version 1.2, 2002/12/09 04:23:05 |
version 1.3, 2003/12/13 12:52:12 |
|
|
% $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} |
|
|
\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} |
|
|