 339: $B$O(B $H=(1,1,1)$ $B$K$D$$$F$NF1<!E*=q$-49$(5,B'$G$"$k(B.
340: $B$3$NNc$O(B $x=z_1, \partial = z_2$ $B$H$7$?(B
341: 1 $BJQ?t$NF1<!2=(B Weyl $BBe?t$K$[$+$J$i$J$$(B.
342: \end{example}
343:
344: $B0J2<(B $H$ $B$N$9$Y$F$N@.J,$O@5$G$"$k$H2>Dj$78GDj$9$k(B.
345: $B$^$?(B$x_1, \ldots, x_n, h $ $B$+$i$J$k%o!<%I$KBP$9$k(B
346: well order $\succ$ $B$r0J2<$R$H$D8GDj$9$k(B.
347: $B=P8=$9$k=q$-49$(5,B'$O$H$/$K$3$H$o$i$J$$8B$jA4$F(B$H$ $B$K$D$$$FF1<!E*=q$-49$(5,B'(B
348: $B$G$"$k(B.
349:
350: \begin{example} \rm
351: $BA0$NNc$N=q$-49$(5,B'(B
1.4 ! noro 352: $ z_2 z_1 \rightarrow z_1 z_2 + h^2 ,
1.1 takayama 353: h z_i \rightarrow z_i h
1.4 ! noro 354: $
1.1 takayama 355: $B$K$5$i$K(B
1.4 ! noro 356: $ z_2^{p+1} \rightarrow 0, z_1 z_2 \rightarrow p h^2 $
1.1 takayama 357: $B$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.
358: $R_p$ $B$O(B $H=(1,1,1)$ $B$K$D$$$F$NF1<!E*=q$-49$(5,B'$G$"$k(B.
359: \end{example}
360:
361: \begin{definition} \rm
362: $n$ $B<!85$N(B weight $B%Y%/%H%k(B $w \in {\bf R}^n $
363: $B$,F1<!E*=q$-49$(5,B'(B
364: $ \{ L_i \rightarrow R_i \} $
365: $B$*$h$S(B $\succ$ $B$K$D$$$F(B
366: $BM-8z(B weight $B%Y%/%H%k(B(admissible weight vector) $B$G$"$k$H$O<!$N>r7o$r$_$?$9(B
367: $B$3$H$G$"$k(B.
368: $B0J2<(B $\tilde w = (w,0) $ ($h$ $B$KBP$9$k(B weight $B$r(B 0 $B$K$7$?$b$N(B)
369: $B$H$*$/(B.
370: \begin{enumerate}
371: \item ${\rm deg}_{\tilde w}(L_i) \geq {\rm deg}_{\tilde w}(R_i)$
372: \item $B:8JU$H1&JU$,F1$8(B $w$-$B<!?t$r$b$D$H$-$O(B $B1&JU$G:8JU$HF1$8(B $w$-weight $B$r;}$D(B
373: $B9`$?$A$O=g=x(B $\succ$ $B$G$+$J$i$:>.$5$$(B.
374: \end{enumerate}
375: \end{definition}
376: %% z_2 z_1 --> z_1 z_2 + z_2 z_1 $BNc(B. z_1 > z_2 (lex) $B$H$9$k(B. $B$3$l$O$@$a(B.
377: $B=q$-49$(5,B'$,$"$k@5?t%Y%/%H%k(B $H$ $B$K$D$$$FF1<!E*$G$"$k$3$H$+$i(B,
1.3 noro 378: $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,
1.1 takayama 379: %%<hyperlink|G-algebra|http://www.singular.uni-kl.de/Manual/latest/sing_407.htm>
1.2 takayama 380: %% z_j z_i \rightarrow c_{ij} z_i z_j + d_{ij}, i<j, c_{ij} \in K^*
381: %% {\rm deg}_w(d_{ij}) \leq w_i + w_j
382: $G$-algebra \cite{LEV} $B$N>r7o$N$&$A(B,
1.1 takayama 383: well order $B$NB8:_>r7o$r2>Dj$7$J$/$F$b(B,
384: $BE,Ev$JF1<!2=(Bweight$B%Y%/%H%k(B, $BM-8z(B weight $B%Y%/%H%k$,B8:_$9$k$J$i$P(B,
385: $h$ $B$r2C$($k@F<!2=(B,
386: $h$ $B$r(B $1$ $B$H$*$/$3$H$K$h$kHs@F2=$K$h$j(B,
1.3 noro 387: $B%0%l%V%J!<4pDl$r7W;;$G$-$k$h$&$K$J$k$HM=A[$5$l$k(B.
