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Annotation of OpenXM/doc/compalg/ex.tex, Revision 1.1

1.1     ! noro        1: \chapter{$B%?%$%_%s%0%G!<%?$*$h$SNc(B}
        !             2:
        !             3: \section{$B%?%$%_%s%0%G!<%?(B : $BJ}Dx<0(B}
        !             4: \begin{tabbing}
        !             5: $MMM\;\;$ \= \kill
        !             6: $C(n)$ \> The cyclic n-roots system of n variables. (Faugere {\it et al.},1993).\\
        !             7:        \> $\{f_1,\cdots,f_n\}$ where
        !             8:          $f_k=
        !             9: \displaystyle{\sum_{i=1}^n\prod_{j=i}^{k+j-1}c_{j \bmod n}-\delta_{k,n}}$.
        !            10: ($\delta$ is the Kronecker symbol.) \\
        !            11: \> The variables and ordering : $c_n \succ c_{n-1} \succ \cdots \succ c_1$\\
        !            12: $K(n)$ \> The Katsura system of n+1 variables. \\
        !            13:       \> $\{u_l - \sum_{i=-n}^n u_i u_{l-i} (l = 0,\cdots, n-1),
        !            14:            \sum_{l=-n}^n u_l - 1\}$\\
        !            15:       \> The variables and ordering : $u_0 \succ u_1 \succ \cdots \succ u_n$.\\
        !            16:       \> Conditions : $u_{-l} = u_l$ and $u_l = 0 (|l| > n)$. \\
        !            17: $R(n)$ \> {\tt e7} in Rouillier (1996). \\
        !            18:       \> $\{-1/2+\sum_{i=1}^n(-1)^{i+1}x_i^k (k=2, \cdots, n+1) \}$\\
        !            19:       \> The variables and ordering : $x_n \succ x_{n-1} \succ \cdots \succ x_1$.\\
        !            20: $D(3)$ \> {\tt e8} in Rouillier (1996). \\
        !            21:        \> $\{f_0,f_1,f_2,\cdots,f_7\}$\\
        !            22:        \> {\scriptsize $f_0=-420y^2-280zy-168uy-140vy-120sy-210ty-105ay+12600y-13440$}\\
        !            23:        \> {\scriptsize $f_1=-840zy-630z^2-420uz-360vz-315sz-504tz-280az+18900z-20160$}\\
        !            24:        \> {\scriptsize $f_2=-630ty-504tz-360tu-315tv-280ts-420t^2-252at+12600t-13440$}\\
        !            25:        \> {\scriptsize $f_3=-5544uy-4620uz-3465u^2-3080vu-2772su-3960tu-2520au+103950u-110880$}\\
        !            26:        \> {\scriptsize $f_4=-4620vy-3960vz-3080vu-2772v^2-2520sv-3465tv-2310av+83160v-88704$}\\
        !            27:        \> {\scriptsize $f_5=-51480sy-45045sz-36036su-32760sv-30030s^2-40040ts-27720as+900900s-960960$}\\
        !            28:        \> {\scriptsize $f_6=-45045ay-40040az-32760au-30030av-27720as-36036at-25740a^2+772200a-823680$}\\
        !            29:        \> {\scriptsize $f_7=-40040by-36036bz-30030bu-27720bv-25740bs-32760bt-24024ba+675675b-720720$}\\
        !            30:        \normalsize
        !            31:          \> The variables and ordering : $b \succ a \succ s \succ v \succ u \succ t \succ z \succ y$.\\
        !            32: $Rose$ \> The Rose system.\\
        !            33: %      \> $\{u_4^4-20/7a_{46}^2, a_{46}^2u_3^4+7/10a_{46}u_3^4+7/48u_3^4-50/27a_{46}^2-35/27a_{46}-49/216,$\\
        !            34: %      \> $a_{46}^5u_4^3+7/5a_{46}^4u_4^3+609/1000a_{46}^3u_4^3+49/1250a_{46}^2u_4^3$\\
