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Annotation of OpenXM/doc/compalg/appgr.tex, Revision 1.3

1.3     ! noro        1: %$OpenXM$
1.1       noro        2: \chapter{$B%0%l%V%J4pDl$N1~MQ(B}
                      3:
                      4: $B%0%l%V%J4pDl$O>C5nK!0J30$K$b$5$^$6$^$J1~MQ$r;}$D(B. $BK\@a$G$O(B, $B$=$l$i$N$$(B
                      5: $B$/$D$+$K$D$$$F2r@b$9$k(B.
                      6:
                      7: \begin{nt}
                      8: $B0J2<$G(B, $B<!$N$h$&$J5-K!$rMQ$$$k(B. \\
                      9: $K$ : $BBN(B.\\
                     10: $X$ = $\{x_1,\cdots,x_n\}$ : $BITDj85(B\\
                     11: $R$ : $K[X]$\\
                     12: $T$ : $R$ $B$N9`A4BN(B\\
                     13: $HT_<(f)$ : $f$ $B$N(B $<$ $B$K4X$9$kF,9`(B.\\
                     14: $HC_<(f)$ : $f$ $B$N(B $<$ $B$K4X$9$kF,78?t(B.\\
                     15: $GB_<(S)$ : $S$ $B$N(B $<$ $B$K4X$9$kHoLs%0%l%V%J4pDl(B.\\
                     16: $NF_<(f,G)$ : $f$ $B$N(B $G$ $B$K4X$9$k@55,7A$N0l$D(B. $G$ $B$,%0%l%V%J4pDl$J$i$P0l0UE*(B
                     17: $B$KDj$^$k(B.
                     18: \end{nt}
                     19:
                     20: \section{$B%$%G%"%k$K4X$9$k1i;;(B}
                     21:
                     22: \begin{pr}($B%$%G%"%k$NAjEy(B)\\
                     23: $B%$%G%"%k(B $I, J \subset R$ $B$K4X$7(B
                     24: $I = J \Leftrightarrow GB(I) = GB(J).$
                     25: \end{pr}
                     26:
                     27: \begin{pr} ($B%$%G%"%k$rK!$H$9$k9gF1(B, $B%a%s%P%7%C%W(B)\\
                     28: $B%$%G%"%k(B $I$, $f, g\in R$ $B$K4X$7(B
                     29: $f \equiv g \bmod I \Leftrightarrow NF(f,GB(I)) = NF(g,GB(I)).$
                     30: $BFC$K(B
                     31: $f \in I \Leftrightarrow NF(f,GB(I)) = 0.$
                     32: \end{pr}
                     33:
                     34: \begin{pr}($B<+L@$J%$%G%"%k(B)\\
                     35: $B%$%G%"%k(B $I \subset R$ $B$K4X$7(B
                     36: $I = R \Leftrightarrow GB(I) = \{1\}.$
                     37: \end{pr}
                     38:
                     39: \begin{pr}(elimination $B%$%G%"%k(B)\\
                     40: $I$ $B$r%$%G%"%k$H$9$k(B.
                     41: $X = (X \setminus U) \cup U$ $B$H$7(B, $B$3$NJ,3d$K$h$j(B
                     42: $B$9$Y$F$N(B $u \in T(U)$, $B$9$Y$F$N(B $x \in T(X \setminus U)$ $B$KBP$7(B $u < x$ $B$J$k(B
                     43: order $<$ $B$rMQ$$$k$H(B,
                     44: $GB(I \cap K[U]) = GB(I) \cap K[U]$
                     45: \end{pr}
                     46: \proof
                     47: $f \in J=I \cap K[U]$ $B$H$9$k(B. $f \in I$ $B$h$j$"$k(B $g \in GB(I)$ $B$,B8:_$7$F(B
                     48: $HT(g) \mid HT(f)$. $HT(f) \in T(U)$$B$h$j(B $HT(g) \in T(U)$.  $B$3$3$G(B, $<$
                     49: $B$N@-<A$h$j(B, $HT(g) \in T(U)$ $B$J$i$P(B $g \in K[U]$. $B$h$C$F(B $g \in GB(I) \cap
                     50: K[U]$ $B$G(B, $GB(I) \cap K[U]$ $B$O(B $J$ $B$N%0%l%V%J4pDl(B. \qed
                     51:
                     52: \begin{pr} ($B%$%G%"%k$N8r$o$j(B)
                     53: $I = Id(f_1,\cdots,f_l)$, $J = Id(g_1,\cdots,g_m)$ $B$H$9$k$H(B,
                     54: $I\cap J = (yIR[y] + (1-y)JR[y])\cap R$
                     55: \end{pr}
                     56: \proof $f \in I\cap J$ $B$H$9$k$H(B, $f = yf+(1-y)f \in (yI + (1-y)J)\cap R$.
