Annotation of OpenXM/doc/Papers/rims2002-noro.tex, Revision 1.1
1.1 ! noro 1: % $OpenXM$
! 2: \documentclass{slides}
! 3: \usepackage{color}
! 4: \usepackage{rgb}
! 5: \usepackage{graphicx}
! 6: \usepackage{epsfig}
! 7: \newcommand{\qed}{$\Box$}
! 8: \newcommand{\mred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}}}
! 9: \newcommand{\tmred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}\limits^{\scriptstyle *}}}
! 10: \def\gr{Gr\"obner basis }
! 11: \def\st{\, s.t. \,}
! 12: \def\ni{\noindent}
! 13: \def\ve{\vfill\eject}
! 14: \textwidth 9.2in
! 15: \textheight 7.2in
! 16: \columnsep 0.33in
! 17: \topmargin -1in
! 18: \def\tc{\color{red}}
! 19: \def\fbc{\bf\color{MediumBlue}}
! 20: \def\itc{\color{brown}}
! 21: \def\urlc{\bf\color{DarkGreen}}
! 22: \def\bc{\color{LightGoldenrod1}}
! 23:
! 24: \def\HT{{\rm HT}}
! 25: \def\HC{{\rm HC}}
! 26: \def\GCD{{\rm GCD}}
! 27: \def\tdeg{{\rm tdeg}}
! 28: \def\pp{{\rm pp}}
! 29: \def\lc{{\rm lc}}
! 30: \def\Z{{\bf Z}}
! 31:
! 32: \title{\tc $B>.I8?tM-8BBN>e$NB?JQ?tB?9`<0$N0x?tJ,2r$K$D$$$F(B ($B$=$N(B 2)}
! 33:
! 34: \author{$BLnO$(B $B@59T(B ($B?@8MBg!&M}(B)}
! 35: \begin{document}
! 36: \large
! 37: \setlength{\parskip}{0pt}
! 38: \maketitle
! 39:
! 40: \begin{slide}{}
! 41: \begin{center}
! 42: \fbox{\fbc \large $BL5J?J}J,2r(B ($BI|=,(B)}
! 43: \end{center}
! 44:
! 45: modification of Bernardin's algorithm [Ber97]
! 46:
! 47: $f \in F[x_1,\ldots,x_n]$, $F$ : $BM-8BBN(B $Char(F) = p$
! 48:
! 49: $f = FGH$, where $F=\prod f_i^{a_i}$,
! 50: $G=\prod g_j^{b_j}$,
! 51: $H=\prod h_k^{c_k}$
! 52:
! 53: $f_i, g_j, h_k$ : $BL5J?J}(B, $B8_$$$KAG(B.
! 54:
! 55: $'$ $B$r(B $d/dx_1$ $B$H$7$F(B
! 56: $f_i' \neq 0$, $p \not{|}a_j$, $p | b_j$, $h_k' = 0$
! 57: $B$H=q$/$H(B
! 58:
! 59: $f' = F'GH$ $B$9$k$H(B $GCD(f,f') = GCD(F,F')GH$
! 60:
! 61: $GCD(F,F') = \prod f_i^{a_i-1}$ $B$@$+$i(B $f/GCD(f,f')=\prod f_i$
! 62:
! 63: $\prod f_i$ $B$G(B $f$ $B$r7+$jJV$73d$k$3$H$G(B, $f_1$ ($B=EJ#EY:G>.(B)
! 64: $B$,5a$^$k(B
! 65:
! 66: $\Rightarrow$ $F$ $B$,A4$FJ,2r$G$-$k(B
! 67:
! 68: \end{slide}
! 69:
! 70: \begin{slide}{}
! 71: \begin{center}
! 72: \fbox{\fbc \large $BL5J?J}J,2r(B ($B$D$E$-(B) }
! 73: \end{center}
! 74:
! 75: $B;D$j(B $f = GH$ $B$G(B, $f' = 0$
! 76:
! 77: $B$3$l$r(B $x_i$ $B$K$D$$$F7+$jJV$7$F;D$C$?(B $f$
! 78:
! 79: $\Rightarrow$ $df/dx_1 = \ldots = df/dx_n = 0$
! 80:
! 81: $\Rightarrow$ $B$3$l$O(B, $BA4$F$N;X?t$,(B $p$ $B$G3d$j@Z$l$k$3$H$r0UL#$9$k(B
! 82:
! 83: $\Rightarrow$ $F$ $B$OM-8BBN$@$+$i(B $f = g^p$ $B$H=q$1$k(B
! 84:
! 85: $\Rightarrow$ $g$ $B$KBP$7$F%"%k%4%j%:%`$rE,MQ(B
! 86: \end{slide}
! 87:
! 88:
! 89: \begin{slide}{}
! 90: \begin{center}
! 91: \fbox{\fbc \large $B<BAu>e$N:$Fq(B : $\GCD(f,f')$ $B$N7W;;(B}
! 92: \end{center}
! 93:
! 94: $BI8?t$,(B 0 $B$N>l9g(B $\GCD(g,f'/g)=1$ $\Rightarrow$ $BB?JQ?t$N(B Hensel $B9=@.$,(B
! 95: $B;H$($k(B
! 96:
! 97: $B@5I8?t$N>l9g(B $$\GCD(g,f'/g) = \GCD(\GCD(F,F')GH,F'/\GCD(F,F'))$$
! 98: $\Rightarrow$ $GH$ $B$NB8:_$N$?$a(B GCD $B$,(B 1 $B$H$O8B$i$J$$(B.