        !            35: %      \> $-27391/800000a_{46}u_4^3-1029/160000u_4^3+3/7a_{46}^5u_3u_4^2+3/5a_{46}^6u_3u_4^2$\\
        !            36: %      \> $+63/200a_{46}^3u_3u_4^2+147/2000a_{46}^2u_3u_4^2+4137/800000a_{46}u_3u_4^2$\\
        !            37: %      \> $-7/20a_{46}^4u_3^2u_4-77/125a_{46}^3u_3^2u_4-23863/60000a_{46}^2u_3^2u_4$\\
        !            38: %      \> $-1078/9375a_{46}u_3^2u_4-24353/1920000u_3^2u_4-3/20a_{46}^4u_3^3-21/100a_{46}^3u_3^3$\\
        !            39: %      \> $-91/800a_{46}^2u_3^3-5887/200000a_{46}u_3^3-343/128000u_3^3 \}$\\
        !            40:     \> $O_1$ : $u_3 \succ u_4 \succ a_{46}$, $O_2$ : $u_3 \succ a_{46} \succ u_4$.\\
        !            41: $Liu$ \> The Liu system.\\
        !            42:       \> $\{y(z-t)-x+a, z(t-x)-y+a, t(x-y)-z+a, x(y-z)-t+a\}$\\
        !            43:       \> The variables and ordering : $x \succ y \succ z \succ t \succ a$.\\
        !            44: $Fate$ \> The Fateman system, appeared on NetNews. \\
        !            45:        \> $\{s^3+2r^3+2q^3+2p^3$, $s^5+2r^5+2q^5+2p^5$,\\
        !            46:        \> $-s^5+(r+q+p)s^4+(r^2+(2q+2p)r+q^2+2pq+p^2)s^3+(r^3+q^3+p^3)s^2$\\
        !            47:        \> $+(3r^4+(2q+2p)r^3+(4q^3+4p^3)r+3q^4+2pq^3+4p^3q+3p^4)s+(4q+4p)r^4$\\
        !            48:        \> $+(2q^2+4pq+2p^2)r^3+(4q^3+4p^3)r^2+(6q^4+4pq^3+8p^3q+6p^4)r$\\
        !            49:           \> $+4pq^4+2p^2q^3+4p^3q^2+6p^4q\}$\\
        !            50:        \> The variables and ordering : $p \succ q \succ r \succ s$.\\
        !            51: $hC(6)$ \> A homogenization of C(6). \\
        !            52:        \> $(C_6\backslash \{c_1c_2c_3c_4c_5c_6-1\})\cup \{c_1c_2c_3c_4c_5c_6-t^6\}$\\
        !            53:        \> The variables and ordering :
        !            54:           $c_1 \succ c_2 \succ c_3 \succ c_4 \succ c_5 \succ c_6 \succ t$.\\
        !            55: \end{tabbing}
        !            56:
        !            57:
        !            58: \section{$B%?%$%_%s%0%G!<%?(B : change of ordering}
        !            59: $B$3$3$G$O(B, $B$5$^$6$^$J(B change of ordering $B%"%k%4%j%:%`$N%?%$%_%s%0%G!<%?(B
        !            60: $B$r<($9(B. $B7WB,$O(B, PC (FreeBSD, 300MHz Pentium II, 512MB of memory) $B$G9T$C$?(B.
        !            61: $BC10L$OIC(B. garbage collection $B;~4V$O=|$$$F$"$k(B.
        !            62:
        !            63: $BM=$a7W;;$7$F$"$k(B DRL \gr $B4pDl$+$i=PH/$7$F(B, LEX \gr $B4pDl7W;;$9$k(B.
        !            64: $BMQ$$$k%"%k%4%j%:%`$O(B,
        !            65: TL ({\it tl\_guess$()$}),
        !            66: HTL ($B@F<!2=(B+{\it tl\_guess$()$}+$BHs@F2=(B),
        !            67: LA ({\it candidate\_by\_linear\_algebra$()$}; 0 $B<!85%7%9%F%`$N$_(B)
        !            68: $B$G$"$k(B.
        !            69: $BI=(B \ref{mcotype} $B$O(B DRL $B$+$i(B LEX $B$X$NJQ49$K$+$+$k;~4V$r$7$a$9(B.
        !            70: {\it DRL} $B$O(B, DRL $B$N7W;;;~4V$r<($9(B. $B%0%l%V%J4pDl%A%'%C%/$N8z2L$r(B
        !            71: $B<($9$?$a$K(B, {\it tl\_check$()$} $B$N;~4V$b<($9(B.
