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} |
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\fbox{\fbc \large $BL5J?J}J,2r(B ($BI|=,(B)} |
\fbox{\fbc \large $BL5J?J}J,2r(B ($BI|=,(B)} |
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\end{center} |
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modification of Bernardin's algorithm [Ber97] |
modification of Bernardin's algorithm [1] |
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$f \in F[x_1,\ldots,x_n]$, $F$ : $BM-8BBN(B $Char(F) = p$ |
$f \in F[x_1,\ldots,x_n]$, $F$ : $BM-8BBN(B $Char(F) = p$ |
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Line 53 $H=\prod h_k^{c_k}$ |
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Line 53 $H=\prod h_k^{c_k}$ |
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$f_i, g_j, h_k$ : $BL5J?J}(B, $B8_$$$KAG(B. |
$f_i, g_j, h_k$ : $BL5J?J}(B, $B8_$$$KAG(B. |
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$'$ $B$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 |
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$f' = F'GH$ $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 |
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Line 190 $g_0$, $h_0$ $B$N(B $x$ $B$K4X$9$k<g78?t$N7h$aJ}(B |
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$BF1MM$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. |
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$B$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. |
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Line 238 Hensel $B9=@.$O(B $\bmod\, y^d$ $B$G9T$&(B. |
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Line 238 Hensel $B9=@.$O(B $\bmod\, y^d$ $B$G9T$&(B. |
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\fbox{\fbc \large Hensel $B9=@.(B} |
\fbox{\fbc \large Hensel $B9=@.(B} |
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$f = g_kh_k \bmod (I^{k+1},y^d)$ $B$@$,(B, |
$f = g_kh_k \bmod (I^{k+1},y^d)$ $B$@$,(B, |
$B==J,Bg$-$$(B $k$ $B$KBP$7(B $f = g_kh_k$ |
$k$ $B==J,Bg(B $\Rightarrow$ $f = g_kh_k$ |
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$u$, $v$ $B$N7W;;(B : Hensel $B9=@.(B |
$u$, $v$ : Hensel $B9=@.(B |
($g_0(a)|_{y=0}$, $h_0(a)_{y=0}$ $B$,8_$$$KAG(B) |
($g_0(a)|_{y=0}$, $h_0(a)|_{y=0}$ $B$,8_$$$KAG(B) |
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$K[y]$ $B>e$N(B Hensel $B9=@.$O<!$N$h$&$K9T$&(B. |
$K[y]$ $B>e$N(B Hensel $B9=@.$O<!$N$h$&$K9T$&(B. |
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Line 293 Asir : $B>.0L?tM-8BBN(B $K$ $B$NBe?t3HBg$rI=8=$9$k7 |
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Line 293 Asir : $B>.0L?tM-8BBN(B $K$ $B$NBe?t3HBg$rI=8=$9$k7 |
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$K[y]/(m(y))$ $B$H$7$FI=8=(B $\Rightarrow$ $m(y)=y^d$ $B$H$7$FN.MQ(B |
$K[y]/(m(y))$ $B$H$7$FI=8=(B $\Rightarrow$ $m(y)=y^d$ $B$H$7$FN.MQ(B |
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$B5U857W;;$K$D$$$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) |
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$\lc_x \neq 0$ $B$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!$G7W;;2DG=(B |
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$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 |
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Line 307 $\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d) |
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Line 307 $\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d) |
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\begin{slide}{} |
\begin{slide}{} |
\begin{center} |
\begin{center} |
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\fbox{\fbc \large Timing data (Wang $B$NNc(B)} |
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{\tt OpenXM\_contrib2/asir2000/lib/fctrdata} $B$N(B |
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{\tt Wang[1],\ldots,Wang[15]} $B$G%F%9%H(B. |
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$B%^%7%s(B : Athlon 1900+ |
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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 |
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$B7k2L(B : $B0lIt(B ({\tt Wang[8]}) $B$r=|$$$FNI9%(B |
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\fbox{\fbc \large Timing data (Maple7)} |
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% & & & & & & & & & & & & & & & \\ \hline |
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{\small |
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\begin{tabular}{c|ccccccccc} \hline |
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$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline |
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2 & N & F & F & F & N & N & 0.01 & 1 & 0.01 \\ \hline |
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3 & 0.07 & 0.1 & 0.07 & N & 0.4 & N & 0.01 & 0.02 & 0.06 \\ \hline |
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5 & N & 0.05 & 0.08 & 3.5 & 0.2 & 0.4 & 0.01 & 0.6 & 0.1 \\ \hline |
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7 & 0.08 & 0.1 & 0.1 & 0.25 & 0.6 & 0.5 & 0.02 & 1 & F \\ \hline |
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\begin{tabular}{c|cccccc} \hline |
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$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline |
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2 & F & N & 0.005 & 0.006 & 0.008 & F \\ \hline |
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3 & 4 & N & 0.004 & 0.007 & 0.14 & 0.02 \\ \hline |
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5 & 0.2 & F & 0.005 & 0.006 & 0.03 & 0.4 \\ \hline |
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7 & 0.6 & 14 & 0.005 & 0.16 & 0.04 & 0.6 \\ \hline |
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\begin{tabular}{c|ccccccccc} \hline |
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$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline |
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547 & 0.2& 0.2& 0.1& 0.3& 1& 1.2& 0.02& 6& F\\ \hline |
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32003& 0.2& 0.2& 0.2& 0.4 & 1 & 1 & 0.02 & 4.2 & F \\ \hline |
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99981793 & 0.5 & 0.6 & 0.5 & 3 & 3 & 4.5 & 0.02 & N & F\\ \hline |
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\begin{tabular}{c|cccccc} \hline |
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$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline |
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547 & 0.9 &3.3 & 0.005 & 0.2 & 0.1 & 0.4 \\ \hline |
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32003 & 1.8 & 4.9 &0.006 & 0.3 & 0.1 & 0.4 \\ \hline |
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99981793 & 2.6 & 11 & 0.006 & 0.9 & 0.5 & 1.4 \\ \hline |
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\end{tabular} |
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} |
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\begin{slide}{} |
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\begin{center} |
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\fbox{\fbc \large Timing data (Asir)} |
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% & & & & & & & & & & & & & & & \\ \hline |
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{\small |
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\begin{tabular}{c|ccccccccc} \hline |
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$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline |
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2(5) & 0.003 & 0.003 & 0.004 & 0.01 & 0.02 & 0.05 & 0.001 & 0.01 & 0.0003 \\ \hline |
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3(5) & 0.003 & 0.002 & 0.005 & 0.003 & 0.003 & 0.1 & 0.002 & 0.001 & 0.003 \\ \hline |
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5(2) & 0.004 & 0.003 & 0.004 & 0.02 & 0.06 & 0.4 & 0.002 & 0.4 & 0.005 \\ \hline |
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7(2) & 0.004 & 0.004 & 0.005 & 0.03 & 0.1 & 0.1 & 0.004 & 1.8 & 0.2 \\ \hline |
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\begin{tabular}{c|cccccc} \hline |
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$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline |
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2(5) & 0.03 & 0.07 & 0.0006 & 0.001 & 0.002 & 0.001 \\ \hline |
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3(5) & 0.04 & 0.2 & 0.0001 & 0.0005 & 0.02 & 0.001 \\ \hline |
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5(2) & 0.01 & 0.2 & 0.001 & 0.001 & 0.004 & 0.01 \\ \hline |
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7(2) & 0.02 & 0.6 & 0.001 & 0.007 & 0.005 & 0.01 \\ \hline |
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\begin{tabular}{c|ccccccccc} \hline |
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$p$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline |
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547 & 0.004 & 0.004 & 0.005 & 0.03 & 0.05 & 0.2 & 0.02& 2& 0.2\\ \hline |
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32003 & 0.004 & 0.004 & 0.005 &0.04 &0.07 & 0.2 & 0.004 & 2 & 0.2 \\ \hline |
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99981793& 0.004 & 0.004& 0.005 & 0.03 & 0.03 & 0.2 & 0.004 & 4 & 0.2 \\ \hline |
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\begin{tabular}{c|cccccc} \hline |
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$p$ & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline |
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547 & 0.04 & 0.3 & 0.001 &0.006 & 0.006 & 0.01 \\ \hline |
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32003 & 0.04 &0.2 &0.001 &0.007 & 0.006 & 0.03 \\ \hline |
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99981793 & 0.04 & 0.3 &0.001 & 0.008 & 0.008 & 0.01 \\ \hline |
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} |
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\begin{center} |
\fbox{\fbc \large $B:#8e$NM=Dj(B} |
\fbox{\fbc \large $B:#8e$NM=Dj(B} |
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Line 318 $\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d) |
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Line 416 $\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d) |
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\item $BBN$NI8?t$,==J,Bg$-$$>l9g$K(B, $BL5J?J}J,2r$rI8?t(B 0 $B$H(B |
\item $BBN$NI8?t$,==J,Bg$-$$>l9g$K(B, $BL5J?J}J,2r$rI8?t(B 0 $B$H(B |
$BF1MM$N(B Hensel $B9=@.$G9T$&$h$&$K$9$k(B. |
$BF1MM$N(B Hensel $B9=@.$G9T$&$h$&$K$9$k(B. |
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\item 2 $BJQ?t$N0x?tJ,2r$K$*$$$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} |
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\end{document} |
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\begin{center} |
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\fbox{\fbc \large $BJ88%(B} |
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[1] Bernardin, L. (1997). |
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\begin{thebibliography}{99} |
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\bibitem{B97-2} |
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Bernardin, L. (1997). |
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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. |
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\bibitem{funny01} |
[2] M. Noro and K. Yokoyama (2002). |
M. Noro and K. Yokoyama (2002). |
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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} |
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\end{document} |
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