! 99:
! 100: $\Rightarrow$ $B$d$`$J$/(B Brown $B$N%"%k%4%j%:%`(B ($BCf9q>jM>DjM}$K$h$k(B GCD $B$N(B
! 101: $B7W;;(B) $B$rMQ$$$F$$$k(B.
! 102: \end{slide}
! 103:
! 104: \begin{slide}{}
! 105: \begin{center}
! 106: \fbox{\fbc \large GCD $B$N7W;;(B}
! 107: \end{center}
! 108:
! 109: \begin{tabbing}
! 110: $BF~NO(B : $f_1,\ldots,f_m \in K[X]$ ($K$ $B$OBN(B, $X$ $B$OJQ?t$N=89g(B)\\
! 111: $B=PNO(B : $\GCD(f_1,\ldots,f_m)$\\
! 112: $y \leftarrow$ $BE,Ev$JJQ?t(B; $Z \leftarrow X\setminus \{y\}$\\
! 113: $< \leftarrow K[Z]$ $B$NE,Ev$J9`=g=x(B; $B0J2<(B $f_i \in K[y][Z]$ $B$H$_$J$9(B\\
! 114: $h_i(y) \leftarrow \HT_<(f_i)$; $h_g(y) \leftarrow \GCD(h_1,\ldots,h_m)$\\
! 115: $g \leftarrow 0$; $M \leftarrow 1$\\
! 116: do \= \\
! 117: \> $a \leftarrow $ $BL$;HMQ$N(B $K$ $B$N85(B\\
! 118: \> $g_a \leftarrow \GCD(f_1|_{y=a},\ldots,f_m|_{y=a})$\\
! 119: \> if \= $g \neq 0$ $B$+$D(B $\HT_<(g) = \HT_<(g_a)$ then \\
! 120: \> \> $adj \leftarrow h_g(a)/\HC_<(g_a)\cdot g_a - g(a)$
! 121: \end{tabbing}
! 122: \end{slide}
! 123:
! 124: \begin{slide}{}
! 125: \begin{tabbing}
! 126: do \= if \= \kill
! 127: \> \> if \= $adj = 0$ $B$+$D(B, $B$9$Y$F$N(B $f_i$ $B$KBP$7(B $g | h_g\cdot f_i$ then \\
! 128: \> \> \> return $\pp(g)$\\
! 129: \> \> endif\\
! 130: \> \> $g \leftarrow g+adj \cdot M(a)^{-1} \cdot M$; $M \leftarrow M\cdot (y-a)$\\
! 131: \> else if $\tdeg(\HT_<(g)) > \tdeg(\HT_<(g_a))$ then \\
! 132: \> \> $g \leftarrow g_a$; $M \leftarrow y-a$\\
! 133: \> else if $\tdeg(\HT_<(g)) = \tdeg(\HT_<(g_a))$ then \\
! 134: \> \> $g \leftarrow 0$; $M \leftarrow 1$\\
! 135: \> endif\\
! 136: end do
! 137: \end{tabbing}
! 138: \end{slide}
! 139:
! 140: \begin{slide}{}
! 141: \begin{center}
! 142: \fbox{\fbc \large $BFsJQ?t$X$N5"Ce(B}
! 143: \end{center}
! 144:
! 145: \begin{itemize}
! 146: \item $B<gJQ?t(B $x$ $B$NA*Br(B
! 147:
! 148: $x$ $B$K4X$9$kHyJ,$,>C$($J$$$h$&$KA*$V(B.
! 149:
! 150: \item $B=>JQ?t(B $y$ $B$NA*Br(B
! 151:
! 152: $K[x,y]$ $B$G0x?tJ,2r$7$F(B, $Z=X\setminus \{x,y\}$ $B$K4X$7$F(B Hensel $B9=@.(B
! 153:
! 154: ($B0lJQ?t$^$GMn$9$H%K%;0x;R$,BgNLH/@8(B)
! 155:
! 156: \item $B$=$l0J30$NJQ?t(B $Z$ $B$X$NBeF~CM$NA*Br(B
! 157:
! 158: $f_a(x,y) = f(x,y,a)$ $B$,L5J?J}$K$J$k$h$&$KA*$V(B.
! 159:
! 160: $B$5$i$K(B, $f_a|_{y=0}$ $B$bL5J?J}$K$J$k$h$&$K(B, $y\leftarrow y+c$ $B$H(B
! 161: $BJ?9T0\F0(B.
! 162: \end{itemize}
! 163:
! 164: \end{slide}
! 165:
! 166: \begin{slide}{}
! 167: \begin{center}
! 168: \fbox{\fbc \large $K[y]$ $B>e$G$N(B Hensel $B9=@.(B ($BA0=hM}(B)}
! 169: \end{center}
! 170:
! 171: $f_a(x,y)$ $B$N0x;R$r(B 2 $BAH$K$o$1(B $f_a(x,y) = g_0(x,y)h_0(x,y)$
! 172: $B$+$i(B $K[y]$ $B>e$G(B Hensel $B9=@.(B
! 173:
! 174: $g_0$, $h_0$ $B$N(B $x$ $B$K4X$9$k<g78?t$N7h$aJ}(B
! 175:
! 176: $B<g78?tLdBj$N2sHr(B : $B??$N0x;R$N<g78?t$H$J$k$Y$/6a$$$b$N$r$"$i$+$8$a8GDj(B
! 177:
! 178: $B??$N0x;R$N<g78?t$O(B $\lc_x(f)$ $B$N0x;R$G$"$k$3$H$r;H$C$?8+@Q$j(B
! 179: \end{slide}
! 180:
! 181: \begin{slide}{}
! 182: \begin{center}
! 183: \fbox{\fbc \large $B<g78?t$N8+@Q$j(B}
! 184: \end{center}
! 185: \begin{enumerate}
! 186: \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).
! 187:
! 188: \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
! 189: $B?t$($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.
! 190: $BF1MM$K(B $h_0$ $B$KBP$7$F$b(B $\lc_h$ $B$r5a$a$k(B.
! 191:
! 192: $B$b$7(B,
! 193: $\lc_x(g_0) \not{|}\, \lc_g(a)$ $B$^$?$O(B $\lc_x(h_0) \not{|}\, \lc_h(a)$
! 194: $B$^$?$O(B,
! 195: $\lc_x(f) \not{|}\, \lc_g \cdot \lc_h$
! 196: $B$J$i$P(B, $B$=$l$O(B, $f_a$ $B$N0x;R$NAH9g$;$,@5$7$/$J$$$3$H$r0UL#$9$k$N$G(B,
! 197: $g_0$, $h_0$ $B$r$H$jD>$9(B.
! 198:
! 199: \item
! 200:
! 201: $g_0 \leftarrow \lc_g(a)/\lc_x(g_0)\cdot g_0$ $B$N<g78?t$r(B $\lc_g$ $B$GCV$-49$($?$b$N(B
! 202:
! 203: $h_0 \leftarrow \lc_h(a)/\lc_x(h_0)\cdot h_0$ $B$N<g78?t$r(B $\lc_h$ $B$GCV$-49$($?$b$N(B
! 204:
! 205: $f \leftarrow \lc_g\cdot \lc_h/\lc_x(f) \cdot f$
! 206:
! 207: $B$H$9$k(B. $B$3$N;~(B, $f = g_0h_0$ $B$H$J$C$F$$$k(B.
! 208: \end{enumerate}
! 209: \end{slide}
! 210:
! 211: \begin{slide}{}
! 212: \begin{center}
! 213: \fbox{\fbc \large $K[y]$ $B>e$G$N(B Hensel $B9=@.(B}
! 214: \end{center}
! 215:
! 216: $f=g_0h_0$ $B$G(B, $g_0$ $B$,@5$7$$0x;R$N<M1F(B, $\lc_x(g_0)$ $B$H2>Dj(B
! 217:
! 218: $g_0$, $h_0$ $B$+$i(B $K[y]$ $B>e$N(BHensel $B9=@.$K$h$j(B,
! 219: $$f=g_kh_k \bmod I^{k+1}$$
! 220: $B$H;}$A>e$2$k(B ($I = \langle z_1-a_1,\ldots,z_{n-2}-a_{n-2} \rangle$).
! 221:
! 222: $z_i \rightarrow z_i+a_i$ $B$J$kJ?9T0\F0$K$h$j(B,
! 223: $I=\langle z_1,\ldots,z_{n-2} \rangle$
! 224:
! 225: $BDL>o$N(B EZ $BK!(B : $\Z/(p^l)$ $B$G7W;;(B ($B78?t$KJ,?t$,8=$l$k$N$rHr$1$k(B)
! 226:
! 227: $B:#2s$N<BAu(B : $\deg_y(f) > d$ $B$J$k(B $d$ $B$KBP$7(B, $K[y]/(y^d)$
! 228: $B>e$G7W;;$9$k(B. ($K[y]$ $B$G$N>&BN$G$N1i;;$rHr$1$k(B)
! 229:
! 230: $B$3$l$O(B, $u g_0(a)+v h_0(a)=1 \bmod y^d$ $B$H$J$k(B $u, v \in K[y]$
! 231: $B$K$h$j2DG=(B
! 232:
! 233: Hensel $B9=@.$O(B $\bmod\, y^d$ $B$G9T$&(B.
! 234: \end{slide}
! 235:
! 236: \begin{slide}{}
! 237: \begin{center}
! 238: \fbox{\fbc \large Hensel $B9=@.(B}
! 239: \end{center}
! 240: $f = g_kh_k \bmod (I^{k+1},y^d)$ $B$@$,(B,
! 241: $B==J,Bg$-$$(B $k$ $B$KBP$7(B $f = g_kh_k$
! 242:
! 243: $u$, $v$ $B$N7W;;(B : Hensel $B9=@.(B
! 244: ($g_0(a)|_{y=0}$, $h_0(a)_{y=0}$ $B$,8_$$$KAG(B)
! 245:
! 246: $K[y]$ $B>e$N(B Hensel $B9=@.$O<!$N$h$&$K9T$&(B.
! 247:
! 248: \begin{enumerate}
! 249: \item $f-g_{k-1}h_{k-1} = \sum_t F_t t \bmod (I^k,y^d)$ $B$H=q$/(B.
! 250: $B$3$3$G(B $t \in I^k$ $B$OC19`<0(B, $F_t \in K[y][x]$.
! 251: \item $G_th_0+H_tg_0 = F_t \bmod y^d$ $B$H$J$k(B $G_t, H_t \in K[y][x]$ $B$r7W;;$9$k(B.
! 252: $B$3$l$O(B $u$, $v$ $B$r;H$C$F:n$l$k(B.
! 253: \item $g_{k+1} \leftarrow g_k + \sum_t G_t t$,
! 254: $h_{k+1} \leftarrow h_k + \sum_t H_t t$ $B$H$9$l$P(B $f = g_{k+1}h_{k+1} \bmod (I^{k+1},y^d)$.
! 255: \end{enumerate}
! 256:
! 257: $\sum_t G_t t = 0$ $B$^$?$O(B $\sum_t H_t t = 0$ $\Rightarrow$ $B;n$73d$j(B
! 258:
! 259: $g_k$ $B$^$?$O(B $h_k$ $B$G(B $f$ $B$r3d$C$F$_$k$3$H$G(B, $B<!?t$N>e8B$^$G(B Hensel $B9=@.(B
! 260: $B$;$:$K(B, $B??$N0x;R$r8!=P$G$-$k(B.
! 261: \end{slide}
! 262:
! 263: \begin{slide}{}
! 264: \begin{center}
! 265: \fbox{\fbc \large $BM-8BBN$NI=8=$K$D$$$F(B}
! 266: \end{center}
! 267:
! 268: $B3F%"%k%4%j%:%`$K$*$$$F(B, $B78?tBN$N0L?t$,==J,Bg$-$$I,MW$"$j(B.
! 269:
! 270: $BBeF~$9$kE@$N?t$,ITB-$9$k(B $\Rightarrow$ $BBe?t3HBg(B
! 271:
! 272: $B7W;;8zN($,Mn$A$J$$$h$&(B, $B86;O:,$rMQ$$$?I=8=$r<BAu(B
! 273:
! 274: $B7gE@(B : $B<BMQE*$J0L?t$,(B $2^{16}$ $BDxEY$K8B$i$l$k(B
! 275:
! 276: $B2~NI(B : $BI8?t$,(B $2^{14}$ $B0J2<$N>l9g$K$O(B, $B86;O:,I=8=(B
! 277:
! 278: $B$=$l0J>e$N>l9g$K$O(B, $BDL>o$NI=8=(B
! 279: ($B<BMQ>e==J,$KBeF~$9$kE@$,F@$i$l$k$+$i(B)
! 280:
! 281: $\Rightarrow$ $B0L?t$,(B $2^{29}$ $BDxEY$^$G$NAGBN>e$G(B, $BB?JQ?tB?9`<0(B
! 282: $B$N0x?tJ,2r$,2DG=(B
! 283: \end{slide}
! 284:
! 285: \begin{slide}
! 286: \begin{center}
! 287: \fbox{\fbc \large $B78?t4D$H$7$F$N(B $K[y]/(y^d)$ $B$K$D$$$F(B}
! 288: \end{center}
! 289:
! 290: Hensel $B9=@.$K$*$$$F(B $K[y]/(y^d)$ $B$r78?t4D$H$7$F07$&(B
! 291:
! 292: Asir : $B>.0L?tM-8BBN(B $K$ $B$NBe?t3HBg$rI=8=$9$k7?(B (GFSN) $B$,$"$k(B
! 293:
! 294: $K[y]/(m(y))$ $B$H$7$FI=8=(B $\Rightarrow$ $m(y)=y^d$ $B$H$7$FN.MQ(B
! 295:
! 296: $B5U857W;;$K$D$$$F$O(B, 0 $B$G$J$$Dj?t9`$r;}$DB?9`<0$O2D5U(B ($B8_=|K!(B)
! 297:
! 298: $\lc_x \neq 0$ $B$h$j(B $K[y]/(y^d)$ $B$,$3$NJ}K!$G$G$-$k(B.
! 299:
! 300: $BB?JQ?tB?9`<0(B : Hensel $B9=@.$N:G=i$G(B, $B$3$N7?$N78?t$r;}$DB?9`<0$KJQ49(B
! 301:
! 302: $d$ $B$r%;%C%H$7$F$*$/(B $\Rightarrow$ $B7k2L$O<+F0E*$K(B $\bmod \, y^d$ $B$5$l$k(B
! 303:
! 304: $\Rightarrow$ $BDL>o$NB?9`<01i;;$K$h$j(B $K[y]/(y^d)$ $B78?t$N(B
! 305: $BB?9`<01i;;$,<B9T$G$-$k(B.
! 306: \end{slide}
! 307:
! 308: \begin{slide}{}
! 309: \begin{center}
! 310: \fbox{\fbc \large $B:#8e$NM=Dj(B}
! 311: \end{center}
! 312:
! 313: \begin{itemize}
! 314: \item $B@5I8?t$N=`AGJ,2r$N<BAu(B.
! 315:
! 316: \item $BBN$N0L?t$,B-$j$J$$>l9g$K(B, $B<+F0E*$K4pACBN$r3HBg$9$k(B.
! 317:
! 318: \item $BBN$NI8?t$,==J,Bg$-$$>l9g$K(B, $BL5J?J}J,2r$rI8?t(B 0 $B$H(B
! 319: $BF1MM$N(B Hensel $B9=@.$G9T$&$h$&$K$9$k(B.
! 320:
! 321: \item 2 $BJQ?t$N0x?tJ,2r$K$*$$$F(B, \cite{funny01} $B$G=R$Y$?(B, $BB?9`<0(B
! 322: $B;~4V%"%k%4%j%:%`$r<+F0E*$KA*Br$7$F<B9T$9$k(B.
! 323: \end{itemize}
! 324: \end{slide}
! 325:
! 326: \end{document}
! 327:
! 328: \begin{thebibliography}{99}
! 329: \bibitem{B97-2}
! 330: Bernardin, L. (1997).
! 331: On square-free factorization of multivariate polynomials over a finite
! 332: field.
! 333: {\em Theoret.\ Comput.\ Sci.\/} {\bf 187}, 105--116.
! 334:
! 335: \bibitem{funny01}
! 336: M. Noro and K. Yokoyama (2002).
! 337: Yet Another Practical Implementation of Polynomial Factorization
! 338: over Finite Fields.
! 339: Proceedings of ISSAC2002, ACM Press, 200--206.
! 340: \end{thebibliography}
! 341:
FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>