        !            72:
        !            73: \begin{table}[hbtp]
        !            74: \caption{Modular change of ordering}
        !            75: \label{mcotype}
        !            76: \begin{center}
        !            77: \begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
        !            78:        & $K(5)$ & $K(6)$ & $K(7)$ & $C(6)$ & $C(7)$ & $R(5)$ & $R(6)$ \\ \hline
        !            79: {\it DRL}&0.84 &8.4    &74     &3.1    &1616   &11     &1775   \\ \hline
        !            80: {\it TL}&$\infty$              &$\infty$               &$\infty$ &$\infty$     &$\infty$       &$\infty$       &$\infty$       \\ \hline
        !            81: {\it HTL}      &16     &1402   &$1.6\times 10^5$       &5.6    &$2\times 10^4$ &383    &$2.1\times 10^5$       \\ \hline
        !            82: {\it LA}       &4.7    &158    &6813   &4      &435    &9.5            &258            \\ \hline
        !            83: {\it tl\_check}        &2.3    &177    &$1.3\times 10^4$       &1.1    &2172   &3      &40             \\ \hline
        !            84: \end{tabular}
        !            85:
        !            86: \begin{tabular}{|c||c|c|c||c|c|c|} \hline
        !            87:        & $D(3)$ & $RoseO_1$ & $RoseO_2$ & $Liu$ & $Fate$ & $hC(6)$ \\ \hline
        !            88: {\it DRL}      &30     &0.19   &0.15   &0.06   &0.5    &7.2    \\ \hline
        !            89: {\it TL}       & $\infty$      &1.7    &354    &$\infty$       &4      &25     \\ \hline
        !            90: {\it HTL}      &$4.1\times 10^4$       &1.7    &36     &18     &4      &25     \\ \hline
        !            91: {\it LA}       &585    &3.3    &12     & --- & --- & --- \\ \hline
        !            92: {\it tl\_check}        &575            &0.6    &13     &17             &26     &24     \\ \hline
        !            93: \end{tabular}
        !            94: \end{center}
        !            95: \end{table}
        !            96:
        !            97: $B@0?t78?tB?9`<0$KBP$7(B, $B$=$N(B {\bf maginitude} $B$r(B, $B78?t$N%S%C%HD9$NOB$GDj5A$9$k(B.
        !            98: {\it TL} $B$H(B {\it HTL} $B$N:9$r8+$k$?$a$K(B,
        !            99: $BI=(B \ref{magnitude} $B$G(B, $B7W;;ESCf$K$*$1$k:GBg(B magnitude $B$r<($9(B.
        !           100:
        !           101: \begin{table}[hbtp]
        !           102: \caption{Maximal magnitude}
        !           103: \label{magnitude}
        !           104: \begin{center}
        !           105: \begin{tabular}{|c||c|c|c|c|c|c|} \hline
        !           106:        & $C(6)$ & $K(5)$ & $K(6)$ & $RoseO_1$ & $RoseO_2$ & Liu \\ \hline
        !           107: {\it TL}& $>$ 735380 & $> 2407737 $ & $>$ 57368231 & 69764 & 947321 & $>$ 327330 \\ \hline
        !           108: {\it HTL}& 1992 & 44187 & 422732 & 37220 & 70018 & 21095 \\ \hline
        !           109: \end{tabular}
        !           110: \end{center}
        !           111: \end{table}
        !           112:
        !           113: $BI=$h$jL@$i$+$K(B, {\it TL} $B$OHs@F<!B?9`<0$KBP$9$k%0%l%V%J4pDl7W;;$KIT8~$-(B
        !           114: $B$G$"$k$3$H$,$o$+$k(B. $B$5$i$K(B, $BI=(B \ref{mcotype} $B$O(B {\it HTL} $B$KBP$9$k(B
        !           115: {\it LA} $B$NM%0L@-$r<($7$F$$$k(B. $B$3$l$O(B, Buchberger $B%"%k%4%j%:%`$,(B
        !           116: Euclid $B$N8_=|K!$KBP1~$7$F$$$F(B, $BCf4V78?tKDD%$G8zN($,:81&$5$l$k$N(B
        !           117: $B$KBP$7(B, modular $B%"%k%4%j%:%`$N8zN($O7k2L$NBg$-$5$N$_$K0MB8$9$k(B
        !           118: $B$3$H$K$h$k(B.
