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Annotation of OpenXM/doc/Papers/rims2002-noro.tex, Revision 1.1

1.1     ! noro        1: % $OpenXM$
        !             2: \documentclass{slides}
        !             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: \def\gr{Gr\"obner basis }
        !            11: \def\st{\, s.t. \,}
        !            12: \def\ni{\noindent}
        !            13: \def\ve{\vfill\eject}
        !            14: \textwidth 9.2in
        !            15: \textheight 7.2in
        !            16: \columnsep 0.33in
        !            17: \topmargin -1in
        !            18: \def\tc{\color{red}}
        !            19: \def\fbc{\bf\color{MediumBlue}}
        !            20: \def\itc{\color{brown}}
        !            21: \def\urlc{\bf\color{DarkGreen}}
        !            22: \def\bc{\color{LightGoldenrod1}}
        !            23:
        !            24: \def\HT{{\rm HT}}
        !            25: \def\HC{{\rm HC}}
        !            26: \def\GCD{{\rm GCD}}
        !            27: \def\tdeg{{\rm tdeg}}
        !            28: \def\pp{{\rm pp}}
        !            29: \def\lc{{\rm lc}}
        !            30: \def\Z{{\bf Z}}
        !            31:
        !            32: \title{\tc $B>.I8?tM-8BBN>e$NB?JQ?tB?9`<0$N0x?tJ,2r$K$D$$$F(B ($B$=$N(B 2)}
        !            33:
        !            34: \author{$BLnO$(B $B@59T(B ($B?@8MBg!&M}(B)}
        !            35: \begin{document}
        !            36: \large
        !            37: \setlength{\parskip}{0pt}
        !            38: \maketitle
        !            39:
        !            40: \begin{slide}{}
        !            41: \begin{center}
        !            42: \fbox{\fbc \large $BL5J?J}J,2r(B ($BI|=,(B)}
        !            43: \end{center}
        !            44:
        !            45: modification of Bernardin's algorithm [Ber97]
        !            46:
        !            47: $f \in F[x_1,\ldots,x_n]$, $F$ : $BM-8BBN(B $Char(F) = p$
        !            48:
        !            49: $f = FGH$, where $F=\prod f_i^{a_i}$,
        !            50: $G=\prod g_j^{b_j}$,
        !            51: $H=\prod h_k^{c_k}$
        !            52:
        !            53: $f_i, g_j, h_k$ : $BL5J?J}(B, $B8_$$$KAG(B.
        !            54:
        !            55: $'$ $B$r(B $d/dx_1$ $B$H$7$F(B
        !            56: $f_i' \neq 0$, $p \not{|}a_j$, $p | b_j$, $h_k' = 0$
        !            57: $B$H=q$/$H(B
        !            58:
        !            59: $f' = F'GH$ $B$9$k$H(B $GCD(f,f') = GCD(F,F')GH$
        !            60:
        !            61: $GCD(F,F') = \prod f_i^{a_i-1}$ $B$@$+$i(B $f/GCD(f,f')=\prod f_i$
        !            62:
        !            63: $\prod f_i$ $B$G(B $f$ $B$r7+$jJV$73d$k$3$H$G(B, $f_1$ ($B=EJ#EY:G>.(B)
        !            64: $B$,5a$^$k(B
        !            65:
        !            66: $\Rightarrow$ $F$ $B$,A4$FJ,2r$G$-$k(B
        !            67:
        !            68: \end{slide}
        !            69:
        !            70: \begin{slide}{}
        !            71: \begin{center}
