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Annotation of OpenXM/doc/compalg/ex.tex, Revision 1.1.1.1

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}