                     57: $B5U$K(B $f=yg + (1-y)h$ ($g \in IR[y], h \in JR[y]$) $B$H$7(B, $f \in R$
                     58: $B$H$9$k(B. $B$3$N;~(B, $y=0$ $B$rBeF~$7$F(B, $f = h|_{y=0} \in J$. $y=1$ $B$r(B
1.2       noro       59: $BBeF~$7$F(B, $f=g|_{y=1} \in I$. \qed
1.1       noro       60:
                     61: \begin{co}
                     62: \label{intersect}
                     63: $I=Id(f_1,\cdots,f_m)$, $J=Id(g_1,\cdots,g_l)$ $B$KBP$7(B
                     64: $GB(I\cap J) = GB(\{yf_1,\cdots,yf_m,(1-y)g_1,\cdots,(1-y)g_l\}) \cap R$
                     65: $B$K$h$j(B $I\cap J$ $B$,7W;;$G$-$k(B. ($B:8JU$O(B $X < y$ $B$J$k(B
                     66: elimination order $B$G7W;;$9$k(B. )
                     67: \end{co}
                     68:
                     69: \begin{df} ($B%$%G%"%k>&(B)
                     70: $B%$%G%"%k(B $I$, $R$ $B$NItJ,=89g(B $S$ $B$KBP$7(B, $B%$%G%"%k>&(B $I:S$ $B$r(B
                     71: $$I:S = \{f \in R \mid fS \subset I\}$$
                     72: $B$GDj5A$9$k(B. $J=Id(S)$ $B$H$9$l$P(B, $I:S = I:J$ $B$G(B,
                     73: $J=Id(f_1,\cdots,f_m)$ $B$J$i$P(B,
                     74: $$I:S = \bigcap_{i=1}^m I:Id(f_i)$$
                     75: \end{df}
                     76:
                     77: \begin{pr}
                     78: $I:Id(f) = {1\over f}(I \cap Id(f))$
                     79: \end{pr}
                     80: \proof $g \in I:Id(f)$ $B$J$i$P(B $gf \in I$ $B$h$j(B $gf \in I \cap Id(f)$.
                     81: $B$h$C$F(B, $g \in {1 \over f}(I \cap Id(f))$.\\
                     82: $B5U$K(B, $g \in {1\over f}(I \cap Id(f))$ $B$J$i$P(B $gf \in I \cap Id(f)$ $B$h$j(B
                     83: $g \in I:Id(f)$. \qed
                     84:
                     85: \begin{co}
                     86: $I \cap Id(f)$ $B$N@8@.85$,5a$^$l$P(B, $B$=$l$i$O(B $f$ $B$r0x;R$K;}$D$N$G(B, $B$=$l(B
                     87: $B$>$l(B $f$ $B$G3d$k$3$H$K$h$j(B $I:Id(f)$ $B$,5a$^$k(B. $B0lHL$N>l9g(B $I:S$ $B$O$=$l(B
                     88: $B$i$N8r$o$j$H$J$k$,(B, $I \cap Id(f)$ $B$r4^$a$F%$%G%"%k$N8r$o$j$N7W;;$O7O(B
                     89: \ref{intersect} $B$K$h$j7W;;$G$-$k$N$G(B, $B%$%G%"%k>&$b7W;;$G$-$k$3$H$K(B
                     90: $B$J$k(B.