        !            72: \fbox{\fbc \large $BL5J?J}J,2r(B ($B$D$E$-(B) }
        !            73: \end{center}
        !            74:
        !            75: $B;D$j(B $f = GH$ $B$G(B,  $f' = 0$
        !            76:
        !            77: $B$3$l$r(B $x_i$ $B$K$D$$$F7+$jJV$7$F;D$C$?(B $f$
        !            78:
        !            79: $\Rightarrow$ $df/dx_1 = \ldots = df/dx_n = 0$
        !            80:
        !            81: $\Rightarrow$ $B$3$l$O(B, $BA4$F$N;X?t$,(B $p$ $B$G3d$j@Z$l$k$3$H$r0UL#$9$k(B
        !            82:
        !            83: $\Rightarrow$ $F$ $B$OM-8BBN$@$+$i(B $f = g^p$ $B$H=q$1$k(B
        !            84:
        !            85: $\Rightarrow$ $g$ $B$KBP$7$F%"%k%4%j%:%`$rE,MQ(B
        !            86: \end{slide}
        !            87:
        !            88:
        !            89: \begin{slide}{}
        !            90: \begin{center}
        !            91: \fbox{\fbc \large $B<BAu>e$N:$Fq(B : $\GCD(f,f')$ $B$N7W;;(B}
        !            92: \end{center}
        !            93:
        !            94: $BI8?t$,(B 0 $B$N>l9g(B $\GCD(g,f'/g)=1$ $\Rightarrow$ $BB?JQ?t$N(B Hensel $B9=@.$,(B
        !            95: $B;H$($k(B
        !            96:
        !            97: $B@5I8?t$N>l9g(B $$\GCD(g,f'/g) = \GCD(\GCD(F,F')GH,F'/\GCD(F,F'))$$
        !            98: $\Rightarrow$ $GH$ $B$NB8:_$N$?$a(B GCD $B$,(B 1 $B$H$O8B$i$J$$(B.
        !            99:
        !           100: $\Rightarrow$ $B$d$`$J$/(B Brown $B$N%"%k%4%j%:%`(B ($BCf9q>jM>DjM}$K$h$k(B GCD $B$N(B
        !           101: $B7W;;(B) $B$rMQ$$$F$$$k(B.
        !           102: \end{slide}
        !           103:
        !           104: \begin{slide}{}
        !           105: \begin{center}
        !           106: \fbox{\fbc \large GCD $B$N7W;;(B}
        !           107: \end{center}
        !           108:
        !           109: \begin{tabbing}
        !           110: $BF~NO(B : $f_1,\ldots,f_m \in K[X]$ ($K$ $B$OBN(B, $X$ $B$OJQ?t$N=89g(B)\\
        !           111: $B=PNO(B : $\GCD(f_1,\ldots,f_m)$\\
        !           112: $y \leftarrow$ $BE,Ev$JJQ?t(B; $Z \leftarrow X\setminus \{y\}$\\
        !           113: $< \leftarrow K[Z]$ $B$NE,Ev$J9`=g=x(B; $B0J2<(B $f_i \in K[y][Z]$ $B$H$_$J$9(B\\
        !           114: $h_i(y) \leftarrow \HT_<(f_i)$; $h_g(y) \leftarrow \GCD(h_1,\ldots,h_m)$\\
        !           115: $g \leftarrow 0$; $M \leftarrow 1$\\
        !           116: do \= \\
        !           117:    \> $a \leftarrow $ $BL$;HMQ$N(B $K$ $B$N85(B\\
        !           118:    \> $g_a \leftarrow \GCD(f_1|_{y=a},\ldots,f_m|_{y=a})$\\
        !           119:    \> if \= $g \neq 0$ $B$+$D(B $\HT_<(g) = \HT_<(g_a)$ then \\
        !           120:    \>    \> $adj \leftarrow h_g(a)/\HC_<(g_a)\cdot g_a - g(a)$
        !           121: \end{tabbing}
        !           122: \end{slide}
        !           123:
        !           124: \begin{slide}{}
        !           125: \begin{tabbing}
        !           126: do \= if \= \kill
        !           127:    \>    \> if \= $adj = 0$ $B$+$D(B, $B$9$Y$F$N(B $f_i$ $B$KBP$7(B $g | h_g\cdot f_i$  then \\
        !           128:    \>    \>    \> return $\pp(g)$\\
        !           129:    \>    \> endif\\
        !           130:    \>    \> $g \leftarrow g+adj \cdot M(a)^{-1} \cdot M$; $M \leftarrow M\cdot (y-a)$\\
        !           131:    \> else if $\tdeg(\HT_<(g)) > \tdeg(\HT_<(g_a))$ then \\
        !           132:    \>    \> $g \leftarrow g_a$; $M \leftarrow  y-a$\\
        !           133:    \> else if $\tdeg(\HT_<(g)) = \tdeg(\HT_<(g_a))$ then \\
        !           134:    \>    \> $g \leftarrow 0$; $M \leftarrow 1$\\
        !           135:    \> endif\\
        !           136: end do
        !           137: \end{tabbing}
        !           138: \end{slide}
        !           139:
        !           140: \begin{slide}{}
        !           141: \begin{center}
        !           142: \fbox{\fbc \large $BFsJQ?t$X$N5"Ce(B}
        !           143: \end{center}
        !           144:
        !           145: \begin{itemize}
        !           146: \item $B<gJQ?t(B $x$ $B$NA*Br(B
        !           147:
        !           148: $x$ $B$K4X$9$kHyJ,$,>C$($J$$$h$&$KA*$V(B.
        !           149:
        !           150: \item $B=>JQ?t(B $y$ $B$NA*Br(B
        !           151:
        !           152: $K[x,y]$ $B$G0x?tJ,2r$7$F(B, $Z=X\setminus \{x,y\}$ $B$K4X$7$F(B Hensel $B9=@.(B
        !           153:
        !           154: ($B0lJQ?t$^$GMn$9$H%K%;0x;R$,BgNLH/@8(B)
        !           155:
        !           156: \item $B$=$l0J30$NJQ?t(B $Z$ $B$X$NBeF~CM$NA*Br(B
        !           157:
        !           158: $f_a(x,y) = f(x,y,a)$ $B$,L5J?J}$K$J$k$h$&$KA*$V(B.
        !           159:
        !           160: $B$5$i$K(B, $f_a|_{y=0}$ $B$bL5J?J}$K$J$k$h$&$K(B, $y\leftarrow y+c$ $B$H(B
        !           161: $BJ?9T0\F0(B.
        !           162: \end{itemize}
        !           163:
        !           164: \end{slide}
        !           165:
        !           166: \begin{slide}{}
        !           167: \begin{center}
        !           168: \fbox{\fbc \large $K[y]$ $B>e$G$N(B Hensel $B9=@.(B ($BA0=hM}(B)}
        !           169: \end{center}
        !           170:
        !           171: $f_a(x,y)$ $B$N0x;R$r(B 2 $BAH$K$o$1(B $f_a(x,y) = g_0(x,y)h_0(x,y)$
        !           172: $B$+$i(B $K[y]$ $B>e$G(B Hensel $B9=@.(B
        !           173:
        !           174: $g_0$, $h_0$ $B$N(B $x$ $B$K4X$9$k<g78?t$N7h$aJ}(B
        !           175:
        !           176: $B<g78?tLdBj$N2sHr(B : $B??$N0x;R$N<g78?t$H$J$k$Y$/6a$$$b$N$r$"$i$+$8$a8GDj(B
        !           177:
        !           178: $B??$N0x;R$N<g78?t$O(B $\lc_x(f)$ $B$N0x;R$G$"$k$3$H$r;H$C$?8+@Q$j(B
        !           179: \end{slide}
        !           180:
        !           181: \begin{slide}{}
        !           182: \begin{center}
        !           183: \fbox{\fbc \large $B<g78?t$N8+@Q$j(B}
        !           184: \end{center}
        !           185: \begin{enumerate}
        !           186: \item $\lc_x(f) = \prod u_i^{n_i}$ $B$H0x?tJ,2r$9$k(B ($u_i \in K[y,Z]$ : $B4{Ls(B).
        !           187:
        !           188: \item $B3F(B $i$ $B$KBP$7(B, $u_i(a) \in K[y]$ $B$,(B $\lc_x(g_0)$ $B$r3d$j@Z$k2s?t$r(B
        !           189: $B?t$($k(B. $B$=$l$r(B $m_i$ $B$H$7$?$H$-(B, $\lc_g = \prod u_i^{m_i}$ $B$H$9$k(B.
        !           190: $BF1MM$K(B $h_0$ $B$KBP$7$F$b(B $\lc_h$ $B$r5a$a$k(B.
        !           191:
        !           192: $B$b$7(B,
        !           193: $\lc_x(g_0) \not{|}\, \lc_g(a)$ $B$^$?$O(B $\lc_x(h_0) \not{|}\, \lc_h(a)$
        !           194: $B$^$?$O(B,
        !           195: $\lc_x(f) \not{|}\, \lc_g \cdot \lc_h$
        !           196: $B$J$i$P(B, $B$=$l$O(B, $f_a$ $B$N0x;R$NAH9g$;$,@5$7$/$J$$$3$H$r0UL#$9$k$N$G(B,
        !           197: $g_0$, $h_0$ $B$r$H$jD>$9(B.
        !           198:
        !           199: \item
        !           200:
        !           201: $g_0 \leftarrow \lc_g(a)/\lc_x(g_0)\cdot g_0$ $B$N<g78?t$r(B $\lc_g$ $B$GCV$-49$($?$b$N(B
        !           202:
        !           203: $h_0 \leftarrow \lc_h(a)/\lc_x(h_0)\cdot h_0$ $B$N<g78?t$r(B $\lc_h$ $B$GCV$-49$($?$b$N(B
        !           204:
        !           205: $f \leftarrow \lc_g\cdot \lc_h/\lc_x(f) \cdot f$
        !           206:
        !           207: $B$H$9$k(B. $B$3$N;~(B, $f = g_0h_0$ $B$H$J$C$F$$$k(B.
        !           208: \end{enumerate}
        !           209: \end{slide}
        !           210:
        !           211: \begin{slide}{}
        !           212: \begin{center}
        !           213: \fbox{\fbc \large $K[y]$ $B>e$G$N(B Hensel $B9=@.(B}
        !           214: \end{center}
        !           215:
        !           216: $f=g_0h_0$ $B$G(B, $g_0$ $B$,@5$7$$0x;R$N<M1F(B, $\lc_x(g_0)$ $B$H2>Dj(B
        !           217:
        !           218: $g_0$, $h_0$ $B$+$i(B $K[y]$ $B>e$N(BHensel $B9=@.$K$h$j(B,
        !           219: $$f=g_kh_k \bmod I^{k+1}$$
        !           220: $B$H;}$A>e$2$k(B ($I = \langle z_1-a_1,\ldots,z_{n-2}-a_{n-2} \rangle$).
        !           221:
        !           222: $z_i \rightarrow z_i+a_i$ $B$J$kJ?9T0\F0$K$h$j(B,
        !           223: $I=\langle z_1,\ldots,z_{n-2} \rangle$
        !           224:
        !           225: $BDL>o$N(B EZ $BK!(B : $\Z/(p^l)$ $B$G7W;;(B ($B78?t$KJ,?t$,8=$l$k$N$rHr$1$k(B)
        !           226:
        !           227: $B:#2s$N<BAu(B : $\deg_y(f) > d$ $B$J$k(B $d$ $B$KBP$7(B, $K[y]/(y^d)$
        !           228: $B>e$G7W;;$9$k(B. ($K[y]$ $B$G$N>&BN$G$N1i;;$rHr$1$k(B)
        !           229:
        !           230: $B$3$l$O(B, $u g_0(a)+v h_0(a)=1 \bmod y^d$ $B$H$J$k(B $u, v \in K[y]$
        !           231: $B$K$h$j2DG=(B
        !           232:
        !           233: Hensel $B9=@.$O(B $\bmod\,  y^d$ $B$G9T$&(B.
        !           234: \end{slide}
        !           235:
        !           236: \begin{slide}{}
        !           237: \begin{center}
        !           238: \fbox{\fbc \large Hensel $B9=@.(B}
        !           239: \end{center}
        !           240: $f = g_kh_k \bmod (I^{k+1},y^d)$ $B$@$,(B,
        !           241: $B==J,Bg$-$$(B $k$ $B$KBP$7(B $f = g_kh_k$
        !           242:
        !           243: $u$, $v$ $B$N7W;;(B : Hensel $B9=@.(B
        !           244: ($g_0(a)|_{y=0}$, $h_0(a)_{y=0}$ $B$,8_$$$KAG(B)
        !           245:
        !           246: $K[y]$ $B>e$N(B Hensel $B9=@.$O<!$N$h$&$K9T$&(B.
        !           247:
        !           248: \begin{enumerate}
        !           249: \item $f-g_{k-1}h_{k-1} = \sum_t F_t t \bmod (I^k,y^d)$ $B$H=q$/(B.
        !           250: $B$3$3$G(B $t \in I^k$ $B$OC19`<0(B, $F_t \in K[y][x]$.
        !           251: \item $G_th_0+H_tg_0 = F_t \bmod y^d$ $B$H$J$k(B $G_t, H_t \in K[y][x]$ $B$r7W;;$9$k(B.
        !           252: $B$3$l$O(B $u$, $v$ $B$r;H$C$F:n$l$k(B.
        !           253: \item $g_{k+1} \leftarrow g_k + \sum_t G_t t$,
        !           254: $h_{k+1} \leftarrow h_k + \sum_t H_t t$ $B$H$9$l$P(B $f = g_{k+1}h_{k+1} \bmod (I^{k+1},y^d)$.
        !           255: \end{enumerate}
        !           256:
        !           257: $\sum_t G_t t = 0$ $B$^$?$O(B $\sum_t H_t t = 0$ $\Rightarrow$ $B;n$73d$j(B
        !           258:
        !           259: $g_k$ $B$^$?$O(B $h_k$ $B$G(B $f$ $B$r3d$C$F$_$k$3$H$G(B, $B<!?t$N>e8B$^$G(B Hensel $B9=@.(B
        !           260: $B$;$:$K(B, $B??$N0x;R$r8!=P$G$-$k(B.
        !           261: \end{slide}
        !           262:
        !           263: \begin{slide}{}
        !           264: \begin{center}
        !           265: \fbox{\fbc \large $BM-8BBN$NI=8=$K$D$$$F(B}
        !           266: \end{center}
        !           267:
        !           268: $B3F%"%k%4%j%:%`$K$*$$$F(B, $B78?tBN$N0L?t$,==J,Bg$-$$I,MW$"$j(B.
        !           269:
        !           270: $BBeF~$9$kE@$N?t$,ITB-$9$k(B $\Rightarrow$ $BBe?t3HBg(B
        !           271:
        !           272: $B7W;;8zN($,Mn$A$J$$$h$&(B, $B86;O:,$rMQ$$$?I=8=$r<BAu(B
        !           273:
        !           274: $B7gE@(B : $B<BMQE*$J0L?t$,(B $2^{16}$ $BDxEY$K8B$i$l$k(B
        !           275:
        !           276: $B2~NI(B : $BI8?t$,(B $2^{14}$ $B0J2<$N>l9g$K$O(B, $B86;O:,I=8=(B
        !           277:
        !           278: $B$=$l0J>e$N>l9g$K$O(B, $BDL>o$NI=8=(B
        !           279: ($B<BMQ>e==J,$KBeF~$9$kE@$,F@$i$l$k$+$i(B)
        !           280:
        !           281: $\Rightarrow$ $B0L?t$,(B $2^{29}$ $BDxEY$^$G$NAGBN>e$G(B, $BB?JQ?tB?9`<0(B
        !           282: $B$N0x?tJ,2r$,2DG=(B
        !           283: \end{slide}
        !           284:
        !           285: \begin{slide}
        !           286: \begin{center}
        !           287: \fbox{\fbc \large $B78?t4D$H$7$F$N(B $K[y]/(y^d)$ $B$K$D$$$F(B}
        !           288: \end{center}
        !           289:
        !           290: Hensel $B9=@.$K$*$$$F(B $K[y]/(y^d)$ $B$r78?t4D$H$7$F07$&(B
        !           291:
        !           292: Asir : $B>.0L?tM-8BBN(B $K$ $B$NBe?t3HBg$rI=8=$9$k7?(B (GFSN) $B$,$"$k(B
        !           293:
        !           294: $K[y]/(m(y))$ $B$H$7$FI=8=(B $\Rightarrow$ $m(y)=y^d$ $B$H$7$FN.MQ(B
        !           295:
        !           296: $B5U857W;;$K$D$$$F$O(B, 0 $B$G$J$$Dj?t9`$r;}$DB?9`<0$O2D5U(B ($B8_=|K!(B)
        !           297:
        !           298: $\lc_x \neq 0$ $B$h$j(B $K[y]/(y^d)$ $B$,$3$NJ}K!$G$G$-$k(B.
        !           299:
        !           300: $BB?JQ?tB?9`<0(B : Hensel $B9=@.$N:G=i$G(B, $B$3$N7?$N78?t$r;}$DB?9`<0$KJQ49(B
        !           301:
        !           302: $d$ $B$r%;%C%H$7$F$*$/(B $\Rightarrow$ $B7k2L$O<+F0E*$K(B $\bmod \, y^d$ $B$5$l$k(B
        !           303:
        !           304: $\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d)$ $B78?t$N(B
        !           305: $BB?9`<01i;;$,<B9T$G$-$k(B.
        !           306: \end{slide}
        !           307:
        !           308: \begin{slide}{}
        !           309: \begin{center}
        !           310: \fbox{\fbc \large $B:#8e$NM=Dj(B}
        !           311: \end{center}
        !           312:
        !           313: \begin{itemize}
        !           314: \item $B@5I8?t$N=`AGJ,2r$N<BAu(B.
        !           315:
        !           316: \item $BBN$N0L?t$,B-$j$J$$>l9g$K(B, $B<+F0E*$K4pACBN$r3HBg$9$k(B.
        !           317:
        !           318: \item $BBN$NI8?t$,==J,Bg$-$$>l9g$K(B, $BL5J?J}J,2r$rI8?t(B 0 $B$H(B
        !           319: $BF1MM$N(B Hensel $B9=@.$G9T$&$h$&$K$9$k(B.
        !           320:
        !           321: \item 2 $BJQ?t$N0x?tJ,2r$K$*$$$F(B, \cite{funny01} $B$G=R$Y$?(B, $BB?9`<0(B
        !           322: $B;~4V%"%k%4%j%:%`$r<+F0E*$KA*Br$7$F<B9T$9$k(B.
        !           323: \end{itemize}
        !           324: \end{slide}
        !           325:
        !           326: \end{document}
        !           327:
        !           328: \begin{thebibliography}{99}
        !           329: \bibitem{B97-2}
        !           330: Bernardin, L. (1997).
        !           331: On square-free factorization of multivariate polynomials over a finite
        !           332: field.
        !           333: {\em Theoret.\ Comput.\ Sci.\/} {\bf 187}, 105--116.
        !           334:
        !           335: \bibitem{funny01}
        !           336: M. Noro and K. Yokoyama (2002).
        !           337: Yet Another Practical Implementation of Polynomial Factorization
        !           338:  over Finite Fields.
        !           339: Proceedings of ISSAC2002, ACM Press, 200--206.
        !           340: \end{thebibliography}
        !           341: