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version 1.2, 2002/12/09 04:23:05 version 1.3, 2003/12/13 12:52:12
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 % $OpenXM: OpenXM/doc/Papers/rims2002-noro.tex,v 1.1 2002/12/09 02:09:23 noro Exp $  % $OpenXM: OpenXM/doc/Papers/rims2002-noro.tex,v 1.2 2002/12/09 04:23:05 noro Exp $
 \documentclass{slides}  \documentclass{slides}
 \usepackage{color}  \usepackage{color}
 \usepackage{rgb}  \usepackage{rgb}
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 \fbox{\fbc \large $BL5J?J}J,2r(B ($BI|=,(B)}  \fbox{\fbc \large $BL5J?J}J,2r(B ($BI|=,(B)}
 \end{center}  \end{center}
   
 modification of Bernardin's algorithm [Ber97]  modification of Bernardin's algorithm [1]
   
 $f \in F[x_1,\ldots,x_n]$, $F$ : $BM-8BBN(B $Char(F) = p$  $f \in F[x_1,\ldots,x_n]$, $F$ : $BM-8BBN(B $Char(F) = p$
   
Line 53  $H=\prod h_k^{c_k}$ 
Line 53  $H=\prod h_k^{c_k}$ 
 $f_i, g_j, h_k$ : $BL5J?J}(B, $B8_$$$KAG(B.  $f_i, g_j, h_k$ : $BL5J?J}(B, $B8_$$$KAG(B.
   
 $'$ $B$r(B $d/dx_1$ $B$H$7$F(B  $'$ $B$r(B $d/dx_1$ $B$H$7$F(B
 $f_i' \neq 0$, $p \not{|}a_j$, $p | b_j$, $h_k' = 0$  $f_i' \neq 0$, $p {\not|}a_j$, $p | b_j$, $h_k' = 0$
 $B$H=q$/$H(B  $B$H=q$/$H(B
   
 $f' = F'GH$ $B$9$k$H(B $GCD(f,f') = GCD(F,F')GH$  $f' = F'GH$ $B$9$k$H(B $GCD(f,f') = GCD(F,F')GH$
Line 190  $g_0$, $h_0$ $B$N(B $x$ $B$K4X$9$k<g78?t$N7h$aJ}(B
Line 190  $g_0$, $h_0$ $B$N(B $x$ $B$K4X$9$k<g78?t$N7h$aJ}(B
 $BF1MM$K(B $h_0$ $B$KBP$7$F$b(B $\lc_h$ $B$r5a$a$k(B.  $BF1MM$K(B $h_0$ $B$KBP$7$F$b(B $\lc_h$ $B$r5a$a$k(B.
   
 $B$b$7(B,  $B$b$7(B,
 $\lc_x(g_0) \not{|}\, \lc_g(a)$ $B$^$?$O(B $\lc_x(h_0) \not{|}\, \lc_h(a)$  $\lc_x(g_0) {\not|}\, \lc_g(a)$ $B$^$?$O(B $\lc_x(h_0) {\not|}\, \lc_h(a)$
 $B$^$?$O(B,  $B$^$?$O(B,
 $\lc_x(f) \not{|}\, \lc_g \cdot \lc_h$  $\lc_x(f) {\not|}\, \lc_g \cdot \lc_h$
 $B$J$i$P(B, $B$=$l$O(B, $f_a$ $B$N0x;R$NAH9g$;$,@5$7$/$J$$$3$H$r0UL#$9$k$N$G(B,  $B$J$i$P(B, $B$=$l$O(B, $f_a$ $B$N0x;R$NAH9g$;$,@5$7$/$J$$$3$H$r0UL#$9$k$N$G(B,
 $g_0$, $h_0$ $B$r$H$jD>$9(B.  $g_0$, $h_0$ $B$r$H$jD>$9(B.
   
Line 238  Hensel $B9=@.$O(B $\bmod\,  y^d$ $B$G9T$&(B. 
Line 238  Hensel $B9=@.$O(B $\bmod\,  y^d$ $B$G9T$&(B. 
 \fbox{\fbc \large Hensel $B9=@.(B}  \fbox{\fbc \large Hensel $B9=@.(B}
 \end{center}  \end{center}
 $f = g_kh_k \bmod (I^{k+1},y^d)$ $B$@$,(B,  $f = g_kh_k \bmod (I^{k+1},y^d)$ $B$@$,(B,
 $B==J,Bg$-$$(B $k$ $B$KBP$7(B $f = g_kh_k$  $k$ $B==J,Bg(B $\Rightarrow$ $f = g_kh_k$
   
 $u$, $v$ $B$N7W;;(B : Hensel $B9=@.(B  $u$, $v$ : Hensel $B9=@.(B
 ($g_0(a)|_{y=0}$, $h_0(a)_{y=0}$ $B$,8_$$$KAG(B)  ($g_0(a)|_{y=0}$, $h_0(a)|_{y=0}$ $B$,8_$$$KAG(B)
   
 $K[y]$ $B>e$N(B Hensel $B9=@.$O<!$N$h$&$K9T$&(B.  $K[y]$ $B>e$N(B Hensel $B9=@.$O<!$N$h$&$K9T$&(B.
   
Line 293  Asir : $B>.0L?tM-8BBN(B $K$ $B$NBe?t3HBg$rI=8=$9$k7
Line 293  Asir : $B>.0L?tM-8BBN(B $K$ $B$NBe?t3HBg$rI=8=$9$k7
   
 $K[y]/(m(y))$ $B$H$7$FI=8=(B $\Rightarrow$ $m(y)=y^d$ $B$H$7$FN.MQ(B  $K[y]/(m(y))$ $B$H$7$FI=8=(B $\Rightarrow$ $m(y)=y^d$ $B$H$7$FN.MQ(B
   
 $B5U857W;;$K$D$$$F$O(B, 0 $B$G$J$$Dj?t9`$r;}$DB?9`<0$O2D5U(B ($B8_=|K!(B)  $B5U857W;;(B : $BDj?t9`$,(B 0 $B$G$J$$B?9`<0$O2D5U(B ($B8_=|K!(B)
   
 $\lc_x \neq 0$ $B$h$j(B $K[y]/(y^d)$ $B$,$3$NJ}K!$G$G$-$k(B.  $\lc_x(g)$ $B$NDj?t9`(B $\neq 0$ $B$h$j$3$NJ}K!$G7W;;2DG=(B
   
 $BB?JQ?tB?9`<0(B : Hensel $B9=@.$N:G=i$G(B, $B$3$N7?$N78?t$r;}$DB?9`<0$KJQ49(B  $BB?JQ?tB?9`<0(B : Hensel $B9=@.$N:G=i$G(B, $B$3$N7?$N78?t$r;}$DB?9`<0$KJQ49(B
   
Line 416  $p$ & 10 & 11 & 12 & 13 & 14 &  15 \\ \hline
Line 416  $p$ & 10 & 11 & 12 & 13 & 14 &  15 \\ \hline
 \item $BBN$NI8?t$,==J,Bg$-$$>l9g$K(B, $BL5J?J}J,2r$rI8?t(B 0 $B$H(B  \item $BBN$NI8?t$,==J,Bg$-$$>l9g$K(B, $BL5J?J}J,2r$rI8?t(B 0 $B$H(B
 $BF1MM$N(B Hensel $B9=@.$G9T$&$h$&$K$9$k(B.  $BF1MM$N(B Hensel $B9=@.$G9T$&$h$&$K$9$k(B.
   
 \item 2 $BJQ?t$N0x?tJ,2r$K$*$$$F(B, \cite{funny01} $B$G=R$Y$?(B, $BB?9`<0(B  \item 2 $BJQ?t$N0x?tJ,2r$K$*$$$F(B, [2] $B$G=R$Y$?(B, $BB?9`<0(B
 $B;~4V%"%k%4%j%:%`$r<+F0E*$KA*Br$7$F<B9T$9$k(B.  $B;~4V%"%k%4%j%:%`$r<+F0E*$KA*Br$7$F<B9T$9$k(B.
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 \end{document}  \begin{slide}{}
   \begin{center}
   \fbox{\fbc \large $BJ88%(B}
   \end{center}
   [1] Bernardin, L. (1997).
   
 \begin{thebibliography}{99}  
 \bibitem{B97-2}  
 Bernardin, L. (1997).  
 On square-free factorization of multivariate polynomials over a finite  On square-free factorization of multivariate polynomials over a finite
 field.  field.
 {\em Theoret.\ Comput.\ Sci.\/} {\bf 187}, 105--116.  {\em Theoret.\ Comput.\ Sci.\/} {\bf 187}, 105--116.
   
 \bibitem{funny01}  [2] M. Noro and K. Yokoyama (2002).
 M. Noro and K. Yokoyama (2002).  
 Yet Another Practical Implementation of Polynomial Factorization  Yet Another Practical Implementation of Polynomial Factorization
  over Finite Fields.   over Finite Fields.
 Proceedings of ISSAC2002, ACM Press, 200--206.  Proceedings of ISSAC2002, ACM Press, 200--206.
 \end{thebibliography}  \end{slide}
   \end{document}
   
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