                     91: \end{co}
                     92:
                     93: \begin{df} (saturation)\\
                     94: $I$ $B$r%$%G%"%k(B, $f\in R$ $B$H$9$l$P(B, $I:f^i$ $B$O%$%G%"%k$NA}BgNs$@$,$i(B, $B$"(B
                     95: $B$k(B $s \in \N$ $B$,B8:_$7$F(B
                     96: $$i\ge s \Rightarrow I:f^i = I:f^s$$
                     97: $B$,@.$jN)$D(B. $B$3$N$H$-(B
                     98: $$I:f^\infty = I:f^s$$
                     99: $B$HDj5A$7(B, $I$ $B$N(B $f$ $B$K4X$9$k(B saturation $B$H8F$V(B.
                    100: \end{df}
                    101:
                    102: \begin{pr}
                    103: $I$ $B$r%$%G%"%k(B, $f \in R$ $B$KBP$7(B,
                    104: $$I:f^\infty = (IR[y]+(1-yf)R[y]) \cap R$$
                    105: $B$9$J$o$A(B $I:f^\infty$ $B$O(B elimination $B%$%G%"%k$K$h$j7W;;$G$-$k(B.
                    106: \end{pr}
                    107: \proof \\
                    108: \underline{$B1&JU(B $\subset$ $B:8JU(B} $g \in$ $B1&JU$H$9$k$H(B, $g = ah+(1-yf)b$ ($a,b \in R[y]$)
                    109: $B$H=q$1$k(B. $B$3$N<0$G(B, $y=1/f$ $B$H$*$$$F(B, $BN>JU$K(B $f^d$ ($d$:$B==J,Bg(B) $B$r3]$1$l$P(B
                    110: $f^dg \in I$ $B$9$J$o$A(B $g \in I:f^d$. $B$h$C$F(B $g \in$ $B:8JU(B.\\
                    111: \underline{$B:8JU(B $\subset$ $B1&JU(B} $g \in$ $B:8JU(B, $B$9$J$o$A$"$k(B $d$ $B$KBP$7(B $f^dg \in I$ $B$H(B
                    112: $B$9$k(B. $B$3$N$H$-(B
                    113: $$g \equiv (yf)^d g \equiv 0 \bmod IR[y]+(1-yf)R[y]$$
                    114: $B$^$?(B, $g \in R$ $B$h$j(B $g \in$ $B1&JU(B. \qed
                    115:
                    116:
                    117:
                    118:
                    119: %*****************************************************
                    120:
                    121: \section{$B>jM>4D(B, $B<!85(B}
                    122:
                    123: \begin{pr}($B>jM>4D$NI=8=(B)\\
                    124: $B%$%G%"%k(B $I$ $B$KBP$7(B, $B>jM>4D(B $R/I$ $B$O(B,$B@55,7A$r85$H$7$FDj5A$5$l$kBe?t(B
                    125: $B9=B$$KF17A$G$"$k!%$9$J$o$A(B, $R/I$ $B$O(B, $B85$N=89g$H$7$F(B
                    126: $$\{NF(f,GB(I))\mid f\in R \}$$ $B$HF10l;k$G$-(B, $B$=$N2CK!(B($\oplus$) $\cdot$ $B>hK!(B
                    127: ($\odot$)$B$H$7$F<!<0$GDj5A$5$l$k$b$N$KF17A$H$J$k(B.
                    128: $$f\oplus g = NF(f+g,GB(I))$$
                    129: $$f\odot g = NF(fg,GB(I))$$
                    130: \end{pr}
                    131:
                    132: \begin{pr}($B>jM>4D$N@~7A6u4V$H$7$F$N4pDl(B)\\
                    133: \label{mbase}
                    134: $B%$%G%"%k(B $I$ $B$KBP$7(B, $B>jM>4D(B $R/I$ $B$O(B $K$-$B@~7A6u4V$H$_$J$;$k$,(B,
                    135: $B$=$N@~7A6u4V$N4pDl$H$7$F%$%G%"%k$N(B $B%0%l%V%J4pDl$KBP$7$F@55,7A$G$"$k(B
                    136: $B9`A4BN$,$H$l$k(B.  $B$9$J$o$A(B, $R/I$ $B$N4pDl$H$7$F(B
                    137: \begin{center}
                    138: $\{ u \in T \mid$ $B$9$Y$F$N(B $f \in GB(I)$ $B$KBP$7(B $HT(f) {\not|} u \}$
                    139: \end{center}
                    140: $B$,$H$l$k(B.
                    141: \end{pr}
                    142:
                    143: \begin{df}
                    144: $I$ $B$r%$%G%"%k$H$9$k(B. $U \subset X$ $B$KBP$7(B, $I \cap K[U] = 0$ $B$,@.$jN)(B
                    145: $B$D$H$-(B $U$ $B$O(B independent modulo $I$ $B$H$$$&(B.
                    146: \end{df}
                    147:
                    148: \begin{df}($B%$%G%"%k$N<!85(B)\\
                    149: $B%$%G%"%k(B $I$ $B$KBP$7(B, $B%$%G%"%k$N<!85(B $\dim(I)$ $B$r(B
                    150: \begin{center}
                    151: $\dim(I)$ = $\max(|U| \mid U \subset X$ independent modulo $I)$
                    152: \end{center}
                    153: $B$GDj5A$9$k(B.
                    154: \end{df}
                    155:
                    156: \begin{re}($B4v2?3XE*0UL#(B)
                    157: \begin{enumerate}
                    158: \item $B%$%G%"%k(B $I$ $B$N<!85$O(B, $K$ $B$NBe?tJDJq>e$G9M$($?Be?tE*=89g(B $V(I)$
                    159: $B$N@.J,$N:GBg<!?t$KEy$7$$(B.
                    160: \item $B$h$j0lHL$K(B, $BAG%$%G%"%k$N8:>/Ns$ND9$5$rMQ$$$F4D$N<!85(B (Krull $B<!85(B)
                    161: $B$,Dj5A$5$l(B, $B>e$NDj5A$H0lCW$9$k$3$H$,<($5$l$k(B.
                    162: \end{enumerate}
                    163: \end{re}
                    164:
                    165: \begin{df}(Hilbert function)\\
                    166: $R=K[X]$ $B$N(B $s$-$B<!@F<!85A4BN$r(B $R_s$ $B$H=q$/$3$H$K$9$k(B. $B%$%G%"%k(B
                    167: $I$ $B$KBP$7(B, $I_s = I \cap K[X]_s$ $B$H=q$/(B.
                    168: $B@F<!%$%G%"%k(B $I$ $B$KBP$7(B, $I$ $B$N(B Hilbert function $H_{R/I}(s)$ $B$r(B
                    169: $$H_{R/I}(s) = \dim_K R_s/I_s$$
                    170: $B$GDj5A$9$k(B.
                    171: \end{df}
                    172:
                    173: \begin{pr}
                    174: $<$ $B$rG$0U$N(B order $B$H$7(B, $J$ $B$r(B $I$ $B$N85$NF,9`$G@8@.$5$l$k%$%G%"%k(B
                    175: $B$H$9$k$H(B, $H_{R/I}(s) = H_{R/J}(s)$
                    176: \end{pr}
                    177:
                    178: \section{$B>C5nK!(B}
                    179:
                    180: \begin{df}
                    181: $B%$%G%"%k(B $I$ $B$KBP$7(B, $I$ $B$N(B radical ($B:,4p(B) $\sqrt{I}$ $B$r(B
                    182: \begin{center}
                    183: $\sqrt{I} = \{f \in R \mid$ $B$"$k(B $e \in \N$ $B$,B8:_$7$F(B $f^e \in I\}$
                    184: \end{center}
                    185: $B$GDj5A$9$k(B. $\sqrt{I}$ $B$b%$%G%"%k$H$J$k(B.
                    186: \end{df}
                    187:
                    188:
                    189: \begin{df}
                    190: $L$ $B$r(B $K$ $B$N3HBgBN$H$7(B, $V \subset K^n$ $B$H$9$k(B. $B$3$N$H$-%$%G%"%k(B $I(V) \subset R$
                    191: $B$r(B
                    192: $$I(V) = \{f \in R \mid f|_V=0 \}$$
                    193: $B$GDj5A$9$k(B.
                    194:
                    195: \end{df}
                    196: $B<!$NDjM}$O>C5nK!$N4pK\$H$J$k(B.
                    197:
                    198: \begin{th}(Nullstellensatz; Hilbert $B$NNmE@DjM}(B)\\
                    199: $K$ $B$rBN(B, $\bar{K}$ $B$r(B $K$ $B$NBe?tJDJq$H$9$k(B. $B%$%G%"%k(B $I \subset K[X]$
                    200: $B$KBP$7(B,
                    201: $I(V_{\bar{K}}(I))=\sqrt{I}$
                    202: \end{th}
                    203:
                    204: \begin{co}
                    205: $B%$%G%"%k(B $I$, $J$ $B$KBP$7(B,
                    206: $V_{\bar{K}}(I)=V_{\bar{K}}(J) \Leftrightarrow \sqrt{I} = \sqrt{J}$
                    207: \end{co}
                    208:
                    209: \begin{pr}(0 $B<!85%$%G%"%k$N@-<A(B)\\
                    210: $BBe?tJDBN(B $K$ $B>e$NB?9`<04D$N%$%G%"%k(B $I$ $B$NNmE@$N8D?t$,M-8B8D(B
                    211: $\Leftrightarrow$ $R/I$ $B$,(B $K$ $B>eM-8B<!85$N@~7A6u4V(B
                    212: \end{pr}
                    213: \proof \\
                    214: $\Rightarrow$)
                    215: $BM-8B8D$N2r$r(B $r_k = (r_{k1}, \cdots, r_{kn})\quad (k = 1,
                    216: \cdots, m)$$B$H$9$k(B. $f_i(x_i) = \prod_k (x_i-r_{ki})$ $B$H$*$/$H(B,
                    217: $f_i(x_i)$ $B$O(B$I$ $B$NNmE@>e$G(B 0 $B$H$J$k$+$i(B, Hilbert $B$NNmE@DjM}$K$h$j$"(B
                    218: $B$k(B $t$ $B$,B8:_$7$F(B $f_i(x_i)^t \in I$.  $B$h$C$F(B, $GB(I)$ $B$K$b(B, $B3F(B $i$ $B$K(B
                    219: $BBP$7(B, $HT(g)$ $B$,(B $x_i$ $B$NQQ$H$J$k$b$N$,B8:_$9$k(B. $B$9$k$H(B, $BL?Bj(B \ref{mbase}
                    220: $B$h$j(B $R/I$ $B$O(B $K$ $B>eM-8B<!85$H$J$k(B.\\
                    221: $\Leftarrow$)
                    222: $B3F(B $i$ $B$KBP$7(B, $BJQ?tCf$G(B $x_i$ $B$,:GDc$N=g=x$K$J$k$h$&$J<-=q<0=g=x$r$H$l$P(B,
                    223: $GB(I)$ $BCf$K(B, $f_i(x_i)$ $B$J$k0lJQ?tB?9`<0$,B8:_$9$k$3$H$,$o$+$k(B.
                    224: $B$h$C$F(B, $B2r$OM-8B8D(B. \qed
                    225:
                    226: $BJQ?t=g=x$H$7$F(B $x_1 < x_2 < \cdots < x_n$ $B$J$k<-=q<0=g=x$r9M$($l$P(B,
                    227: $B%$%G%"%k(B $I$ $B$NNmE@$,M-8B8D$N>l9g$O(B, $B$9$Y$F$N(B $i$ $B$KBP$7(B, $B$"$k(B
                    228: $f_i \in GB(I) \cap (K[x_1,\cdots,x_i]\setminus K[x_1,\cdots,x_{i-1}])$
                    229: $B$,B8:_$9$k(B. $B$h$C$F(B
                    230: $f_1(x_1)$$B$+$i:,(B $\alpha_1$ $B$r5a$a(B, $f_2(\alpha_1,x_2)$ $B$+$i:,(B
                    231: $\alpha_2$ $B$r5a$a(B, $B$H$$$&A`:n$r7+$jJV$;$P(B, $F$ $B$N6&DLNmE@$r$9$Y$F5a$a(B
                    232: $B$k$3$H$,$G$-$k(B.
                    233:
                    234: \section{$B2C72$N%0%l%V%J4pDl(B}
                    235:
                    236: $B<+M32C72(B $K[X]^l$ $B$*$h$S(B, $B$=$NItJ,2C72(B $M \subset F$ $B$K(B
                    237: $BBP$7$F$b%0%l%V%J4pDl$,Dj5A$5$l$k(B. $B$3$N>l9g(B, $B9`$H$7$F$O(B,
                    238: $te_i$ ($t \in T(X)$; $e_i = (0,\cdots,1,\cdots,0)$ : $BBh(B $i$ $B@.J,$N$_(B 1)
                    239: $B$r$H$j(B,
                    240: \begin{enumerate}
                    241: \item $B$9$Y$F$N(B $t \in T$, $B$9$Y$F$N(B $F$ $B$N9`(B $m$ $B$KBP$7(B $m \le tm$
                    242: \item $B$9$Y$F$N(B $t \in T$, $B$9$Y$F$N(B $F$ $B$N9`$NAH(B $m_1, m_2$ $B$KBP$7(B
                    243: $m_1 \le m_2$ $\Rightarrow$ $tm_1 \le tm_2$
                    244: \end{enumerate}
                    245: $B$rK~$?$9A4=g=x$rF~$l$k(B. $B%b%N%$%G%"%k(B $E(S)$ $B$bF1MM$KDj5A$5$l(B, $B%0%l%V%J(B
                    246: $B4pDl$b(B, $E(G)$ $B$,(B $E(M)$ $B$r@8@.$9$k$b$N$H$7$FDj5A$5$l$k(B. Buchberger $B%"(B
                    247: $B%k%4%j%:%`$O(B, $S$-$BB?9`<0$r(B, $BF,9`$N(B $F$ $B$K$*$1$k0LCV$,Ey$7$$(B ($B$9$J$o$A(B,
                    248: $HT(a)=t_ae_a$, $HT(b)=t_be_b$ $B$N$H$-(B $a=b$)$B$KBP$7$FDL>o$NB?9`<0$HF1MM(B
                    249: $B$KDj5A$7(B, $B$=$l0J30$O(B 0 $B$HDj5A$9$l$PA4$/F1MM$K$G$-$k(B. $B2C72$N%0%l%V%J4p(B
                    250: $BDl$O(B, syzygy $B$N7W;;$rDL$7$F(B, $B2C72$N<+M3J,2r(B (free resolution) $B$rM?$($k(B.
                    251: $B$3$l$K$h$j(B, $B2C72$N%[%b%m%8!<$N7W;;$,2DG=$K$J$k$,(B, $B$3$3$G$O=R$Y$J(B
                    252: $B$$(B.
                    253:
                    254: \section{$BNc(B : $BAPBP6J@~$N7W;;(B}
                    255: elimination $B%$%G%"%k$N1~MQ$H$7$F(B, $BAPBP6J@~$N7W;;$NNc$r<($9(B.
                    256: $f(x_1,x_2) \in \Q[x_1,x_2]$ $B$H$7(B, $F$ $B$N(B total degree $B$r(B $d$ $B$H$9$l$P(B,
                    257: $F(x_0,x_1,x_2)=x_0^df(x_1/x_0,x_2/x_0)$
                    258: $B$O(B $d$ $B<!F1<!B?9`<0$G(B, $F$ $B$NDj5A$9$kBe?t6J@~$NAPBP6J@~$O(B,
                    259: $$\left\{
                    260: \parbox[c]{8in}{
                    261: $u_i={{\partial F}\over {\partial x_i}} (x_0,x_1,x_2)$ $(i=0,1,2)$\\
                    262: $F(x_0,x_1,x_2)=0$
                    263: }
                    264: \right.$$
                    265: $B$+$i(B $x_0, x_1, x_2$ $B$r>C5n$7$FF@$i$l$k(B. $B>C5nK!$N0l$D$H$7$F%0%l%V%J4pDl(B
                    266: $B$K$h$k>C5n$,2DG=$G$"$k(B.
                    267: $$I = Id(
                    268: u_0-{{\partial F}\over {\partial x_0}},
                    269: u_1-{{\partial F}\over {\partial x_1}},
                    270: u_2-{{\partial F}\over {\partial x_2}},
                    271: F)$$
                    272: $B$H$9$k;~(B, $\{x_0, x_1, x_2\}$ $\succ$ $\{u_0, u_1, u_2\}$ $B$J$kG$0U$N>C(B
                    273: $B5n=g=x$K$h$j(B $I$ $B$N%0%l%V%J4pDl(B $GB(I)$ $B$r7W;;$9$l$P(B,
                    274: $$I \cap \Q[u_0,u_1,u_2] = Id(GB(I) \cap Q[u_0,u_1,u_2]).$$
                    275: $B0J2<$NNc$G(B, $V(g_i)$ $B$O(B $V(f_i)$ $B$NAPBP6J@~$G$"$k(B.
                    276:
                    277: \vskip\baselineskip
                    278: $\left\{
                    279: \parbox[c]{6in}{
                    280: $f_1=x^5-x^3+y^2$\\
                    281: $g_1=108x^7-108x^5+1017y^2x^4-16y^4x^3-4250y^2x^2+1800y^4x-108y^6+3125y^2$
                    282: }
                    283: \right.$
                    284:
                    285: \vskip\baselineskip
                    286: $\left\{
                    287: \parbox[c]{6in}{
                    288: $f_2=x^6+3y^2x^4+(3y^4-4y^2)x^2+y^6$\\
                    289: $g_2=-256x^6+(64y^4-192y^2+864)x^4+(-192y^4+1620y^2-729)x^2-256y^6+864y^4-729y^2$
                    290: }
                    291: \right.$
                    292:
                    293: \vskip\baselineskip
                    294: $\left\{
                    295: \parbox[c]{6in}{
                    296: $f_3=2x^4-3yx^2+y^4-2y^3+y^2$\\
                    297: $g_3=-12x^6+(-y^2+178y-37)x^4+(12y^3-768y^2+2208y+4608)x^2-32y^4+1024y^3-7680y^2-8192y-2048$
                    298: }
                    299: \right.$
                    300:
                    301: \vskip\baselineskip
                    302: \begin{figure}[hbtp]
                    303: \begin{tabular}{cc}
                    304: \begin{minipage}{.5\hsize}
                    305: \begin{center}
                    306: \epsfxsize=7cm
                    307: \epsffile{ps/1.ps}
                    308: \end{center}
                    309: \caption{$f_1=0$}
                    310: \label{f2}
                    311: \end{minipage} &
                    312:
                    313: \begin{minipage}{.5\hsize}
                    314: \begin{center}
                    315: \epsfxsize=7cm
                    316: \epsffile{ps/1d.ps}
                    317: \end{center}
                    318: \caption{$g_1=0$}
                    319: \label{g2}
                    320: \end{minipage}
                    321: \end{tabular}
                    322: \end{figure}
                    323:
                    324:
                    325: \begin{figure}[hbtp]
                    326: \begin{tabular}{cc}
                    327: \begin{minipage}{.5\hsize}
                    328: \begin{center}
                    329: \epsfxsize=7cm
                    330: \epsffile{ps/2.ps}
                    331: \end{center}
                    332: \caption{$f_2=0$}
                    333: \label{f3}
                    334: \end{minipage} &
                    335:
                    336: \begin{minipage}{.5\hsize}
                    337: \begin{center}
                    338: \epsfxsize=7cm
                    339: \epsffile{ps/2d.ps}
                    340: \end{center}
                    341: \caption{$g_2=0$}
                    342: \label{g3}
                    343: \end{minipage}
                    344: \end{tabular}
                    345: \end{figure}
                    346:
                    347: \begin{figure}[hbtp]
                    348: \begin{tabular}{cc}
                    349: \begin{minipage}{.5\hsize}
                    350: \begin{center}
                    351: \epsfxsize=7cm
                    352: \epsffile{ps/4.ps}
                    353: \end{center}
                    354: \caption{$f_3=0$}
                    355: \label{f5}
                    356: \end{minipage} &
                    357:
                    358: \begin{minipage}{.5\hsize}
                    359: \begin{center}
                    360: \epsfxsize=7cm
                    361: \epsffile{ps/4d.ps}
                    362: \end{center}
                    363: \caption{$g_3=0$}
                    364: \label{g5}
                    365: \end{minipage}
                    366: \end{tabular}
                    367: \end{figure}