1.1 takayama 388:
389: $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
390: $w$ $B$r8+$D$1$kI,MW$,$"$k(B.
391: $B$?$H$($P0lJQ?t%o%$%kBe?t$N>l9g(B $w_1 + w_2 \geq 0$ $B$N>r7o$r$_$?$5$J$$$H(B
392: $BM-8z(B weight $B%Y%/%H%k$H$J$i$J$$(B.
393: $B$3$N$H$-F1;~2=(B weight $B%Y%/%H%k$rMQ$$$F=q$-49$(5,B'$N1&JU$r@F<!2=(B
394: $B$9$l$P(B, $BF1<!E*=q$-49$(5,B'$,F@$i$l$k(B.
395:
396: $B8=:_$N<BAu$K$*$$$F$O(B, weight $B%Y%/%H%k$,@_Dj$5$l$J$$8B$j(B, weight $B$K(B
397: $B$h$kHf3S$O9T$o$J$$(B.
398: $B4X?t(B {\tt qt\_set\_weight()} $B$K$h$j(B
399: $B0lIt$NITDj85$KBP$7$F(B weight $B$,@_Dj$5$l$k$H(B,
400: $BB>$NITDj85$N(B weight $B$O<+F0E*$K(B 0 $B$H$J$k(B.
401: $B$3$N(B weight $B$rMQ$$$?(B $B<!?t$NHf3S8e$K8=:_@_Dj(B
402: $B$5$l$F$$$kC19`<0=g=x$,E,MQ$5$l$k(B.
403:
404: \begin{example}
405: \begin{verbatim}
406:
407: [300] qt_set_ord([z1,z2,h])$
408: [301] qt_set_weight([[z1,-1],[z2,1]])$
409: [302] Rule1=[[`h*z1,`z1*h], [`h*z2,`z2*h], [`z2*z1,`z1*z2+h^2]] $
410: [303] Rule2=[[`z2*z2,`0], [`z1*z2,`h^2]]$
411: [304] F=`z2^2*(h^2+z1^2)$
412: [305] qt_rewrite(F,Rule1,2);
413: ((z2)*(z2)*(h)*(h))+((z1)*(z1)*(z2)*(z2))+((4)*(z1)*(z2)*(h)*(h))+((2)*(h)*(h)*(h)*(h))
414: \end{verbatim}
415: \end{example}
416:
417: \begin{remark}
418: $BM-8z(B weight $B%Y%/%H%k$,Ii$N@.J,$r$b$D$HHs@F<!2=$7$?$"$H$N(B
419: reduction $B$NDd;_@-$O$$$($J$$(B.
420: \end{remark}
421:
422:
423: \section{$B=q$-49$(5,B'$NNc(B}
424:
425: $B0J2<$K(B, $B=q$-49$(5,B'$NNc$r$$$/$D$+>R2p$9$k(B.
426:
427: \begin{example}[$B2D49@-(B]
428: \begin{verbatim}
429:
430: [246] qt_normalize(`(x+y-z)^2,1);
431: ((x)^(2))+((x)*(y))+((-1)*(x)*(z))+((y)*(x))+((y)^(2))+((-1)*(y)*(z))
432: +((-1)*(z)*(x))+((-1)*(z)*(y))+((z)^(2))
433: [247] Rcomm=[[`X*Y,`nqt_comp(Y*X,X*Y)>0,`Y*X]]$
434: [248] qt_rewrite(`(x+y-z)^2,Rcomm,1);
435: ((x)^(2))+((2)*(x)*(y))+((-2)*(x)*(z))+((y)^(2))+((-2)*(y)*(z))+((z)^(2))
436: \end{verbatim}
437: {\tt nqt\_comp()} $B$OHf3S4X?t$G$"$k(B.
