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Annotation of OpenXM/doc/Papers/rims2004-noro-ohp.tex, Revision 1.1

1.1     ! noro        1: \documentclass{slides}
        !             2: %\documentclass[pdf,distiller,slideColor,colorBG,azure]{prosper}
        !             3: \usepackage{color}
        !             4: \usepackage{rgb}
        !             5: \usepackage{graphicx}
        !             6: \usepackage{epsfig}
        !             7: \newcommand{\qed}{$\Box$}
        !             8: \newcommand{\mred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}}}
        !             9: \newcommand{\tmred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}\limits^{\scriptstyle *}}}
        !            10: \newtheorem{prop}{\redc $BL?Bj(B}
        !            11: \def\gr{Gr\"obner basis }
        !            12: \def\st{\, s.t. \,}
        !            13: \def\ni{\noindent}
        !            14: \def\init{{\rm in}}
        !            15: \def\Q{{\bf Q}}
        !            16: \def\Z{{\bf Z}}
        !            17: \def\Spoly{{\rm Spoly}}
        !            18: \def\Span{{\rm Span}}
        !            19: \def\Supp{{\rm Supp}}
        !            20: \def\StdMono{{\rm StdMono}}
        !            21: \def\Im{{\rm Im}}
        !            22: \def\Ker{{\rm Ker}}
        !            23: \def\NF{{\rm NF}}
        !            24: \def\HT{{\rm HT}}
        !            25: \def\LT{{\rm LT}}
        !            26: \def\ini{{\rm in}}
        !            27: \def\rem{{\rm rem}}
        !            28: \def\Id#1{\langle #1 \rangle}
        !            29: \def\ve{\vfill\eject}
        !            30: \textwidth 9.2in
        !            31: \textheight 7.2in
        !            32: \columnsep 0.33in
        !            33: \topmargin -1in
        !            34: \def\tc{\color{orange}}
        !            35: \def\fbc{\bf\color{orange}}
        !            36: %\def\itc{\color{LimeGreen}}
        !            37: \def\itc{\color{DarkGreen}}
        !            38: %\def\urlc{\bf\color{DarkGreen}}
        !            39: \def\urlc{\bf\color{LimeGreen}}
        !            40: \def\goldc{\color{goldenrod3}}
        !            41: \def\redc{\color{orange}}
        !            42: \def\vs{\vskip 1cm}
        !            43: \def\vsh{\vskip 0.5cm}
        !            44: \def\ns{\itc\LARGE}
        !            45: \title{\tc\bf\ns $BBe?tBN>e$N%$%G%"%k$N%0%l%V%J!<4pDl7W;;$K$D$$$F(B}
        !            46:
        !            47: %\slideCaption{$BBe?tBN>e$N%$%G%"%k$N%0%l%V%J!<4pDl7W;;$K$D$$$F(B}
        !            48: \author{{\bf\Large $BLnO$(B $B@59T(B\\ $B?@8MBg3XM}3XIt(B}}
        !            49: \date{\bf\Large Dec. 16, 2004}
        !            50: \begin{document}
        !            51: \setlength{\parskip}{20pt}
        !            52: \maketitle
        !            53:
        !            54: %\itc: item color
        !            55: %\fbc: fbox color
        !            56: %\urlc: URL color
        !            57: %\goldc: bold color a
        !            58: %\redc: bold color b
        !            59:
        !            60: \large
        !            61: \bf
        !            62: \setlength{\parskip}{0pt}
        !            63:
        !            64: \begin{slide}{\ns $BBe?tBN$N85$NI=8=J}K!(B}
        !            65:
        !            66: $B86;O85$K$h$kI=8=$O2DG=(B : $F=\Q[t]/(m(t))$
        !            67: $\Rightarrow$ $B78?t$NA}Bg$r>7$/(B
        !            68: $\Rightarrow$ $BC`<!3HBg$,8=<BE*(B
        !            69: $$F_0=\Q,\quad F_i = F_{i-1}(\alpha_i)\quad (i=1,\ldots,n),\quad F=F_m$$
        !            70: $m_i(t,\alpha_1,\ldots,\alpha_{i-1})$ : $\alpha_i$ $B$N:G>.B?9`<0(B $/F_{i-1}$
        !            71:
        !            72: $m_i$ $B$N4{Ls@-%A%'%C%/$O$+$D$F:$Fq(B
        !            73: $\Rightarrow$ knapsack factorization $B$K$h$j$=$&$G$b$J$/$J$C$?(B. $B0J2<(B,
        !            74: $$I=\langle m_1(x_1),m_2(x_1,x_2),\ldots,m_n(x_1,\ldots,x_n)\rangle$$
        !            75: $B$G(B, $I$ $B$,(B $\Q[X]=\Q[x_1,\ldots,x_n]$ $B$N6KBg%$%G%"%k$H$9$k(B.
        !            76: \end{slide}
        !            77:
        !            78: \begin{slide}{\ns $BBe?tE*?t$N4JLs(B}
        !            79:
        !            80: $m_i$ : $B<gJQ?t(B $x_i$ $B$K4X$9$k<g78?t$OM-M}?t$H$7$F$h$$(B.
        !            81: $\Rightarrow$
        !            82: $G=\{m_1,\ldots,m_n\}$ $B$O(B,
        !            83: $x_n > x_{n-1} > \cdots > x_1$ $B$J$k<-=q<0=g=x$K4X$9$k(B $I$ $B$N%0%l%V%J!<4pDl(B.
        !            84:
        !            85: $h(x) \bmod I \in Q[X]/I$ $B$KBP$7(B,
        !            86:
        !            87: $h_0 \equiv h \bmod I$, $\deg_{x_i}(h_0) < d_i$
        !            88: $\Rightarrow$ $h=h\NF_G(h)$
        !            89:
        !            90: $h_0=\rem_{x_1}(\rem_{x_2}(\cdots \rem_{x_n}(h,m_n)\cdots),m_2),m_1)$
        !            91:
        !            92: $B$G$b$"$k$,(B, \underline{$B$3$N=g$G7W;;$7$F$O$$$1$J$$(B!!}
        !            93:
        !            94: ($f(a) \bmod b$ $B$r(B $c=f(a)$ $\Rightarrow$ $c \bmod b$ $B$H7W;;$9$k$h$&$J$b$N(B)
        !            95: \end{slide}
        !            96:
        !            97: \begin{slide}{\ns $BC19`4JLs$K$h$kBe?tE*?t$N4JLs(B}
        !            98:
        !            99: $B$+$H$$$C$F(B, $B=g=x$rJQ$($l$P$h$$(B, $B$H$$$&$o$1$G$b$J$$(B
        !           100:
        !           101: \underline{$BB?9`<0>jM><+BN$,4m81(B} : $BB>$NJQ?t$N<!?t$,5^$K>e$,$k(B
        !           102:
        !           103: $\Rightarrow$ $G$ $B$K$h$kC19`4JLs$rMQ$$$k(B
        !           104:
        !           105: $B3F4JLs%9%F%C%W$KMQ$$$k(B $m_i$ : $i$ $B$N>.$5$$$b$NM%@h(B
        !           106: \end{slide}
        !           107:
        !           108: \begin{slide}{\ns $B4JLs$NNc(B}
        !           109:
        !           110: $m_1(x_1)=x_1^7-7x_1+3$,\\
        !           111: $m_2(x_1,x_2)=x_2^6+x_1x_2^5+x_1^2x_2^4+x_1^3x_2^3+x_1^4x_2^2+\cdots$\\
        !           112: $m_3(x_1,x_2,x_3)=63x_3^4+\cdots$
        !           113:
        !           114: $B$GDj5A$5$l$k(B $F=\Q(\alpha_1,\alpha_2,\alpha_3)$ $B$O(B $m_1(x_1)$ $B$N(B
        !           115: $B:G>.J,2rBN(B.
        !           116:
        !           117: $(\alpha_1+\alpha_2+\alpha_3)^{20}$ $B$N4JLs(B
        !           118:
        !           119: $i$ $B$,>.$5$$(B $m_i$ $BM%@h$G4JLs(B : 0.1 $BIC(B
        !           120:
        !           121: $i$ $B$,Bg$-$$(B $m_i$ $BM%@h$G4JLs(B : 260 $BIC(B
        !           122: \end{slide}
        !           123:
        !           124: \begin{slide}{\ns $B5U857W;;(B}
        !           125:
        !           126: $B5U857W;;$O%\%H%k%M%C%/$N0l$D(B
        !           127:
        !           128: $B3HBg<!?t$r(B $d$ $B$H$9$k(B.
        !           129:
        !           130: $BC13HBg(B $\Rightarrow$ $O(d^2)$ ($B3HD%(B Euclid $B8_=|K!(B)
        !           131:
        !           132: $BC`<!3HBg$K$b:F5"E*$KE,MQ2DG=(B
        !           133:
        !           134: $h(x_1,\ldots,x_n) \bmod I$ $B$N5U85$r7W;;(B
        !           135:
        !           136: $x_n$ $B$K4X$7(B $h$, $m_n$ $B$K3HD%(B Euclid $B8_=|K!$rE,MQ$9$k(B
        !           137:
        !           138: $\Rightarrow$ $\exists a,\exists b, \exists r$, $ah+bm_n=r(x_1,\ldots,x_{n-1})$
        !           139:
        !           140: $\Rightarrow$ $r$ $B$N5U85$r7W;;$9$l$P(B, $h$ $B$N5U85$,5a$^$k(B
        !           141: \end{slide}
        !           142:
        !           143: \begin{slide}{\ns $B$3$NJ}K!$NLdBjE@(B}
        !           144:
        !           145: \begin{itemize}
        !           146: \item $BCf4V<0KDD%(B
        !           147:
        !           148: $B:G=*7k2L$KHf3S$7$F(B, $r$ $B$N5U85$,5pBg$K$J$k>l9g$,$"$k(B.
        !           149: ($B7W;;J}K!$K0MB8(B)
        !           150:
        !           151: \item $B4JLs2=$H$N4X78(B ($BItJ,=*7k<0;;K!$r;H$&>l9g(B)
        !           152:
        !           153: \begin{itemize}
        !           154: \item $h \in (\Q[x_1,\ldots,x_{n-1}])[x_n]$ $B$H8+$J$9(B
        !           155:
        !           156: $B78?t$N=|;;$OB?9`<0$N@0=|$H$J$k$,(B, $B$3$l$i$K(B
        !           157: $B8=$o$l$kJQ?t$KBP$9$k4JLs2=$,9T$o$l$J$$$N$G(B, $B0lHL$KBg$-$JB?9`<0(B
        !           158: $B$,78?t$K8=$o$l$k(B.
        !           159:
        !           160: \item $h\in (\Q(\alpha_1,\ldots,\alpha_{n-1})[x_n]$ $B$H8+$J$9(B
        !           161:
        !           162: $B78?t=|;;$,BN1i;;$H$J$j(B, $B5U857W;;$,I,MW$H$J$k(B.
        !           163: \end{itemize}
        !           164: \end{itemize}
        !           165: \end{slide}
        !           166:
        !           167: \begin{slide}{\ns $B%b%8%e%i!<7W;;$K$h$k5U857W;;(B}
        !           168:
        !           169: $BBe?tE*?t$N7W;;$K$O%b%8%e%i!<7W;;$,M-8z(B
        !           170:
        !           171: [NORO96], [HOEIJ02]).
        !           172:
        !           173: \begin{itemize}
        !           174: \item $BCf9q>jM>DjM}(B
        !           175:
        !           176: $B==J,B?$/$NK!(B $p$ $B$KBP$7(B, $BK!(B $p$ $B$G$N5U85$r7W;;(B
        !           177:
        !           178: $BCf9q>jM>DjM}(B,  $B@0?t(B-$BM-M}?tJQ49$K$h$j5U85$rF@$k(B
        !           179:
        !           180: $BM-8B8D$N(B $p$ $B$r=|$$$FK!(B $p$ $B$G$N5U85$O7W;;$G$-$k(B.
        !           181:
        !           182: \item $BL$Dj78?tK!(B
        !           183: $$M=\{x_1^{e_1}\cdots x_n^{e_n} \bmod I \,|\, 0 \le e_i \le d_i-1
        !           184: (i=1,\ldots,n)\}$$ $B$K$h$j5U85$r(B $u = \sum_{t \in M} a_t t$ $B$HI=$7(B, $hu
        !           185: \equiv 1 \bmod I$ $B$+$i(B $a_t$ $B$N@~7AJ}Dx<07O$r:n$C$F2r$/(B
        !           186: \end{itemize}
        !           187: \end{slide}
        !           188:
        !           189: \begin{slide}{\ns $BL$Dj78?tK!(B + Hensel lifting}
        !           190:
        !           191: $BL$Dj78?tK!(B $B$O(B $O(d^3)$ $B$@$,(B, $B@~7AJ}Dx<07O$r(B Hensel lifting+$B@0?t(B-$BM-M}?tJQ49(B
        !           192: $B$G2r$1$k(B
        !           193:
        !           194: $\Rightarrow$
        !           195:
        !           196: \begin{itemize}
        !           197: \item $O(d^3)$ $B$OM-8BBN>e$N(B LU $BJ,2r$N$_(B
        !           198:
        !           199: $B7k2L$,Bg$-$$78?t$r$b$D$J$i$P(B, $B7W;;;~4V$O(B
        !           200: Hensel lifting (1 step $B$"$?$j(B $O(d^2)$) $B$,(B dominant
        !           201:
        !           202: \item $\NF_G(th)$ ($t \in M$) $B$N7W;;$N$_$G@~7AJ}Dx<0$,$G$-$k(B
        !           203:
        !           204: $BCf4V<0KDD%$dBN=|;;$K$h$kLdBj$O8=$o$l$J$$(B.
        !           205: \end{itemize}
        !           206: \end{slide}
        !           207:
        !           208: \begin{slide}{\ns $BBe?tBN>e$N%0%l%V%J!<4pDl7W;;(B}
        !           209:
        !           210: \underline{$BDjM}(B}
        !           211:
        !           212: $F = \Q[\alpha_1,\ldots,\alpha_l] = \Q[T]/I$
        !           213:
        !           214: $I=\langle m_1(t_1),\ldots,m_l(t_1,\ldots,t_l)\rangle$
        !           215:
        !           216: $J =\langle B \rangle \subset R = F[x_1,\ldots,x_n]$ : $R$ $B$N??$N%$%G%"%k(B
        !           217:
        !           218: $<$ : $R$ $B$N9`=g=x(B
        !           219:
        !           220: $<_F$ : $\Q[X]$ $B>e$G(B $<$ $B$KEy$7$/(B, $X >> T$ $B$G$"$k%V%m%C%/=g=x(B
        !           221:
        !           222: $B_F = B \cup \{m_1,\ldots,m_l\}$
        !           223:
        !           224: $G_F = \langle B_F\rangle$ $B$N(B  $<_F$ $B$K4X$9$k%0%l%V%J!<4pDl(B
        !           225:
        !           226: $\Rightarrow$ $G=(G_F \setminus \Q[T]) \bmod I$ $B$O(B $J$ $B$N(B $<$ $B$K4X$9$k%0%l%V%J!<4pDl(B
        !           227: \end{slide}
        !           228:
        !           229: \begin{slide}{\ns monic $B2=$N$?$a$K(B S-$BB?9`<0$,A}$($k(B}
        !           230:
        !           231: $BDjM}$K$h$j(B, $F$ $B>e$N%0%l%V%J!<4pDl7W;;$O(B $\Q$ $B>e$N$=$l$K5"Ce(B
        !           232:
        !           233: $B<B9T$r4Q;!$9$k$H(B $\cdots$ $B@8@.$5$l$kCf4V4pDl$NF,9`$N(B $t$ $BJQ?t$,$@$s$@$s>CLG(B
        !           234: = $BF,78?t$N5U857W;;$r(B S-$BB?9`<0$HC19`4JLs$G<B9T(B
        !           235:
        !           236: $\Rightarrow$ S-$BB?9`<0$N?t$,A}Bg$7$F$$$k(B
        !           237:
        !           238: $BJ@32(B : $BITE,@Z$J=g=x$G(B S-$BB?9`<0$,=hM}$5$l$k2DG=@-$bA}$($k(B.
        !           239:  ($BITI,MW$J78?tKDD%$N2DG=@-(B)
        !           240: \end{slide}
        !           241:
        !           242: \begin{slide}{\ns $B@55,7A$r(B monic $B2=(B}
        !           243:
        !           244: \begin{itemize}
        !           245: \item $BDL>o$N=hM}(B
        !           246:
        !           247: $S(f,g) \tmred{G} h \neq 0$ $B$J$i$P(B, $G \leftarrow G \cup \{h\}$
        !           248:
        !           249: \item $BJQ998e$N=hM}(B
        !           250:
        !           251: $S(f,g) \tmred{G} h(x,t) \neq 0$ $B$J$i$P(B,
        !           252: $h(x,\alpha)$ $B$r(B monic $B2=$7$?$b$N(B
        !           253: $\tilde{h}(x,\alpha)$ $B$r:n$j(B,
        !           254: $G \leftarrow G \cup \{\tilde{h}(x,t)\}$
        !           255: \end{itemize}
        !           256:
        !           257: $B$3$N$h$&$JJQ99$r9T$C$F$b(B $G_F$ $B$,7W;;$G$-$k$3$H$O$"$-$i$+(B
        !           258:
        !           259: \end{slide}
        !           260:
        !           261: \begin{slide}{\ns trace $B%"%k%4%j%:%`(B}
        !           262:
        !           263: $\bmod p$ $B$G$N(Btrace $B%"%k%4%j%:%`$NB39T$KI,MW$J$3$H(B
        !           264:
        !           265: \begin{itemize}
        !           266: \item $\cdots$ $\Q$ $B>e$N4JLs2=$K$"$i$o$l$kJ,Jl$,(B $p$ $B$G3d$j@Z$l$J$$(B
        !           267:
        !           268: \item $B@55,7A$NF,78?t$,(B $p$ $B$G3d$j@Z$l$J$$(B
        !           269: \end{itemize}
        !           270:
        !           271: $\Rightarrow$ monic $B2=$K(B $p$ $B$G3d$l$kJ,Jl$,$"$i$o$l$J$$(B, $B$H4K$a$i$l$k(B
        !           272:
        !           273: ($B$"$i$+$8$a%$%G%"%k$N@8@.85$N0l$D$G$"$C$?$H9M$($l$P$h$$(B)
        !           274:
        !           275: \underline{$BB>$N<B8=J}K!(B}
        !           276:
        !           277: $\overline{I}=I \bmod p$ $B$,:,4p%$%G%"%k(B
        !           278:
        !           279: $\Rightarrow$ $GF(p)[t]/\overline{I}$ $B$OM-8BBN$ND>OB(B
        !           280:
        !           281: $I_p$ : $I$ $B$N78?t$r(B $\Q_{<p>}=\{a/b\,|\, a, b \in Z, p \not{|} b\}$
        !           282: $B$K@)8B(B
        !           283:
        !           284: $\phi$ : $I_p$ $B$+$i$"$kD>OB@.J,$X$N<M1F(B
        !           285:
        !           286: $B$H$7$F$b$h$$(B
        !           287:
        !           288: $\Rightarrow$ $BM-8BBN>e$N(B trace $B7W;;$N<j4V$r8:$i$;$k2DG=@-$,$"$k(B.
        !           289: \end{slide}
        !           290:
        !           291: \begin{slide}{\ns Risa/Asir $B>e$G$N<BAu(B : $BC`<!Be?t3HBg(B}
        !           292:
        !           293: {\tt Alg} : $B%\%G%#It$,:F5"I=8=B?9`<0(B ($B4{B8(B)
        !           294:
        !           295: {\tt DAlg} : $B%\%G%#It$,J,;6I=8=B?9`<0(B ($B?75,(B) -- Alg $B$+$iJQ49(B
        !           296: \begin{verbatim}
        !           297: typedef struct oDAlg {
        !           298:         short id;
        !           299:         char nid;
        !           300:         char pad;
        !           301:         struct oDP *nm;  /* $B<B:]$K$O@0?t78?t(B */
        !           302:         struct oQ *dn;   /* $B<B:]$K$O@0?t(B */
        !           303: } *DAlg;
        !           304: \end{verbatim}
        !           305: $BBe?tE*?t$H$7$F$O(B {\tt nm/dn} : $BJ,Jl$rDLJ,$7$F(B {\tt dn} $B$H$9$k(B.
        !           306:
        !           307: $BB>$K(B, $B4JLsMQ$K3HBgBN%G!<%?$r9=B$BN$H$7$FJ];}(B
        !           308:
        !           309: {\tt set\_field()} $B$G@_Dj$G$-$k(B.
        !           310: \end{slide}
        !           311:
        !           312: \begin{slide}{\ns Risa/Asir $B>e$G$N<BAu(B : $B%0%l%V%J!<4pDl7W;;(B}
        !           313:
        !           314: \begin{itemize}
        !           315: \item {\tt nd\_gr} $B$*$h$S(B {\tt nd\_gr\_trace} $B$r2~B$(B
        !           316:
        !           317: $BF~NO$K:G>.B?9`<0$rDI2C$7$F<B9T(B + $B@55,7A$N(B monic $B2=(B
        !           318: $BF~NO$O(B {\tt Alg} $B7?$r78?t$K4^$`B?9`<0=89g$G$h$$(B.
        !           319:
        !           320: \item $BFbItI=8=(B
        !           321:
        !           322: $BBe?tE*?t$O(B, $B85$NB?9`<0JQ?t$HF1Ey(B, $B@55,2=7W;;$OM-M}?tBN>e$G(B
        !           323: $\Rightarrow$ $B78?t$N(B content $B=|5n$,<+F0E*$KE,MQ$5$l$k(B.
        !           324:
        !           325: \item monic $B2=(B
        !           326:
        !           327: $B$N:]$K$N$_(B, $BK\Mh$N78?t$,Be?tE*?t(B ({\tt DAlg $B7?(B}) $B$H(B
        !           328: $B$7$F<h$j=P$5$l(B, $B5U857W;;$J$I$,9T$o$l$k(B.
        !           329:
        !           330: \item weight
        !           331:
        !           332: $BBe?tE*?t$KBP1~$9$k(Bweight $B$r(B 0 $B$K@_Dj(B $\Rightarrow$ sugar $B$r(B
        !           333: $BE,@5$K$9$k$?$a(B
        !           334: \end{itemize}
        !           335: \end{slide}
        !           336:
        !           337: \begin{slide}{\ns $BB>$N<BAuK!(B}
        !           338:
        !           339: \underline{$BBe?tE*?t$r40A4$K78?t$H$7$F07$&(B}
        !           340:
        !           341: $B78?t$K4X$7$F4JLs2=$*$h$S5U857W;;$r9T$&(B
        !           342:
        !           343: $B<+A3$J<BAu$H8@$($k(B.
        !           344:
        !           345: content $B=|5n$KAjEv$9$kJ}K!$r?7$?$K9M0F$9$kI,MW(B
        !           346: $B$,$"$j(B, $B:#8e$N2]Bj(B.
        !           347: \end{slide}
        !           348:
        !           349: \begin{slide}{\ns $B7W;;$NNc(B}
        !           350:
        !           351: $\langle \sqrt{2}x^2+(\sqrt{2}+\sqrt{3})xy+\sqrt{3}y^2-\sqrt{3},(\sqrt{2}-2\sqrt{3})x^2+2\sqrt{3}xy+2\sqrt{2}x^2+\sqrt{2}+\sqrt{3}\rangle$ $B$N%0%l%V%J!<7W;;(B
        !           352: \begin{verbatim}
        !           353: [0] S2=newalg(x^2-2);
        !           354: (#0)
        !           355: [1] S3=newalg(x^2-3);
        !           356: (#1)
        !           357: [2] F1=S2*x^2+(S2+S3)*x*y+S3*y^2-S3$
        !           358: F2=(S2-2*S3)*x^2+2*S3*x*y+2*S2*x^2+S2+S3]$
        !           359: [3] nd_gr_trace([F1,F2],[x,y],1,1,2);
        !           360: [90*y^4+(-21*#0*#1-246)*y^2+(16*#0*#1+144),
        !           361: 20*x+(15*#0*#1-60)*y^3+(-7*#0*#1+83)*y]
        !           362: \end{verbatim}
        !           363: \end{slide}
        !           364:
        !           365: \begin{slide}{\ns $B<B83(B : $BC1:,E:2C(B}
        !           366:
        !           367: {\small
        !           368: \begin{eqnarray*}
        !           369: Cyc&=&\{f_1,f_2,f_3,f_4,f_5,f_6,f_7\}\\
        !           370: f_1&=&\omega c_5c_4c_3c_2c_1c_0-1\\
        !           371: f_2&=&(((((c_5+\omega )c_4+\omega c_5)c_3+\omega c_5c_4)c_2+\omega c_5c_4c_3)c_1+\omega c_5c_4c_3c_2)c_0+\omega c_5c_4c_3c_2c_1\\
        !           372: f_3&=&((((c_4+\omega )c_3+\omega c_5)c_2+\omega c_5c_4)c_1+\omega c_5c_4c_3)c_0+c_5c_4c_3c_2c_1+\omega c_5c_4c_3c_2\\
        !           373: f_4&=&(((c_3+\omega )c_2+\omega c_5)c_1+\omega c_5c_4)c_0+c_4c_3c_2c_1+c_5c_4c_3c_2+\omega c_5c_4c_3\\
        !           374: f_5&=&((c_2+\omega )c_1+\omega c_5)c_0+c_3c_2c_1+c_4c_3c_2+c_5c_4c_3+\omega c_5c_4\\
        !           375: f_6&=&(c_1+\omega )c_0+c_2c_1+c_3c_2+c_4c_3+c_5c_4+\omega c_5\\
        !           376: f_7&=&c_0+c_1+c_2+c_3+c_4+c_5+\omega
        !           377: \end{eqnarray*}}
        !           378: $Cyc$: cyclic-7 $B$N(B $c_6$ $B$K(B 1 $B$N86;O(B 7 $B>h:,$rBeF~(B
        !           379:
        !           380: \underline{$\Q(\omega)$ $B>e$G$N(B GB $B7W;;(B}: $B@F<!2=(B trace $B%"%k%4%j%:%`$K$h$j(B 22 $BIC(B
        !           381: (monic $B2=$K(B 2.2 $BIC(B ($B5U857W;;(B 0.2$BIC(B))
        !           382:
        !           383: \underline{$B:G>.B?9`<0$rE:2C$7$F(B $\Q$ $B>e$G7W;;(B} : 220 $BIC(B
        !           384:
        !           385:
        !           386: \end{slide}
        !           387:
        !           388: \begin{slide}{\ns $B<B83(B : 2 $B:,E:2C(B}
        !           389: {\small
        !           390: \begin{eqnarray*}
        !           391: Cap&=&\{f_1,f_2,f_3,f_4\}\\
        !           392: f_1&=&(2ty-2)x-(\alpha+\beta)zy^2-z\\
        !           393: f_2&=&2\beta\alpha^4zx^3+(4ty+\beta)x^2+(4zy^2+4z)x+2ty^3-10y^2-10ty+2\alpha^2+\beta^2\\
        !           394: f_3&=&(t^2-1)x+(\beta\alpha^4+\beta^3\alpha^3)tzy-2z\\
        !           395: f_4&=&(-z^2+4t^2+\beta\alpha+2\beta^3)zx+(4tz^2+2t^3-10t)y+4z^2-10t^2+\beta\alpha^3\\
        !           396: m_1&=&u^7-7u+3\\
        !           397: m_2&=&u^6+\alpha u^5+\alpha^2u^4+\alpha^3u^3+\alpha^4u^2+\alpha^5u+\alpha^6-7
        !           398: \end{eqnarray*}}
        !           399: $Cap$ : $Caprasse$ [SYMBDATA] $B$N78?t$r%i%s%@%`$KBe?tE*?t$KCV$-49$($?(B
        !           400:
        !           401: $\alpha$, $\beta$ : $t^7-7t+3$ $B$N(B 2 $B:,(B
        !           402:
        !           403: \underline{$\Q(\alpha,\beta)$ $B>e$G$N(B GB $B7W;;(B} : $B@F<!2=(B trace $B%"%k%4%j%:%`$G(B 589 $BIC(B
        !           404: (monic $B2=(B 36 $BIC(B)
        !           405:
        !           406: \underline{$\Q$ $B>e$G$N7W;;(B} : 1 $B;~4VBT$C$F$b=*N;$7$J$$(B.
        !           407: \end{slide}
        !           408:
        !           409: \begin{slide}{\ns $B<B83(B : 3 $B:,E:2C(B}
        !           410:
        !           411: $f(x)=x^7-7x+3$ $B$N:G>.J,2rBN(B $F$ $B$O(B 3 $B:,E:2C$G<B8=$5$l$k(B ($B4{=P(B).
        !           412:
        !           413: $f(x)$ $B$N(B $F$ $B>e0x?tJ,2r$K8=$l$k(B, $F$ $B>e$N(B 2 $B$D$N(B 2 $B<!<0$N(B
        !           414: GCD $B7W;;(B (GCD $B$O(B 1 $B<!<0(B) $B%0%l%V%J!<4pDl$G7W;;$9$k(B.
        !           415:
        !           416: \underline{$F$ $B>e$N(B GB $B7W;;(B} : 0.8 $BIC(B ($B5U857W;;(B 1 $B2sJ,$,BgItJ,(B)
        !           417:
        !           418: \underline{$\Q$ $B>e$G7W;;(B} : 60 $BIC(B
        !           419:
        !           420: \end{slide}
        !           421:
        !           422: \begin{slide}{\ns $B$*$o$j$K(B}
        !           423:
        !           424: \begin{itemize}
        !           425: \item DCGB $B$H$N4X78(B
        !           426:
        !           427: $B:4F#$i(B [SATO01] $B$K$h$k(B DCGB $B$H$NHf3S(B
        !           428:
        !           429: \item change of ordering, RUR
        !           430:
        !           431: FGLM $B$d(B RUR $B$N7W;;$NBe?tBN>e$X$N3HD%(B
        !           432:
        !           433: ($BBe?tBN>e(B? or $BM-M}?tBN>e(B?)
        !           434:
        !           435: \item $BBe?tBN1i;;$N<BAu$N8zN(2=(B
        !           436:
        !           437: $BBe?tBN$NI=8=$r(B {\tt DP} $B$+$i(B, $B$h$j8zN($h$$(B
        !           438: $B<BAu$KJQ99$9$k(B
        !           439: \end{itemize}
        !           440:
        !           441: \end{slide}
        !           442:
        !           443: \begin{slide}{\ns $BJ88%(B}
        !           444:
        !           445: [HOEIJ02] M.v. Hoeij, M. Monagan, A Modular GCD algorithm over Number Fields presented with Multiple Extensions. Proc. ISSAC'02 (2002), 109-116.
        !           446:
        !           447: [NORO96] $BLnO$(B, $BC`<!Be?t3HBgBN>e$G$N(B 1 $BJQ?tB?9`<0$N(B GCD $B$K$D$$$F(B. $B?tM}8&9V5fO?(B 920 (1996), 1-8.
        !           448:
        !           449: [SATO01] Y. Sato, A. Suzuki,  Discrete Comprehensive Gr\"obner Bases.
        !           450: Proc. ISSAC'01 (2001), 292-296.
        !           451:
        !           452: [SYMBDATA] {\tt http://www.SymbolicData.org}.
        !           453: \end{slide}
        !           454:
        !           455: %\begin{slide}{\ns }
        !           456: %\end{slide}
        !           457:
        !           458: \end{document}
        !           459: