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Annotation of OpenXM/doc/Papers/rims-2002-noro-ja.tex, Revision 1.2

1.2     ! noro        1: % $OpenXM: OpenXM/doc/Papers/rims-2002-noro-ja.tex,v 1.1 2002/12/04 08:57:21 noro Exp $
1.1       noro        2: \documentclass[theorem]{jarticle}
                      3: \usepackage{jssac}
                      4:
                      5: \def\HT{{\rm HT}}
                      6: \def\HC{{\rm HC}}
                      7: \def\GCD{{\rm GCD}}
                      8: \def\tdeg{{\rm tdeg}}
                      9: \def\pp{{\rm pp}}
                     10: \def\lc{{\rm lc}}
                     11: \def\Z{{\bf Z}}
                     12:
                     13: \title{$BM-8BBN>e$NB?JQ?tB?9`<0$N0x?tJ,2r$K$D$$$F(B ($B$=$N(B 2)}
                     14:
                     15: \author{$BLnO$(B $B@59T(B ($B?@8MBg!&M}(B)}
                     16:
                     17: \begin{document}
                     18: \maketitle
                     19:
                     20: \section{$B$O$8$a$K(B}
                     21:
                     22: $BK\9F$G$O(B, \cite{funny01} $B$G=R$Y$?(B, $BM-8BBN>e$G$N(B 2 $BJQ?tB?9`<0$N0x?tJ,2r(B
                     23: $B$r4pAC$H$7$F(B, $B0lHL$NB?JQ?tB?9`<0$N(B GCD, $BL5J?J}J,2r(B, $B0x?tJ,2r%"%k%4%j%:%`(B
                     24: $B$*$h$S$=$N<BAu$K$D$$$F=R$Y$k(B.
                     25:
                     26: \section{$BB?JQ?tB?9`<0$NL5J?J}J,2r$H(B GCD}
                     27:
                     28: $B8=:_(B Risa/Asir $B$GMQ$$$F$$$k%"%k%4%j%:%`$O(B,
                     29: $B0J2<$K=R$Y$k$h$&$K(B,
                     30: Bernardin $B$NL5J?J}J,2r%"%k%4%j%:%`(B \cite{B97-2} $B$r(B
                     31: modify $B$7$?$b$N$G$"$k(B.
                     32:
                     33: $F$ $B$rI8?t(B $p$ $B$NM-8BBN$H$7(B, $f \in F[x_1,\ldots,x_n]$ $B$H(B
                     34: $B$9$k(B. $'$ $B$,(B $d/dx_1$ $B$rI=$9$H$9$k(B.
                     35: $$f = FGH, F=\prod f_i^{a_i}, G=\prod g_j^{b_j},
                     36: H=\prod h_k^{c_k}$$
                     37: ($f_i, g_j, h_k$ $B$OL5J?J}(B, $B8_$$$KAG$G(B,
                     38: $f_i' \neq 0$, $p \not{|}a_j$, $p | b_j$, $h_k' = 0$)
                     39: $B$H=q$/$H(B $f' = F'GH$ $B$,@.$jN)$D(B.
                     40: $B$9$k$H(B $$\GCD(f,f') = \GCD(F,F')GH$$ $B$G(B,
                     41: $\GCD(F,F') = \prod f_i^{a_i-1}$ $B$@$+$i(B
                     42: $$f/\GCD(f,f')=\prod f_i.$$
                     43: $\prod f_i$ $B$G(B $f$ $B$r7+$jJV$73d$j(B, $B3d$j@Z$l$J$/$J$C$?(B
                     44: $BCJ3,$G(B, $B$=$N>&$H(B $\prod f_i$ $B$N(B GCD $B$r7W;;$9$k$3$H$G(B,
                     45: $F$ $BCf$N=EJ#EY:G>.$N0x;R(B $f_1$ $B$,5a$^$k(B. $B$3$l$r7+$jJV$9$H(B
                     46: $F$ $B$,A4$FL5J?J}J,2r$G$-$k(B.
                     47: $B;D$j$N(B $f$ $B$O(B $f = GH$ $B$H=q$1$k(B. $B$3$3$G(B $f' = 0$ $B$,(B
                     48: $B@.$jN)$D$3$H$KCm0U$9$k(B. $B0J>e$N<jB3$-$r(B
                     49: $B3F(B $x_i$ $B$K$D$$$F7+$jJV$7$F;D$C$?(B $f$ $B$O(B,
                     50: $$df/dx_1 = \ldots = df/dx_n = 0$$
                     51: $B$rK~$?$9(B. $B$3$l$O(B, $BA4$F$N;X?t$,(B $p$
                     52: $B$G3d$j@Z$l$k$3$H$r0UL#$9$k(B. $B$9$k$H(B, $F$ $B$OI8?t(B $p$ $B$N(B
                     53: $BM-8BBN$@$+$i(B, $f = g^p$ $B$H=q$1$k$3$H$K$J$k(B.
                     54: $B$3$N(B $g$ $B$KBP$7$F(B, $B0J>e$N<jB3$-$r:F5"E*$KE,MQ$9$k(B
                     55: $B$3$H$G(B, $f$ $B$NL5J?J}J,2r$,F@$i$l$k(B.
                     56:
                     57: $B$3$N%"%k%4%j%:%`$K$*$$$F(B, $g=\GCD(f,f')$ $B$N7W;;$,I,MW$H$J$k(B.
                     58: $BI8?t$,(B 0 $B$N>l9g(B, $\GCD(g,f'/g)=1$ $B$,(B
                     59: $BJ]>Z$5$l$k$?$a(B, $B$3$N7W;;$KB?JQ?t$N(B Hensel $B9=@.$rMQ$$$k$3$H$,$G$-$k(B.
                     60: $B$7$+$7(B, $B@5I8?t$N>l9g$K$O(B,
                     61: $\GCD(g,f'/g) = \GCD(\GCD(F,F')GH,F'/\GCD(F,F'))$ $B$H$J$j(B,
                     62: $B0x;R(B $GH$ $B$NB8:_$N$?$a(B, $B$3$N(B GCD $B$,(B 1 $B$H$O8B$i$J$$(B. $B$3$N$?$a(B,
                     63: $B$d$`$J$/(B Brown $B$N%"%k%4%j%:%`(B ($BCf9q>jM>DjM}$K$h$k(B GCD $B$N7W;;(B)
                     64: $B$rMQ$$$F$$$k(B.
                     65:
                     66: \begin{tabbing}
                     67: GCD $B$N7W;;(B \\
                     68: $BF~NO(B : $f_1,\ldots,f_m \in K[X]$ ($K$ $B$OBN(B, $X$ $B$OJQ?t$N=89g(B)\\
                     69: $B=PNO(B : $\GCD(f_1,\ldots,f_m)$\\
                     70: $y \leftarrow$ $BE,Ev$JJQ?t(B; $Z \leftarrow X\setminus \{y\}$\\
                     71: $< \leftarrow K[Z]$ $B$NE,Ev$J9`=g=x(B; $B0J2<(B $f_i \in K[y][Z]$ $B$H$_$J$9(B\\
                     72: $h_i(y) \leftarrow \HT_<(f_i)$; $h_g(y) \leftarrow \GCD(h_1,\ldots,h_m)$\\
                     73: $g \leftarrow 0$; $M \leftarrow 1$\\
                     74: do \= \\
                     75:    \> $a \leftarrow $ $BL$;HMQ$N(B $K$ $B$N85(B\\
                     76:    \> $g_a \leftarrow \GCD(f_1|_{y=a},\ldots,f_m|_{y=a})$\\
                     77:    \> if \= $g \neq 0$ $B$+$D(B $\HT_<(g) = \HT_<(g_a)$ then \\
                     78:    \>    \> $adj \leftarrow \cdot h_g(a)/\HC_<(g_a)\cdot g_a - g(a))$\\
                     79:    \>    \> if \= $adj = 0$ $B$+$D(B, $B$9$Y$F$N(B $f_i$ $B$KBP$7(B $g | hg\cdot f_i$  then \\
                     80:    \>    \>    \> return $\pp(g)$\\
                     81:    \>    \> endif\\
                     82:    \>    \> $g \leftarrow g+adj \cdot M(a)^{-1} \cdot M$; $M \leftarrow M\cdot (y-a)$\\
                     83:    \> else if $\tdeg(\HT_<(g)) > \tdeg(\HT_<(g_a)$ then \\
                     84:    \>    \> $g \leftarrow g_a$; $M \leftarrow  y-a$\\
                     85:    \> else if $\tdeg(\HT_<(g)) = \tdeg(\HT_<(g_a))$ then \\
                     86:    \>    \> $g \leftarrow 0$; $M \leftarrow 1$\\
                     87:    \> endif\\
                     88: end do
                     89: \end{tabbing}
                     90:
                     91: $\HT$ $B$OF,9`(B, $\HC$ $B$OF,78?t(B, $\tdeg$ $B$OA4<!?t(B, $\pp$ $B$O86;OE*ItJ,(B
                     92: $B$rI=$9(B.
                     93:
                     94: $B$3$N%"%k%4%j%:%`$G$O(B, $f_1, \ldots, f_m$ $B$KBP$7(B, $B$"$kJQ?t(B $y$ $B$K$5$^$6$^$J(B
                     95: $BCM(B $a$
                     96: $B$rBeF~$7$F(BGCD $g_a$ $B$r7W;;$7(B, $B$=$l$i$rCf9q>jM>DjM}$G7k9g$9$k(B. $B$=$N:](B,
                     97: $B;D$j$NJQ?t(B $Z$ $B$K4X$7$FE,Ev$J9`=g=x(B $<$ $B$r@_Dj$7(B, $B$=$N9`=g=x$K4X$9$kF,(B
                     98: $B9`$,Ey$7$$4V$O(B, $B??$N(B GCD $B$N@5$7$$<M1F$K$J$C$F$$$k$H(B
                     99: $B2>Dj$9$k(B. $B$=$l$^$G$H0[$J$kF,9`$r;}$D(B $g_a$ $B$,F@$i$l$?>l9g(B,
                    100: $B$=$NF,9`$NA4<!?t$,(B, $B$=$l$^$G$NF,9`$NA4<!?t$h$j??$KBg$-$$>l9g$K$O(B,
                    101: $BL@$i$+$K@5$7$/$J$$$N$G(B, $B<N$F$k(B. $B$^$?(B, $B??$K>.$5$$>l9g$K$O(B, $B$3$l$^$G(B
                    102: $B$N7k2L$O@5$7$/$J$$$3$H$K$J$j(B, $B?7$?$K(B $g_a$ $B$+$i%j%9%?!<%H$9$k(B.
                    103: $B$b$7(B, $BA4<!?t$,Ey$7$1$l$P(B, $BN>J}$N7k2L$,@5$7$/$J$$$3$H$K$J$j(B,
                    104: $BN>J}$r<N$F$F%j%9%?!<%H$9$k(B.
                    105:
                    106: $B$^$?(B, $B??$N(B GCD $B$N(B $<$ $B$K4X$9$kF,78?t$,(B
                    107: $f_i$ $B$NF,78?t$N(B GCD $B$G$"$k(B $h_g$ $B$N0x;R$K$J$C$F$$$k$3$H$rMQ$$(B, $BCf9q>jM>DjM}$r(B
                    108: $BE,MQ$9$k:](B, $BF,78?t$,(B $h_g$ $B$K$J$k$h$&$KD4@a$7$F$$$k(B.
                    109:
                    110: $B?7$?$J(B $a$ $B$GF@$?(B $g_a$ $B$H(B, $B$3$l$^$GF@$?(B $g$ $B$N(B $a$ $B$G$NCM$,(B
                    111: $BEy$7$$>l9g$K(B, $B<B:]$K(B $g$ $B$,(B $h_gf_i$ $B$r$9$Y$F3d$j@Z$k$+%A%'%C%/$r(B
                    112: $B9T$C$F$$$k(B. $B<B:]$N<BAu$K$*$$$F$O(B, $BE,Ev$J(B $f_i$ $B$rA*$s$G$*$-(B,
                    113: cofactor $B$N$[$&$bCf9q>jM>DjM}$G9=@.$7$F$$$-(B, $B$=$A$i$G$b3d;;%A%'%C%/(B
                    114: $B$r9T$&$3$H$G(B, GCD, cofactor $B$$$:$l$+$,I|85$G$-$?;~E@$G%"%k%4%j%:%`(B
                    115: $B$,=*N;$G$-$k(B.
                    116:
                    117: $B$3$N%"%k%4%j%:%`$G$O(B, GCD $B$N(B $y$ $B$K4X$9$k<!?t$h$jB?$$(B $K$ $B$N85$,I,MW(B
                    118: $B$K$J$k(B. $K$ $B$N0L?t$,$3$l$KK~$?$J$$>l9g$K$O(B, $K$ $B$r3HBg$9$k(B. $B$3$l$K(B
                    119: $B$D$$$F$O8e$G=R$Y$k(B.
                    120:
                    121: \section{$BL5J?J}B?9`<0$N0x?tJ,2r(B}
                    122:
                    123: $B0J2<(B, $f \in K[X]$ $B$OL5J?J}$H$9$k(B. $f$ $B$N0x?tJ,2r$O<!$N$h$&$K9T$&(B.
                    124:
                    125: \subsection{$B<gJQ?t(B $x$ $B$NA*Br(B}
                    126:
                    127: $f$ $B$N0x?tJ,2r$O(B, $B$"$kJQ?t(B $x$ $B0J30$NJQ?t$KE,Ev$JCM$rBeF~$7$FF@$i$l$?(B
                    128: $x$ $B$N0lJQ?tB?9`<0$N0x?tJ,2r$r%?%M$+$i=g<!(B Hensel $B9=@.$K$h$j7W;;$9$k(B (EZ $BK!(B).
                    129: $x$ $B$O(B, $x$ $B$K4X$9$kHyJ,$,>C$($J$$$h$&$KA*$VI,MW$,$"$k(B.
                    130: $B$^$?(B, $x$ $B$K4X$9$k<!?t$,Bg$-$$Dx(B, $B%?%M$H$J$k0x;R$N?t$,Bg$-$/$J$k2DG=@-$,(B
                    131: $B$"$k$?$a(B, $B$J$k$Y$/<!?t$,>.$5$$$h$&$J(B $x$ $B$rA*$s$G$$$k(B.
                    132:
                    133: \subsection{$B=>JQ?t(B $y$ $B$NA*Br(B}
                    134:
                    135: $B=>JQ?t$H$OL/$JMQ8l$G$"$k$,(B, $B$3$3$G$O(B, $BB?JQ?t$N0x?tJ,2r$r(B, 2 $BJQ?t$N0x?t(B
                    136: $BJ,2r$+$i(B Hensel $B9=@.$K$h$jF@$k$?$a(B, $B$=$N$?$a$NJQ?t$r$b$&0l$DA*$s$G$*$/(B
                    137: $BI,MW$,$"$k(B. $B$9$J$o$A(B, $B$"$kJQ?t(B $y$ $B$rA*$S(B, $x$, $y$ $B0J30$NJQ?t$K(B
                    138: $BE,Ev$JCM$rBeF~$7$?(B 2 $BJQ?tB?9`<0$N0x?tJ,2r$r(B \cite{funny01} $B$G=R$Y$?(B
                    139: $BJ}K!$K$h$j7W;;$9$k(B. $B$=$l$r4p$K(B, $B;D$j$NJQ?t$K4X$7$F(B Hensel $B9=@.$r(B
                    140: $B9T$&(B. $B$3$3$G(B, Hensel $B9=@.$O(B $K[X] = K[y][x,Z]$
                    141: ($Z=X\setminus\{x,y\} = \{z_1,\ldots,z_{n-2}\}$) $B$H$_$J$7$F(B
                    142: $B9T$&(B. $B$9$J$o$A(B, $B0lJQ?tB?9`<04D(B $K[y]$ $B$r(B, $BM-M}?tBN>e$NB?JQ?tB?9`<0(B
                    143: $B$N0x?tJ,2r$K$*$1$k(B, $B@0?t4D$NN`;w$H$_$J$9$o$1$G$"$k(B. $B$3$&$9$k$3$H$K(B
                    144: $B$h$j(B, 1 $BJQ?t$NJ,2r$+$i=PH/$7$?>l9g$K@8$:$kBgNL$N%K%;0x;R$K$h$k:$Fq$r(B
                    145: $BHr$1$k$3$H$,$G$-$k(B. $B$"$H$G<($9$h$&$K(B, Hensel $B9=@.<+BN$b(B, $B@0?t>e$N(B
                    146: $B>l9g$NN`;w$NJ}K!$K$h$j9T$&$3$H$,$G$-$k(B.