438: \end{example}
439:
440: \begin{example}[$B30@QBe?t(B]
441: \begin{verbatim}
442:
443: [249] Rext0=[`X*Y,`qt_is_var(X) && qt_is_var(Y) && nqt_comp(Y,X)>0,`-Y*X]$
444: [250] Rext1=[`X^N,`eval_quote(N)>=2,`0]$
445: [251] Rext2=[`X*X,`0]$
446: [252] Rext=[Rext0,Rext1,Rext2]$
447: [253] qt_set_coef([a,b,c])$
448: [254] qt_rewrite(`(a*x+b*y+c*z)*(b*x+c*y+a*z)*(c*x+a*y+b*z),Rext,1);
449: (-a^3+3*c*b*a-b^3-c^3)*(x)*(y)*(z)
450: \end{verbatim}
451: $B9TNs<0$N7W;;$KAjEv$9$k(B. $BJQ?t$N@Q$r8rBeE*$K=q$-49$($k5,B'$rDj5A$7$F$$$k(B.
452: \end{example}
453:
454: \begin{example}[$BHyJ,(B]
455: \begin{verbatim}
456:
457: [255] qt_set_coef([a])$
458: [256] Rd1=[`d(X+Y),`d(X)+d(Y)]$
459: [257] Rd2=[`d(X*Y),`d(X)*Y+X*d(Y)]$
460: [258] Rd3=[`d(N),`qt_is_coef(N),`0]$
461: [259] Rd=[Rd1,Rd2,Rd3]$
462: [260] qt_rewrite(`d((x+a*y)^2),Rd,1);
463: (d((x)^(2)))+((a)*(d(x))*(y))+((a^2)*(d((y)^(2))))+((a)*(d(y))*(x))
464: +((a)*(x)*(d(y)))+((a)*(y)*(d(x)))
465: \end{verbatim}
466: \end{example}
467:
468: \begin{example}[Weyl $BBe?t(B]
469:
470: \begin{verbatim}
471: def member(V,L) {
472: for ( I = 0; L != [] && V != car(L); L = cdr(L), I++ );
473: return L==[] ? -1 : I;
474: }
475: def qt_weyl_vmul(X,K,Y,L) {
476: extern WeylV, WeylDV;
477: if ( member(X,WeylV) >= 0 || member(Y,WeylDV) >= 0 ) return Y^L*X^K;
478: if ( WeylV[I=member(X,WeylDV)] != Y ) return Y^L*X^K;
479: else {
480: K = eval_quote(K); L = eval_quote(L); M = K>L?L:K;
481: for ( T = 1, I = 0; I <= M; T = idiv(T*K*L,I+1), I++, L--, L-- )
482: R += T*Y^L*X^K;
483: return R;
484: }
485: }
486:
487: [256] WeylV=[`x,`y,`z]$
488: [257] WeylDV=[`dx,`dy,`dz]$
489: [258] qt_set_ord(map(eval_quote,append(WeylV,WeylDV)))$
490: [259] Rweyl=[[`X^K*Y^L,`qt_is_var(X)&&qt_is_var(Y)&&nqt_comp(Y,X)>0,
491: `qt_weyl_vmul(X,K,Y,L)]]$
492: [260] qt_rewrite(`((x*dy+y*dx)^2),Rweyl,1);
493: (((x)^(2))*((dy)^(2)))+((2)*(x)*(y)*(dx)*(dy))+((x)*(dx))
494: +(((y)^(2))*((dx)^(2)))+((y)*(dy))
495: \end{verbatim}
496: $Action$ $B$K%f!<%6Dj5A4X?t$rMQ$$$k$3$H$K$h$j(B, Weyl $BBe?t$N=q$-49$(5,B'$r0l$D(B
497: $B$K$^$H$a$F$$$k(B.
498: \end{example}
499:
500: \section{$B$^$H$a(B}
501:
502: Risa/Asir $B$K$*$1$k?t<0$NCf4VE*I=8=$G$"$k(B \FN $B$r%f!<%68@8l$+$iA`:n(B
503: $B$9$k$?$a$N%$%s%?%U%'!<%9$r<BAu$7$?(B. $B$3$l$K$h$j(B, $B%f!<%6$,Dj5A$9$k(B
504: $B=q$-49$(5,B'$K$h$k?t<0$N=q$-49$($,2DG=$H$J$C$?(B. $B=q$-49$($N8zN($K$D$$$F$O(B
505: $B$[$H$s$I9MN8$G$-$F$$$J$$(B. $BFC$K(B, $BI8=`7A$X$NJQ49$H=q$-49$($rJB9T$7$F(B
506: $B9T$&$3$H$,I,MW$H9M$($F$*$j(B, $B:#8e$N2]Bj$N0l$D$G$"$k(B. $B$^$?(B, $B%Q%?!<%s(B
507: $B%^%C%A%s%0<+BN$b$^$@40A4$J$b$N$H$O$$$($:(B, $B2~NI$9$Y$-E@$,B?$/$"$k(B.
508: $BL\E*$K1~$8$?I8=`E*$J=q$-49$(5,B'=89g$r%G%U%)%k%H$GDs6!$9$k$3$H$b(B
509: $BI,MW$G$"$k(B.
510:
511: $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,
1.3 noro 512: $B<+M37k9gBe?t$K$*$1$k0lHLE*$J=q$-49$(7W;;$rO@$8$?(B.
1.1 takayama 513: $B$3$3$GDs0F$7$?0lHL2=$O(B Weyl $BBe?t$NF1<!2=$NM}O@$r4^$`(B. Risa/Asir $B$G?7(B
514: $B$7$/F3F~$7$?(B, \QT $B$KBP$9$k0lHLE*$J(B weight $B%Y%/%H%k$N%a%+%K%:%`(B
515: \verb@ qt_set_weight @ $B$K$h$j$o$l$o$l$NM}O@$H%"%k%4%j%:%`$N%W%m%H%?%$(B
516: $B%W$rMF0W$K;n$9$3$H$,2DG=$G$"$k(B. V. Levandovskyy \cite{LEV} $B$O(B
1.3 noro 517: $G$-algebra $B$N35G0$rF3F~$7$F(B, Singular $B$K<BAu$7$?(B.
518: $B$o$l$o$l$N%"%W%m!<%A$rH/E8$5$;(B,
519: $BF1<!2=$r$H$*$7$F(B, well order $B$G$J$$>l9g$K$bE,MQ$G$-$k(B
520: $B%0%l%V%J!<4pDl$NM}O@$r9=@.$9$l$P(B,
1.1 takayama 521: $G$-algebra $B$h$j9-$$HO0O$N(B algebra $B$r07$&$3$H$,2DG=$H$J$k(B.
522: $B0lHLE*$JOHAH$_$N1~MQ$H$7$F(B, $B>-MhE*$K$O(B $D$-$B2C72$N%"%k%4%j%:%`$r3HD%$7(B,
523: Calderon-Moreno $BEy$NF3F~$7$?(B algebra $B$r6I=jE*$K07$&$J$I$N1~MQ$,8+9~$^(B
524: $B$l$k(B.
1.2 takayama 525: $B$^$?(B, \FN $B$r%f!<%68@8l$h$jA`:n$9$k4X?t$rMQ$$$k$3$H$K$h$j(B,
526: $BF~NO(B, $B=PNO$N%f!<%6%$%s%?%U%'!<%9$rBgI}$K2~A1$G$-$k$3$H$K$b(B
527: $BCm0U$7$F$*$-$?$$(B.
1.1 takayama 528:
529:
530: \begin{thebibliography}{99}
531: \bibitem{MMA}
532: S. Wolfram, The MATHEMATICA Book, Fourth Edition. Cambridge University Press (1999).
533:
534: \bibitem{LEV}
535: %V. Levandovskyy, H. Sch\"onemann:
536: %PLURAL - a Computer Algebra System for Noncommutative Polynomial Algebras.
537: %In Proc. ISSAC 2003, ACM Press (2003).
538: V. Levandovskyy, Non-commutative Computer Algebra for Polynomial Algebras:
539: Gr\"obner Bases, Applications and Implementation.
540: Dissertation, Universit\"at Kaiserslautern (2005).
541:
542: \end{thebibliography}
543: \end{document}
544:
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