        !           119:
        !           120: \section{$B%?%$%_%s%0%G!<%?(B : RUR}
        !           121:
        !           122: RUR $B$N(B modular $B7W;;$N%?%$%_%s%0%G!<%?$r<($9(B. $B7W;;4D6-$OA0@a$HF1MM$G$"(B
        !           123: $B$k(B. $B$3$3$G$O(B, $BM=$a(B modular $B7W;;$K$h$j(B separating element$B$r5a$a$F$"(B
        !           124: $B$k(B. $B$3$l$i$rMQ$$$F(B, $B$=$l$>$l<!$N$h$&$JB?9`<0$rE:2C$7$?%$%G%"%k$KBP$7(B,
        !           125: $w$ $B$K4X$9$k(B RUR $B7W;;$r9T$&(B.  $BI=$G(B, Quick Test $B$O(B modular $B7W;;$G(B $w$
        !           126: $B$,(B separating element $B$H$J$k$3$H$r%A%'%C%/$9$k;~4V(B, Normal Form $B$O(B,
        !           127: $B@~7AJ}Dx<0$r@8@.$9$k$?$a$N(B, monomial $B$N@55,7A$N7W;;(B, Linear Equation $B$O(B,
        !           128: $B@~7AJ}Dx<05a2r$N;~4V$G$"$k(B. $BI=(B \ref{maxblen} $B$G$O(B, LEX $B4pDl$H(B RUR $B$G(B
        !           129: $B78?t$NBg$-$5$,$I$N$/$i$$0c$&$+$r<($7$F$$$k(B.
        !           130:
        !           131: \begin{tabbing}
        !           132: $MMM\;\;$ \= \kill
        !           133: $C(6)$ \> $w-(c_1+3c_2+9c_3+27c_4+81c_5+243c_6)$\\
        !           134: $C(7)$ \> $w-(c_1+3c_2+9c_3+27c_4+81c_5+243c_6+729c_7)$\\
        !           135: $K(n)$ \> $w-u_n$\\
        !           136: $R(5)$ \> $w-(x_1-3x_2-2x_3+3x_4+2x_5)$\\
        !           137: $R(6)$ \> $w-(x_1-3x_2-2x_3+3x_4+2x_5-4x_6)$\\
        !           138: $D(3)$ \> $w-y$
        !           139: \end{tabbing}
        !           140:
        !           141: \begin{table}[h]
        !           142: \caption{$BF~NO%$%G%"%k$K4X$9$k%G!<%?(B}
        !           143: \begin{center}
        !           144: \begin{tabular}{|c||c|c|c|c||c|c|c|c|c|} \hline
        !           145:        & $K(5)$        & $K(6)$        & $K(7)$        & $K(8)$        & $C(6)$& $C(7)$        & $R(5)$ & $R(6)$ & $D(3)$ \\ \hline
        !           146: $\dim_{\Q} R/I$        & 32    & 64    & 128   & 256   & 156   & 924   &144    &576    & 128 \\ \hline
        !           147: DRL GB& 0.8    & 7.2   & 68    & 798   & 3.1   & 1616  & 11    & 1775  & 30    \\ \hline
        !           148: \end{tabular}
        !           149: \end{center}
        !           150: \end{table}
        !           151:
        !           152: \begin{table}[h]
        !           153: \caption{$B7W;;;~4V(B ($BIC(B)}
        !           154: \begin{center}
        !           155: \begin{tabular}{|c|c|c|c||c|c|c|c|c|} \hline
        !           156:        & $K(6)$& $K(7)$& $K(8)$& $C(6)$& $C(7)$& $R(5)$ & $R(6)$ & $D(3)$ \\ \hline
        !           157: Total  & 7.4   & 69    & 1209  & 4.6   & 1643  & 52    & 8768  & 67    \\ \hline
        !           158: Quick test& 0.4        & 3.2   & 26    & 0.5   & 57    & 6.5   & 384   & 3.1   \\ \hline
        !           159: Normal form& 1.1       & 12    & 308   & 1.4   & 762   & 15    & 2861  & 7.3   \\ \hline
        !           160: Linear equation& 4.1   & 43    & 775   & 1.4   & 641   & 22    & 3841  & 45    \\ \hline
        !           161: Garbage collection& 1.7        & 10    & 100   & 1.2   & 181   & 7.8   & 1681  & 11    \\ \hline
        !           162: \end{tabular}
        !           163: \end{center}
        !           164: \end{table}
        !           165:
        !           166: \begin{table}[h]
        !           167: \label{maxblen}
        !           168: \caption{Maximal bit length of coefficients in LEX basis and the RUR}
        !           169: \begin{center}
        !           170: \begin{tabular}{|c||c|c|c|c|c|} \hline
        !           171: & $K(5)$ & $K(6)$      & $K(7)$        & $K(8)$ & $D(3)$ \\ \hline
        !           172: LEX & 1421 & 6704 & 36181 & --- & 6589 \\ \hline
        !           173: RUR & 120 & 249 & 592 & 1258 & 821 \\ \hline
        !           174: \end{tabular}
        !           175: \end{center}
        !           176: \end{table}
        !           177:
        !           178: \section{$BNc(B : $B=`AGJ,2r(B}
        !           179:
        !           180: $B<!$NNc$O(B, symplectic integrator $B$H8F$P$l$k0BDj$J@QJ,%9%-!<%`$N(B
        !           181: $B?tCM7W;;K!$K4X$7$F8=$l$?J}Dx<07O$G$"$k(B \cite{SYMP}.
        !           182:
        !           183: \vskip\baselineskip
        !           184: {\small
        !           185: $\left\{
        !           186: \parbox[c]{6in}{
        !           187: $d_1+d_2+d_3+d_4=1, c_1+c_2+c_3+c_4=1,$\\
        !           188: $(6d_1c_2+(6d_1+6d_2)c_3+(6d_1+6d_2+6d_3)c_4)c_1
        !           189:  +(6d_2c_3+(6d_2+6d_3)c_4)c_2+6d_3c_4c_3=1,$\\
        !           190: $(3d_1^2+(6d_2+6d_3+6d_4)d_1+3d_2^2+(6d_3+6d_4)d_2+3d_3^2+6d_4d_3+3d_4^2)c_1$\\
        !           191: $+(3d_2^2+(6d_3+6d_4)d_2+3d_3^2+6d_4d_3+3d_4^2)c_2+(3d_3^2+6d_4d_3+3d_4^2)c_3+3d_4^2c_4=1,$\\
        !           192: $(3d_1+3d_2+3d_3+3d_4)c_1^2+((6d_2+6d_3+6d_4)c_2+(6d_3+6d_4)c_3+6d_4c_4)c_1$\\
        !           193: $+(3d_2+3d_3+3d_4)c_2^2+((6d_3+6d_4)c_3+6d_4c_4)c_2+(3d_3+3d_4)c_3^2+6d_4c_4c_3+3d_4c_4^2=1,$\\
        !           194: $(24d_2d_1c_3+(24d_2+24d_3)d_1c_4)c_2+(24d_3d_1+24d_3d_2)c_4c_3=1,$\\
        !           195: $(12d_2^2+(24d_3+24d_4)d_2+12d_3^2+24d_4d_3+12d_4^2)d_1c_2
        !           196: +((12d_3^2+24d_4d_3+12d_4^2)d_1$\\
        !           197: $+(12d_3^2+24d_4d_3+12d_4^2)d_2)c_3
        !           198: +(12d_4^2d_1+12d_4^2d_2+12d_4^2d_3)c_4=1,$\\
        !           199: $4d_1c_2^3+(12d_1c_3+12d_1c_4)c_2^2+(12d_1c_3^2+24d_1c_4c_3
        !           200: +12d_1c_4^2)c_2+(4d_1+4d_2)c_3^3$\\
        !           201: $+(12d_1+12d_2)c_4c_3^2+(12d_1+12d_2)c_4^2c_3+(4d_1+4d_2+4d_3)c_4^3=1$
        !           202: }
        !           203: \right.$}
        !           204:
        !           205: \vskip\baselineskip
        !           206: \noindent
        !           207: $B$3$l$r=`AGJ,2r$K$+$1$k$H(B, $B<!$NJ,2r$,F@$i$l$k(B.
        !           208:
        !           209: \vskip\baselineskip
        !           210: $\left\{
        !           211: \parbox[c]{8in}{
        !           212: $24c_4^2-6c_4+1=0$\\
        !           213: $c_1=-c_4+{1\over 4}$,
        !           214: $c_2=-c_4+{1\over 2}$,
        !           215: $c_3=c_4+{1\over 4}$
        !           216: $d_1=-2c_4+{1\over 2}$,
        !           217: $d_2={1\over 2}$,
        !           218: $d_3=2c_4$,
        !           219: $d_4=0$}
        !           220: \right.$
        !           221:
        !           222: $\left\{
        !           223: \parbox[c]{8in}{
        !           224: $6c_4^3-12c_4^2+6c_4-1=0$\\
        !           225: $c_1=0$,
        !           226: $c_2=c_4$,
        !           227: $c_3=-2c_4+1$
        !           228: $d_1={1\over 2}c_4$,
        !           229: $d_2=-{1\over 2}c_4+{1\over 2}$,
        !           230: $d_3=-{1\over 2}c_4+{1\over 2}$,
        !           231: $d_4={1\over 2}c_4$}
        !           232: \right.$
        !           233:
        !           234: $\left\{
        !           235: \parbox[c]{8in}{
        !           236: $48c_4^3-48c_4^2+12c_4-1=0$\\
        !           237: $c_1=c_4$,
        !           238: $c_2=-c_4+{1\over 2}$,
        !           239: $c_3=-c_4+{1\over 2}$
        !           240: $d_1=2c_4$,
        !           241: $d_2=-4c_4+1$,
        !           242: $d_3=2c_4$,
        !           243: $d_4=0$}
        !           244: \right.$
        !           245:
        !           246: $\left\{
        !           247: \parbox[c]{8in}{
        !           248: $6c_4^2-3c_4+1=0$\\
        !           249: $c_1=0$,
        !           250: $c_2=-c_4+{1\over 2}$,
        !           251: $c_3={1\over 2}$
        !           252: $d_1=-{1\over 2}c_4+{1\over 4}$,
        !           253: $d_2=-{1\over 2}c_4+{1\over 2}$,
        !           254: $d_3={1\over 2}c_4+{1\over 4}$,
        !           255: $d_4={1\over 2}c_4$}
        !           256: \right.$
        !           257:
        !           258: \section{$BNc(B : $BAPBP6J@~$N7W;;(B}
        !           259: $f(x_1,x_2) \in \Q[x_1,x_2]$ $B$H$7(B, $F$ $B$N(B total degree $B$r(B $d$ $B$H$9$l$P(B,
        !           260: $F(x_0,x_1,x_2)=x_0^df(x_1/x_0,x_2/x_0)$
        !           261: $B$O(B $d$ $B<!F1<!B?9`<0$G(B, $F$ $B$NDj5A$9$kBe?t6J@~$NAPBP6J@~$O(B,
        !           262: $$\left\{
        !           263: \parbox[c]{8in}{
        !           264: $u_i={{\partial F}\over {\partial x_i}} (x_0,x_1,x_2)$ $(i=0,1,2)$\\
        !           265: $F(x_0,x_1,x_2)=0$
        !           266: }
        !           267: \right.$$
        !           268: $B$+$i(B $x_0, x_1, x_2$ $B$r>C5n$7$FF@$i$l$k(B. $B>C5nK!$N0l$D$H$7$F%0%l%V%J4pDl(B
        !           269: $B$K$h$k>C5n$,2DG=$G$"$k(B.
        !           270: $$I = Id(
        !           271: u_0-{{\partial F}\over {\partial x_0}},
        !           272: u_1-{{\partial F}\over {\partial x_1}},
        !           273: u_2-{{\partial F}\over {\partial x_2}},
        !           274: F)$$
        !           275: $B$H$9$k;~(B, $\{x_0, x_1, x_2\}$ $\succ$ $\{u_0, u_1, u_2\}$ $B$J$kG$0U$N>C(B
        !           276: $B5n=g=x$K$h$j(B $I$ $B$N%0%l%V%J4pDl(B $GB(I)$ $B$r7W;;$9$l$P(B,
        !           277: $$I \cap \Q[u_0,u_1,u_2] = Id(GB(I) \cap Q[u_0,u_1,u_2]).$$
        !           278: $B0J2<$NNc$G(B, $V(g_i)$ $B$O(B $V(f_i)$ $B$NAPBP6J@~$G$"$k(B.
        !           279:
        !           280: \vskip\baselineskip
        !           281: $\left\{
        !           282: \parbox[c]{6in}{
        !           283: $f_1=x^5-x^3+y^2$\\
        !           284: $g_1=108x^7-108x^5+1017y^2x^4-16y^4x^3-4250y^2x^2+1800y^4x-108y^6+3125y^2$
        !           285: }
        !           286: \right.$
        !           287:
        !           288: \vskip\baselineskip
        !           289: $\left\{
        !           290: \parbox[c]{6in}{
        !           291: $f_2=x^6+3y^2x^4+(3y^4-4y^2)x^2+y^6$\\
        !           292: $g_2=-256x^6+(64y^4-192y^2+864)x^4+(-192y^4+1620y^2-729)x^2-256y^6+864y^4-729y^2$
        !           293: }
        !           294: \right.$
        !           295:
        !           296: \vskip\baselineskip
        !           297: $\left\{
        !           298: \parbox[c]{6in}{
        !           299: $f_3=2x^4-3yx^2+y^4-2y^3+y^2$\\
        !           300: $g_3=-12x^6+(-y^2+178y-37)x^4+(12y^3-768y^2+2208y+4608)x^2-32y^4+1024y^3-7680y^2-8192y-2048$
        !           301: }
        !           302: \right.$
        !           303:
        !           304: \vskip\baselineskip
        !           305: \begin{figure}[hbtp]
        !           306: \begin{tabular}{cc}
        !           307: \begin{minipage}{.5\hsize}
        !           308: \begin{center}
        !           309: \epsfxsize=7cm
        !           310: \epsffile{ps/1.ps}
        !           311: \end{center}
        !           312: \caption{$f_1=0$}
        !           313: \label{f2}
        !           314: \end{minipage} &
        !           315:
        !           316: \begin{minipage}{.5\hsize}
        !           317: \begin{center}
        !           318: \epsfxsize=7cm
        !           319: \epsffile{ps/1d.ps}
        !           320: \end{center}
        !           321: \caption{$g_1=0$}
        !           322: \label{g2}
        !           323: \end{minipage}
        !           324: \end{tabular}
        !           325: \end{figure}
        !           326:
        !           327:
        !           328: \begin{figure}[hbtp]
        !           329: \begin{tabular}{cc}
        !           330: \begin{minipage}{.5\hsize}
        !           331: \begin{center}
        !           332: \epsfxsize=7cm
        !           333: \epsffile{ps/2.ps}
        !           334: \end{center}
        !           335: \caption{$f_2=0$}
        !           336: \label{f3}
        !           337: \end{minipage} &
        !           338:
        !           339: \begin{minipage}{.5\hsize}
        !           340: \begin{center}
        !           341: \epsfxsize=7cm
        !           342: \epsffile{ps/2d.ps}
        !           343: \end{center}
        !           344: \caption{$g_2=0$}
        !           345: \label{g3}
        !           346: \end{minipage}
        !           347: \end{tabular}
        !           348: \end{figure}
        !           349:
        !           350: \begin{figure}[hbtp]
        !           351: \begin{tabular}{cc}
        !           352: \begin{minipage}{.5\hsize}
        !           353: \begin{center}
        !           354: \epsfxsize=7cm
        !           355: \epsffile{ps/4.ps}
        !           356: \end{center}
        !           357: \caption{$f_3=0$}
        !           358: \label{f5}
        !           359: \end{minipage} &
        !           360:
        !           361: \begin{minipage}{.5\hsize}
        !           362: \begin{center}
        !           363: \epsfxsize=7cm
        !           364: \epsffile{ps/4d.ps}
        !           365: \end{center}
        !           366: \caption{$g_3=0$}
        !           367: \label{g5}
        !           368: \end{minipage}
        !           369: \end{tabular}
        !           370: \end{figure}