                    147:
                    148: \subsection{2 $BJQ?t$N0x?tJ,2r(B}
                    149:
                    150: $x$, $y$ $B$,7h$^$C$?$i(B, $f_a(x,y) = f(x,y,a)$ $B$,(B,
                    151: $BL5J?J}$K$J$k$h$&$K(B $Z$ $B$KBeF~$9$kCM$N%Y%/%H%k(B
                    152: $a = (a_1,\ldots,a_{n-2}) \in K^{n-2}$ $B$rA*$S(B, $f_a$ $B$r0x?tJ,2r$9$k(B.
                    153: $B$3$3$G(B, $f_a$ $B$N(B $x$ $B$K4X$9$k<g78?t(B
                    154: ($B$3$l$O(B $y$ $B$NB?9`<0(B) $B$NDj?t9`$,(B 0 $B$G$J$/(B, $B$+$D(B
                    155: $f_a|_{y=0}$ $B$,L5J?J}$G$"$k$h$&(B,
                    156: $BI,MW$,$"$l$P(B $y\rightarrow y+c$ $B$H$$$&J?9T0\F0$r9T$&(B.
                    157:
                    158: \subsection{$K[y]$ $B>e$G$N(B Hensel $B9=@.(B ($BA0=hM}(B)}
                    159:
                    160: $f_a(x,y)$ $B$N0x?tJ,2r$N7k2L$h$j(B, $B0x;R$r(B 2 $BAH$K$o$1(B
                    161: $f_a(x,y) = g_0(x,y)h_0(x,y)$ $B$H$7$?>e$G(B, $K[y]$ $B>e$G(B Hensel $B9=@.$r(B
                    162: $B9T$&(B. $B$3$N:](B, $BLdBj$H$J$k$N$,(B $g_0$, $h_0$ $B$N(B $x$ $B$K4X$9$k<g78?t$N(B
                    163: $B7h$aJ}$G$"$k(B. $B$$$o$f$k<g78?tLdBj$r2sHr$9$k$?$a$K(B, $B??$N0x;R$N<g78?t(B
                    164: $B$H$J$k$Y$/6a$$$b$N$r$"$i$+$8$a8GDj$7$F$*$/$N$,$h$$(B. $B>/$J$/$H$b(B,
                    165: $B$=$l$O(B, $f$ $B$N(B, $x$ $B$K4X$9$k<g78?t(B $\lc_x(f)$ $B$N0x;R$G$O$"$k$,(B,
                    166: $\lc_x(f)$ $B$=$N$b$N$r$H$k$3$H$O0lHL$K(B overestimate $B$G$"$k(B.
                    167: $BM-M}?tBN>e$N>l9g(B, P. S. Wang $B$K$h$j<g78?t$N7hDjJ}K!$,Ds0F$5$l$F(B
                    168: $B$$$k$,(B, $B$3$3$G$O<!$N$h$&$K8+@Q$b$k(B:
                    169:
                    170: \begin{enumerate}
                    171: \item $\lc_x(f) = \prod u_i^{n_i}$ $B$H0x?tJ,2r$9$k(B ($u_i \in K[y,Z]$ : $B4{Ls(B).
                    172:
                    173: \item $B3F(B $i$ $B$KBP$7(B, $u_i(a) \in K[y]$ $B$,(B $\lc_x(g_0)$ $B$r3d$j@Z$k2s?t$r(B
                    174: $B?t$($k(B. $B$=$l$r(B $m_i$ $B$H$7$?$H$-(B, $\lc_g = \prod u_i^{m_i}$ $B$H$9$k(B.
                    175: $BF1MM$K(B $h_0$ $B$KBP$7$F$b(B $\lc_h$ $B$r5a$a$k(B.
                    176:
                    177: $B$b$7(B,
                    178: $\lc_x(g_0) \not{|}\, \lc_g(a)$ $B$^$?$O(B $\lc_x(h_0) \not{|}\, \lc_h(a)$
                    179: $B$^$?$O(B,
                    180: $\lc_x(f) \not{|}\, \lc_g \cdot \lc_h$
                    181: $B$J$i$P(B, $B$=$l$O(B, $f_a$ $B$N0x;R$NAH9g$;$,@5$7$/$J$$$3$H$r0UL#$9$k$N$G(B,
                    182: $g_0$, $h_0$ $B$r$H$jD>$9(B.
                    183:
                    184: \item
                    185:
                    186: $g_0 \leftarrow \lc_g(a)/\lc_x(g_0)\cdot g_0$ $B$N<g78?t$r(B $\lc_g$ $B$GCV$-49$($?$b$N(B
                    187:
                    188: $h_0 \leftarrow \lc_h(a)/\lc_x(h_0)\cdot h_0$ $B$N<g78?t$r(B $\lc_h$ $B$GCV$-49$($?$b$N(B
                    189:
                    190: $f \leftarrow \lc_g\cdot \lc_h/\lc_x(f) \cdot f$
                    191:
                    192: $B$H$9$k(B. $B$3$N;~(B, $f = g_0h_0$ $B$H$J$C$F$$$k(B.
                    193: \end{enumerate}
                    194: $B$3$NCJ3,$G(B, $B$b$7(B $g_0$ $B$,??$N0x;R$N<M1F$H$J$C$F$$$l$P(B, Hensel $B9=@.(B
                    195: $B$K$h$j(B, $f$ $B$N0x;R$K;}$A>e$k$O$:$G$"$j(B, $\lc_x(g_0)$ $B$O4{$K(B, $B??$N0x;R(B
                    196: $B$N<g78?t$KEy$7$/$J$C$F$$$k(B. $B$3$N>l9g(B, $h_0$ $B$bF1MM$N@-<A$rK~$?$9(B.
                    197:
                    198: \subsection{$K[y]$ $B>e$G$N(B Hensel $B9=@.(B}
                    199:
                    200: $BA09`$K$h$j(B, $f=g_0h_0$ $B$K$*$$$F(B,
                    201: $g_0$ $B$,@5$7$$0x;R$N<M1F$J$i$P(B, $\lc_x(g_0)$ $B$O(B
                    202: $B4{$K??$N0x;R$N<g78?t$KEy$7$$(B. $B$3$3$G$O(B, $g_0$, $h_0$ $B$+$i(B $K[y]$ $B>e$N(B
                    203: Hensel $B9=@.$K$h$j(B, $f=g_kh_k \bmod I^{k+1}$, $B$?$@$7(B
                    204: $I = <z_1-a_1,\ldots,z_{n-2}-a_{n-2}>$, $B$H$J$k(B $g_k$, $h_k$ $B$r(B EZ $BK!$K(B
                    205: $B$h$j7W;;$9$k(B. $B$^$:(B, $z_i \rightarrow z_i+a_i$ $B$J$kJ?9T0\F0$K$h$j(B,
                    206: $I=<z_1,\ldots,z_{n-2}>$ $B$H$7$F$*$/(B. $BDL>o$N(B EZ $BK!$G$O(B, $B78?t$KJ,?t$,(B
                    207: $B8=$l$k$N$rHr$1$k$?$a(B, $B0x;R$N78?t$NBg$-$5$NI>2A$+$iDj$a$i$l$k(B,
                    208: $B$"$kBg$-$JAG?t6R(B $p^l$ $B$rK!$H$7$F(B $\Z/(p^l)$ $B>e$G7W;;$9$k(B.
                    209: $B$3$3$G$O(B, $f$ $B$N(B $y$ $B$K4X$9$k<!?t$r1[$($k@0?t(B $d$ $B$KBP$7(B,
                    210: $K[y]/(y^d)$ $B$G$N1i;;$rF3F~$9$k$3$H$G(B, $K[y]$ $B$G$N>&BN$G$N1i;;$rHr$1$k(B.
                    211: $B$9$J$o$A(B, $u g_0(a)+v h_0(a)=1 \bmod y^d$ $B$H$J$k(B $u, v \in K[y]$ $B$r(B
                    212: $B7W;;$7$F$*$-(B, Hensel $B9=@.$N78?t$N7W;;$O(B $u, v$ $B$rMQ$$$F(B $\bmod\,  y^d$
                    213: $B$G9T$&$N$G$"$k(B. $B$3$N$H$-(B, $f = g_kh_k \bmod (I^{k+1},y^d)$ $B$@$,(B,
                    214: $d$ $B$,==J,Bg$-$$$?$a(B, $g_0$ $B$,??$N0x;R$N<M1F$J$i$P(B, Hensel $B9=@.$N0l0U(B
                    215: $B@-$K$h$j(B, $B==J,Bg$-$$(B $k$ $B$KBP$7(B $f = g_kh_k$ $B$H$J$k(B. $u$, $v$ $B$N7W;;$O(B,
                    216: $g_0(a)|_{y=0}$, $h_0(a)_{y=0}$ $B$,8_$$$KAG$G$"$k$3$H$rMxMQ$7$F(B,
                    217: $B$d$O$j(B Hensel $B9=@.$K$h$j7W;;$G$-$k(B.
                    218:
                    219: $K[y]$ $B>e$N(B Hensel $B9=@.$O<!$N$h$&$K9T$&(B.
                    220:
                    221: \begin{enumerate}
                    222: \item $f-g_{k-1}h_{k-1} = \sum_t F_t t \bmod (I^k,y^d)$ $B$H=q$/(B.
                    223: $B$3$3$G(B $t \in I^k$ $B$OC19`<0(B, $F_t \in K[y][x]$.
                    224: \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.
                    225: $B$3$l$O(B $u$, $v$ $B$r;H$C$F:n$l$k(B.
                    226: \item $g_{k+1} \leftarrow g_k + \sum_t G_t t$,
                    227: $h_{k+1} \leftarrow h_k + \sum_t H_t t$ $B$H$9$l$P(B $f = g_{k+1}h_{k+1} \bmod (I^{k+1},y^d)$.
                    228: \end{enumerate}
                    229:
                    230: $B$3$NA`:n$K$*$$$F(B, $\sum_t G_t t$ $B$^$?$O(B $\sum_t H_t t$ $B$,(B 0 $B$H$J$C$?>l9g$K(B
                    231: $g_k$ $B$^$?$O(B $h_k$ $B$G(B $f$ $B$r3d$C$F$_$k$3$H$G(B, $B<!?t$N>e8B$^$G(B Hensel $B9=@.(B
                    232: $B$;$:$K(B, $B??$N0x;R$r8!=P$9$k$3$H$,$G$-$k(B.
                    233:
                    234:
                    235: \section{$B<BAu$K$D$$$F(B}
                    236:
                    237: \subsection{$BM-8BBN$NI=8=$K$D$$$F(B}
                    238:
                    239: $B$3$3$G=R$Y$?3F%"%k%4%j%:%`$K$*$$$F$O(B, $B78?tBN$N0L?t$,==J,Bg$-$$I,MW$,$"(B
                    240: $B$k(B. \cite{funny01} $B$G=R$Y$?$h$&$K(B, $BBeF~$9$kE@$N?t$,ITB-$9$k>l9g$N$?$a(B
                    241: $B$K(B, $BBe?t3HBg$7$F$b7W;;8zN($,Mn$A$J$$$h$&$J(B, $B86;O:,$rMQ$$$?I=8=$r<BAu$7(B,
                    242: $B$=$NM-8z@-$r<($7$?(B. $B7gE@$H$7$F$O(B, $B$3$NI=8=$K$*$$$F(B, $B<BMQE*$J(B
                    243: $B0L?t$,(B $2^{16}$ $BDxEY$K8B$i$l$k$3$H$G$"$C$?(B. $B:#2s(B, $B$=$N2~NI$H$7$F(B, $BI8?t(B
                    244: $B$,(B $2^{14}$ $B0J2<$N>l9g$K$O(B, $B86;O:,I=8=$r5v$7(B, $B$=$l0J>e$N>l9g$K$O(B, $BDL>o$N(B
                    245: $BI=8=$r$H$k$3$H$K$7$?(B. $B$3$l$O(B, $B8e<T$G$O<BMQ>e==J,$KBeF~$9$kE@$,F@$i$l$k(B
                    246: $B$+$i$G$"$k(B. $B$3$l$K$h$j(B, $B0L?t$,(B $2^{29}$ $BDxEY$^$G$NAGBN>e$G(B, $BB?JQ?tB?9`<0(B
                    247: $B$N0x?tJ,2r$,9T$($k$h$&$K$J$C$?(B.
                    248:
1.2     ! noro      249: \subsection{$B78?t4D$H$7$F$N(B $K[y]/(y^d)$ $B$K$D$$$F(B}
1.1       noro      250:
1.2     ! noro      251: Hensel $B9=@.$K$*$$$F$O(B, $K[y]/(y^d)$ $B$r(B, $BB?9`<0$N78?t4D$H$7$F(B
        !           252: $B07$&I,MW$,$"$k(B. Asir $B$K$*$$$F$O(B, $B4{$K(B, $B>.0L?tM-8BBN(B $K$ $B$NBe?t3HBg(B
        !           253: $B$rI=8=$9$k7?$,$"$k$,(B, $B$3$l$O(B, $K[y]/(m(y))$ ($m(y)$ $B$O:G>.B?9`<0(B)
        !           254: $B$H$7$FI=8=$5$l$F$$$k$?$a(B, $m(y)=y^d$ $B$H$9$l$P(B, $B2C8:>h;;$ON.MQ(B
        !           255: $B$G$-$k(B. $B$^$?(B, $B=|;;$K8=$l$k5U857W;;$K$D$$$F$O(B, 0 $B$G$J$$Dj?t9`$r;}$D(B
        !           256: $BB?9`<0$GI=8=$5$l$k85$K8B$l$P(B, $B$=$l$O(B $y^d$ $B$H8_$$$KAG$J$N$G(B
        !           257: $B5U85$r;}$A(B, $B8_=|K!$G7W;;$G$-$k(B. $B4{$K=R$Y$?$h$&$K(B, $x$ $B$K4X$9$k<g78?t(B
        !           258: $B$,(B 0 $B$G$J$$Dj?t9`$r;}$D$h$&$KJ?9T0\F0$7$F$"$k$N$G(B, $K[y]/(y^d)$ $B$N(B
        !           259: $B7W;;$r(B, $B$3$3$G=R$Y$?J}K!$G9T$&$3$H$,$G$-$k(B.
        !           260: $BB?JQ?tB?9`<0$O(B, Hensel $B9=@.$N:G=i$G(B, $B$3$N(B
1.1       noro      261: $B7?$N78?t$r;}$DB?9`<0$KJQ49$5$l$k(B. $B$"$i$+$8$a(B $d$ $B$r%;%C%H$7$F(B
                    262: $B$*$/$3$H$K$h$j(B, $B1i;;$O<+F0E*$K(B $\bmod \, y^d$ $B$5$l$k$?$a(B,
                    263: $BDL>o$NB?9`<01i;;$N8F$S=P$7$r9T$&$@$1$G(B, $K[y]/(y^d)$ $B78?t$N(B
                    264: $BB?9`<01i;;$,<B9T$G$-$k(B.
1.2     ! noro      265:
        !           266: \section{$B%?%$%_%s%0%G!<%?(B}
        !           267:
        !           268: $BM-8BBN>e$NB?JQ?tB?9`<0$N0x?tJ,2r$rDs6!$7$F$$$k%7%9%F%`$O?t>/$J$$(B.
        !           269: $BI.<T$NCN$kM#0l$N$b$N$O(B Maple $B$J$N$G(B, Maple $B$HHf3S$r9T$&(B. Maple $B$O(B,
        !           270: $B$3$N5!G=$K4X$7$F$O(B, kernel $B$K$*$$$F@lMQ$NFC<l$J%G!<%?7?$rB?MQ$7$F(B
        !           271: $B8zN($r>e$2$F$$$k$?$a(B, $BHf3S$9$k$3$H$O%"%s%U%'%"$G$O$J$$$H9M$($k(B.
        !           272:
        !           273: \section{$B$*$o$j$K(B}
        !           274:
        !           275: $B:#8e$NM=Dj$H$7$F(B, $B<!$N$h$&$J$3$H$r9M$($F$$$k(B.
        !           276:
        !           277: \begin{itemize}
        !           278: \item $B@5I8?t$N=`AGJ,2r(B
        !           279:
        !           280: \item $BBN$N0L?t$,B-$j$J$$>l9g$K(B, $B<+F0E*$K4pACBN$r3HBg$9$k(B
        !           281:
        !           282: \item $B3FItJ,$N8zN(2=(B
        !           283: \end{itemize}
1.1       noro      284:
                    285: \begin{thebibliography}{99}
                    286: \bibitem{B97-2}
                    287: Bernardin, L. (1997).
                    288: On square-free factorization of multivariate polynomials over a finite
                    289: field.
                    290: {\em Theoret.\ Comput.\ Sci.\/} {\bf 187}, 105--116.
                    291:
                    292: \bibitem{funny01}
                    293: M. Noro and K. Yokoyama (2002).
                    294: Yet Another Practical Implementation of Polynomial Factorization
                    295:  over Finite Fields.
                    296: Proceedings of ISSAC2002, ACM Press, 200--206.
                    297: \end{thebibliography}
                    298: \end{document}
                    299: