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version 1.1, 2002/12/09 02:09:23 version 1.3, 2003/12/13 12:52:12
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 % $OpenXM$  % $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}
Line 42 
Line 42 
 \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 307  $\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d)
Line 307  $\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d)
   
 \begin{slide}{}  \begin{slide}{}
 \begin{center}  \begin{center}
   \fbox{\fbc \large Timing data (Wang $B$NNc(B)}
   \end{center}
   
   {\tt OpenXM\_contrib2/asir2000/lib/fctrdata} $B$N(B
   
    {\tt Wang[1],\ldots,Wang[15]} $B$G%F%9%H(B.
   
   $B%^%7%s(B : Athlon 1900+
   
   Maple7 $B$HHf3S(B --- Maple7 $B$b(B Kernel $B$G=t1i;;$r%5%]!<%H$7$F$$$k$N$G(B, $B%"%s%U%'%"$G$O$J$$$@$m$&(B
   
   $B7k2L(B : $B0lIt(B ({\tt Wang[8]}) $B$r=|$$$FNI9%(B
   \end{slide}
   
   \begin{slide}{}
   \begin{center}
   \fbox{\fbc \large Timing data (Maple7)}
   \end{center}
   
   \begin{center}
   % &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \hline
   {\small
   \begin{tabular}{c|ccccccccc} \hline
   $p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
   2 & N & F & F & F & N & N & 0.01 & 1 & 0.01 \\ \hline
   3 & 0.07 & 0.1 & 0.07 & N & 0.4 & N & 0.01 & 0.02 & 0.06 \\ \hline
   5 & N & 0.05 & 0.08 & 3.5 & 0.2 & 0.4 & 0.01 & 0.6 & 0.1 \\ \hline
   7 & 0.08  & 0.1  & 0.1  & 0.25 & 0.6  & 0.5  & 0.02  & 1  & F \\ \hline
   \end{tabular}
   
   \begin{tabular}{c|cccccc} \hline
   $p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
   2 & F & N & 0.005 & 0.006 & 0.008 & F \\ \hline
   3 & 4 & N & 0.004 & 0.007 & 0.14  & 0.02 \\ \hline
   5 & 0.2  & F  & 0.005 & 0.006 & 0.03  &  0.4 \\ \hline
   7 & 0.6  & 14  & 0.005  & 0.16  & 0.04 & 0.6 \\ \hline
   \end{tabular}
   
   \begin{tabular}{c|ccccccccc} \hline
   $p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
   547 & 0.2& 0.2& 0.1& 0.3& 1& 1.2& 0.02& 6& F\\ \hline
   32003&  0.2& 0.2&  0.2& 0.4 & 1 & 1 & 0.02 & 4.2 & F  \\ \hline
   99981793 & 0.5 & 0.6 & 0.5 & 3 & 3 & 4.5 & 0.02 & N &  F\\ \hline
   \end{tabular}
   
   \begin{tabular}{c|cccccc} \hline
   $p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
   547 & 0.9 &3.3  & 0.005 & 0.2 & 0.1 & 0.4  \\ \hline
   32003 & 1.8 & 4.9 &0.006  & 0.3 & 0.1  & 0.4 \\ \hline
   99981793 & 2.6  & 11  & 0.006 & 0.9 & 0.5  & 1.4  \\ \hline
   \end{tabular}
   }
   \end{center}
   \end{slide}
   
   \begin{slide}{}
   \begin{center}
   \fbox{\fbc \large Timing data (Asir)}
   \end{center}
   
   \begin{center}
   % &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\ \hline
   {\small
   \begin{tabular}{c|ccccccccc} \hline
   $p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
   2(5) & 0.003 & 0.003 & 0.004 & 0.01 & 0.02 & 0.05 & 0.001 & 0.01 & 0.0003 \\ \hline
   3(5) & 0.003 & 0.002 & 0.005 & 0.003 & 0.003 & 0.1 & 0.002 & 0.001 & 0.003  \\ \hline
   5(2) & 0.004 & 0.003 & 0.004 & 0.02 & 0.06 & 0.4 & 0.002 & 0.4 & 0.005 \\ \hline
   7(2) & 0.004  & 0.004  & 0.005 & 0.03 & 0.1  & 0.1  & 0.004  & 1.8  & 0.2 \\ \hline
   \end{tabular}
   
   \begin{tabular}{c|cccccc} \hline
   $p$ & 10 & 11 & 12 & 13 & 14 &  15 \\ \hline
   2(5) & 0.03 & 0.07 & 0.0006 & 0.001 & 0.002 & 0.001  \\ \hline
   3(5) & 0.04 & 0.2 & 0.0001  & 0.0005 & 0.02  & 0.001 \\ \hline
   5(2) & 0.01  & 0.2  & 0.001 & 0.001 & 0.004  & 0.01  \\ \hline
   7(2) & 0.02  & 0.6  & 0.001  & 0.007  & 0.005 & 0.01 \\ \hline
   \end{tabular}
   
   \begin{tabular}{c|ccccccccc} \hline
   $p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
   547 & 0.004 & 0.004 & 0.005 & 0.03 & 0.05 & 0.2 & 0.02& 2& 0.2\\ \hline
   32003 & 0.004  &  0.004 & 0.005  &0.04  &0.07  & 0.2 & 0.004 & 2 & 0.2 \\ \hline
   99981793& 0.004 & 0.004& 0.005 & 0.03 & 0.03 & 0.2 & 0.004 & 4 & 0.2  \\ \hline
   \end{tabular}
   
   \begin{tabular}{c|cccccc} \hline
   $p$ & 10 & 11 & 12 & 13 & 14 &  15 \\ \hline
   547 & 0.04 & 0.3 & 0.001 &0.006  & 0.006 & 0.01  \\ \hline
   32003 & 0.04 &0.2  &0.001  &0.007  & 0.006  & 0.03 \\ \hline
   99981793 & 0.04  & 0.3  &0.001  & 0.008 & 0.008  & 0.01  \\ \hline
   \end{tabular}
   }
   \end{center}
   \end{slide}
   
   \begin{slide}{}
   \begin{center}
 \fbox{\fbc \large $B:#8e$NM=Dj(B}  \fbox{\fbc \large $B:#8e$NM=Dj(B}
 \end{center}  \end{center}
   
Line 318  $\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d)
Line 416  $\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d)
 \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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