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Revision 1.1, Wed Feb 2 06:54:59 2005 UTC (19 years, 5 months ago) by takayama
Branch: MAIN
CVS Tags: RELEASE_1_2_3

New library to parametrize algebraic curves of genus 0.
The library is written by Shuhei Todo.

@comment $OpenXM: OpenXM/src/asir-contrib/packages/doc/todo_parametrize_ja.tex,v 1.1 2005/02/02 06:54:59 takayama Exp $
@comment    Copyright (c)  2005, Shuhei Todo,
@comment    Permission is granted to copy, distribute and/or modify this document
@comment    under the terms of the GNU Free Documentation License, Version 1.1
@comment    or any later version published by the Free Software Foundation;
@comment    with the Invariant Sections being LIST THEIR TITLES, with the
@comment    Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.
@comment    A copy of the license is included in the section entitled "GNU
@comment    Free Documentation License".
@comment
\input texinfo
@comment \input jtexinfo
@iftex
@catcode`@#=6
@def@fref#1{@xrefX[#1,,@code{#1},,,]}
@def@b#1{{@bf@gt #1}}
@catcode`@#=@other
@end iftex
@overfullrule=0pt
@c -*-texinfo-*-
@comment %**start of header
@comment --- $B$*$^$8$J$$=*$j(B ---

@comment --- GNU info $B%U%!%$%k$NL>A0(B ---
@setfilename parametrize

@comment --- $B%?%$%H%k(B ---
@settitle Risa/Asir $BBe?t6J@~O@MQ%Q%C%1!<%8(B

@comment %**end of header
@comment %@setchapternewpage odd

@comment --- $B$*$^$8$J$$(B ---
@ifinfo
@macro fref{name}
@ref{\name\,,@code{\name\}}
@end macro
@end ifinfo

@iftex
@comment @finalout
@end iftex

@titlepage
@comment --- $B$*$^$8$J$$=*$j(B ---

@comment --- $B%?%$%H%k(B, $B%P!<%8%g%s(B, $BCx<TL>(B, $BCx:n8"I=<((B ---
@title Risa/Asir $BBe?t6J@~O@MQ%Q%C%1!<%8@bL@=q(B
@subtitle $BMxMQ@bL@=q(B
@subtitle 1.0 $BHG(B
@subtitle 2004 $BG/(B 8 $B7n(B

@author  by Shuhei Todo
@page
@vskip 0pt plus 1filll
Copyright @copyright{} Risa/Asir committers
2001. All rights reserved.
@end titlepage

@comment --- $B$*$^$8$J$$(B ---
@synindex vr fn
@comment --- $B$*$^$8$J$$=*$j(B ---

@comment --- @node $B$O(B GNU info, HTML $BMQ(B ---
@comment --- @node  $B$N0z?t$O(B node-name,  next,  previous,  up --- 
@node Top,, (dir), (dir)

@menu
* $B4X?t4J0W%^%K%e%"%k(B::
* Index::
@end menu

@node $B4X?t4J0W%^%K%e%"%k(B,,, Top
@chapter $B4X?t4J0W%^%K%e%"%k(B

@menu
* $B35MW(B::
* Notation::
* $B<g$J4X?t(B::
* $B$=$NB>$N4X?t(B::
@end menu

@node $B35MW(B,,, $B4X?t4J0W%^%K%e%"%k(B
@section $B35MW(B

@comment --- $B=qBN;XDj$K$D$$$F(B ---
@comment --- @code{} $B$O%?%$%W%i%$%?BNI=<((B ---
@comment --- @var{} $B$O<P;zBNI=<((B ---
@comment --- @b{} $B$O%\!<%k%II=<((B ---
@comment --- @samp{} $B$O%U%!%$%kL>$J$I$NI=<((B ---

$B$3$N%Q%C%1!<%8$K$O!"Be?t6J@~$N=t@-<A$rD4$Y$k$?$a$N4X?t$,(B
$B=8$a$i$l$F$$$k!#<g$J5!G=$O!"Be?t6J@~$KBP$7$FDj5A$5$l$k0J2<(B
$B$NBP>]$r7W;;$G$-$k$3$H$G$"$k!'(B
@itemize @bullet
@item 2$B6J@~$N8rE@$N:BI8(B
@item $BFC0[E@$N:BI8(B
@item neighborhood graph$B!JFs<!JQ49$K$h$C$FFC0[E@$,$I$N$h$&$K(B
$BJ,2r$5$l$k$+$rI=$9(Btree$B!K(B
@item $B4{Ls6J@~$N<o?t(B
@item $B?oH<6J@~(B(adjoint curves)
@item $BFs<!6J@~>e$NM-M}E@(B
@item $BM-M}6J@~!J<o?t#0$N6J@~!K$r%Q%i%a%H%i%$%:$9$kM-M}4X?t(B
@end itemize
$B$=$NB>!"B?9`<0$NA4<!?t$r7W;;$9$k$H$$$C$?$h$&$JM=HwE*$J4X?t72(B
$B$,MQ0U$5$l$F$$$k!#%f!<%6!<$NF~NO$9$kBe?t6J@~$NDj5AB?9`<0$OI,$:(B
$BM-M}?tBN>e$NJQ?t(B@var{x,y,z} $B$N(B@b{$B@F<!(B}$BB?9`<0$G$J$1$l$P$J$i$J$$!#(B

@node Notation,,,$B4X?t4J0W%^%K%e%"%k(B
@section Notation

$BK\=q$GMQ$$$i$l$k5-9f$K$D$$$F!"<!$N$h$&$JLsB+$r$7$F$*$/!#(B
@itemize @bullet
@item $BE@(B@code{[x,y,z]} $B$H$O<M1FJ?LL$NE@$N@F<!:BI8(B
@var{(x:y:z)}$B$r0UL#$7!"FC$KCG$j$,$J$1$l$P!"(B@var{z=0}$B$G$J$$(B
$B$H$-$OI,$:(B@var{z=1}$B$H$J$k$h$&$K@55,2=$5$l$F$$$k!#(B
@item Q $B$OM-M}?tBN!"(B
@tex $\overline{Q}$ @end tex
$B$OBe?tE*?tA4BN$N$J$9BN$r0UL#$9$k!#(B
@end itemize

@node $B<g$J4X?t(B,,, $B4X?t4J0W%^%K%e%"%k(B
@section $B<g$J4X?t(B

@menu
* intersect::
* sing::
* nbh::
* genus::
* adjoint1,adjoint2::
* intpt::
* parametrize::
@end menu

@node intersect,,, $B<g$J4X?t(B
@subsection @code{intersect}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex intersect

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item intersect(@var{F},@var{G}) 
:: 2$B6J@~(B@var{F=0},@var{G=0} $B$N8rE@$N:BI8$+$i$J$k%j%9%H$rJV$9(B.
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
$B%j%9%H(B
@item F G
$BJQ?t(Bx,y,z $B$N@F<!B?9`<0(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item 2$B6J@~(B@var{F=0},@var{G=0} $B$N8rE@(B@code{[x,y,z]}$B$+$i$J$k(B
$B%j%9%H$rJV$9!#(B
@item @var{F},@var{G}$B$O6&DL0x;R$r;}$C$F$$$F$O$$$1$J$$!#(B
@end itemize

@comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
@example
[1] intersect(y^2-x*z,(x^2+y^2)^3-4*x^2*y^2*z^2);
[[0,0,1],[(#4),(#5),1]]
[2] defpoly(alg(4));
t#4^3+3*t#4^2+3*t#4-3
[3] defpoly(alg(5));
t#5^2-t#4
[4] intersect(x^2-y^2,x^3+y*x^2+(y^2-z^2)*x+y^3-z^2*y);
***two curve have common components***
@end example


@node sing,,, $B<g$J4X?t(B
@subsection @code{sing}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex sing

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item sing(@var{F}) 
:: $B6J@~(B@var{F=0} $B$NFC0[E@$N:BI8$+$i$J$k%j%9%H$rJV$9(B.
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
$B%j%9%H(B
@item F
$BJQ?t(Bx,y,z $B$N@F<!B?9`<0(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item $B6J@~(B@var{F=0} $B$NFC0[E@(B@code{[x,y,z]}$B!J(B
@tex
$F_x(x,y,z)=F_y(x,y,z)=F_z(x,y,z)=0$
@end tex
$B$rK~$?$9E@!K$+$i$J$k%j%9%H(B
$B$rJV$9!#(B
@item @var{F}$B$O=EJ#0x;R$r;}$C$F$$$F$O$$$1$J$$!JDj5A$h$j(B
$B=EJ#0x;R$NNmE@$O$9$Y$FFC0[E@$G$"$k!K!#(B
@end itemize

@comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
@example
[1] sing(16*x^6-24*z^2*x^4+9*z^4*x^2+4*z^2*y^4-4*z^4*y^2);
[[0,0,1],[(#4),0,1],[1/2,(#3),1],[-1/2,(#3),1],[0,1,0]]
[2] defpoly(alg(3));
2*t#3^2-1
[3] defpoly(alg(4));
4*t#4^2-3
[4] sing((x-y)*(y^2-x*z));
[[1,1,1],[0,0,1]]
[5] sing((x-y)^2*(y^2-x*z));
***Argument has multiple divisor***
@end example

@comment --- $B;2>H(B($B%j%s%/(B)$B$r=q$/(B ---
@table @t
@item $B;2>H(B
@ref{nbh}
@ref{multia}
@end table


@node nbh,,, $B<g$J4X?t(B
@subsection @code{nbh}
@findex nbh

@comment --- $B4X?t$N4JC1$J@bL@(B ---

@table @t
@item nbh(@var{F}) 
:: $B6J@~(B@var{F}=0 $B$N(Bneighborhood graph $B$rJV$9!#(B
@end table

@table @var
@item return
$B%j%9%H(B
@item F
$BJQ?t(Bx,y,z $B$N@F<!B?9`<0(B
@end table

@itemize @bullet
@item $B6J@~(B@var{F=0} $B$N(Bneigborhood graph $B$rI=$9%j%9%H$rJV$9!#(B
neighborhood graph $B$H$OFs<!JQ49$K$h$C$FFC0[E@$,$I$N$h$&$K(B
$BJ,2r$5$l$k$+$rI=$9(Btree$B$G$"$k!#J,2r$K$h$C$F8=$l$kE@$N$3$H$r(B
$BNY@\E@$H8F$V!#FC0[E@!"NY@\E@$N>pJs$O!"$=$l$>$l<!$N$h$&$J(B
@b{$B%Y%/%H%k(B}$B$K$h$C$FI=$5$l$k!#(B

@table @code
@item $BFC0[E@(B
[  $BE@$N8D?t(B, $BE@$N:BI8(B, [$B=EJ#EY(B, $BDL>oFC0[E@(B(=1)$B$+$=$&$G$J$$(B(=-1)$B$+(B], [$B$3$N!J$3$l$i$N!KFC0[E@$+$i=P$F$/$kNY@\E@$N>pJs(B($B$3$l0J>eNY@\E@$,8=$l$J$$>l9g$O(B''terminal'')]  ]
@item $BNY@\E@(B
[  $BE@$N8D?t(B, [$B=EJ#EY(B, $BDL>oFC0[E@(B(=1)$B$+$=$&$G$J$$$+(B(=-1)$B$+(B], [$B$3$N!J$3$l$i$N!KNY@\E@$+$i=P$F$/$kNY@\E@$N>pJs(B($B$3$l0J>eNY@\E@$,8=$l$J$$>l9g$O(B''terminal'')]  ]
@item $B0lHL$K!"FC0[E@$N:BI8$OBe?tE*?t$K$J$k!#$3$N>l9g!"Be?tE*?t$r6&Lr$JBe?tE*?t$GCV$-49$($FF@$i$l$kE@$b$^$?!"FC0[E@$K$J$k!#$3$N@-<A$rMxMQ$7$FJ#?t$NFC0[E@$r0lEY$KI=<($9$k$N$G$"$k$,!"FC0[E@%Y%/%H%k$N:G=i$N0z?t!VE@$N8D?t!W$O$3$N$h$&$JI=<($K$h$C$F!"$$$/$D$NFC0[E@$,I=$5$l$F$$$k$+$r<($7$F$$$k!#$7$?$,$C$F!"FC0[E@$,M-M}E@$J$i$P!"E@$N8D?t(B=1 $B$G$"$k!#NY@\E@%Y%/%H%k$N:G=i$N0z?t$G$"$k!VE@$N8D?t!W$O?F%Y%/%H%k$NI=$93FE@$+$i!"$3$N?t$@$1F1$8%?%$%W$NNY@\E@$,=P$F$/$k$3$H$r0UL#$9$k!#(B
@end table

@item neighborhood graph $B$O$3$l$i$N%Y%/%H%k$rF~$l;R$K$7$?%j%9%H$K$h$C$FI=8=$5$l$F$$$k!#(B

@example
[1] F=x^6+3*y^2*x^4+(3*y^4-4*z^2*y^2)*x^2+y^6;
x^6+3*y^2*x^4+(3*y^4-4*z^2*y^2)*x^2+y^6
[2] sing(F);
[[0,0,1],[(#0),1,0]]
[3] nbh(F);
[ 1 [0,0,1] [4,-1] [[ 1 [2,1] [terminal] ],[ 1 [2,1] [terminal] ]] ]
[ 2 [(#0),1,0] [2,-1] [[ 1 [1,1] [terminal] ]] ]
@end example
$BFC0[E@(B@code{[0,0,1]} $B$O=EJ#EY(B4 $B$NDL>o$G$J$$FC0[E@$G$"$j!"(B
2$B$D$NNY@\E@$r$b$D!#$=$l$i$O$I$A$i$H$b=EJ#EY(B2 $B$NDL>oFC0[E@(B
$B$G$"$k!#FC0[E@(B@code{[(#0),1,0]}$B$NNY@\E@$OC1=cE@$G$"$k!#(B
@item @var{F}$B$O=EJ#0x;R$r;}$C$F$$$F$O$$$1$J$$!#(B
@end itemize

@table @t
@item $B;2>H(B
@ref{sing}
@end table


@node genus,,, $B<g$J4X?t(B
@subsection @code{genus}
@findex genus

@table @t
@item genus(@var{F}) 
:: $B6J@~(B@var{F=0} $B$NFC0[E@$N:BI8$+$i$J$k%j%9%H$rJV$9(B.
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
0$B0J>e$N@0?t(B
@item F
$BJQ?t(Bx,y,z $B$N@F<!B?9`<0(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item $B6J@~(B@var{F=0} $B$N<o?t$rJV$9!#(B
@item @var{F} $B$O(B@tex $\overline{Q}[x,y,z]$ @end tex
$B$K$*$$$F4{Ls$G$J$1$l$P$J$i$J$$!#$3$N>r7o$N2<$G$7$+@53N$JCM$,JV$5$l$kJ]>Z$,$J$$!#(BQ[x,y,z] $B$K$*$$$F4{Ls$G$"$C$?$H$7$F$b!"(B
@tex $\overline{Q}[x,y,z]$ @end tex
$B$G4{Ls$H$O8B$i$J$$$N$GCm0U$rMW$9$k!#F~NO$,$3$N>r7o$rK~$?$7$F(B
$B$$$k$+$I$&$+$O%A%'%C%/$5$l$J$$!#(B
@end itemize

@comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
@example
[1] genus(x^6+3*y^2*x^4+(3*y^4-4*z^2*y^2)*x^2+y^6);
0
[2] genus(y^2*z-x^3-z^3);
1
[3] genus(x^2+y^2+z^2-x*y-y*z-z*x);
-1
[4] fctr(x^2+y^2+z^2-x*y-y*z-z*x);
[[1,1],[x^2+(-y-z)*x+y^2-z*y+z^2,1]]
[5] irr_conic(x^2+y^2+z^2-x*y-y*z-z*x);
reducible
@end example

@table @t
@item $B;2>H(B
@ref{irr_conic}
@end table

@node adjoint1 adjoint2,,, $B<g$J4X?t(B
@subsection @code{adjoint1},@code{adjoint2}
@findex adjoint1
@findex adjoint2

@table @t
@item adjoint1(@var{F})
@itemx adjoint2(@var{F})
:: $B$=$l$>$l6J@~(B@var{F=0}$B$N(Bn-1$B<!(B,n-2$B<!$N?oH<6J@~(B(adjoint
 curve)$B$rJV$9(B(n=deg(F))$B!#(B
@end table

@table @var
@item return
$B@~7A$N%Q%i%a!<%?!<$r4^$`JQ?t(Bx,y,z $B$N@F<!B?9`<0(B
@item F
$BJQ?t(Bx,y,z $B$N@F<!B?9`<0(B
@end table

@itemize @bullet
@item n-2 $B<!$N6J@~(B@var{G=0}$B$,6J@~(B@var{F=0} $B$N=EJ#EY(Br $B$NE@$r>/$J$/$H$b=EJ#EY(Br-1 $B$K$b$D$H$-!"6J@~(B@var{G=0}$B$r6J@~(B@var{F=0} $B$N(Bn-2 $B<!$N?oH<6J@~(B(adjoint curve)$B$H8F$V!#(Bn-1 $B8D$N?oH<6J@~(B
@tex
$G_0=0,G_1=0, \ldots ,G_{n-2}=0$
@end tex
$B$,B8:_$7$F!"(Bn-2 $B<!$N?oH<6J@~$NDj5AB?9`<0A4BN$O(B
@tex
$c_0G_0+c_1G_1+ \ldots +c_{n-2}G_{n-2}$ ($c_{i}$ $B$O78?tBN$N85(B)
@end tex
$B$HI=$5$l$k!#(B@code{adjoint2}(@var{F}) $B$O!"$3$N(Bn-1 $B8D$N@~7A$N%Q%i%a!<%?!<$r4^$s$@@F<!B?9`<0$rJV$9!#(Bn-1 $B<!$N?oH<6J@~$bF1MM$KDj5A$5$l$k!#(Bn-1 $B<!$N?oH<6J@~$NDj5AB?9`<0A4BN$b>e$HF1MM$K!"(B2n-1 $B8D$N@~7A%Q%i%a!<%?!<$r4^$s$@(Bn-1 $B<!$N@F<!B?9`<0$GI=$5$l$k!#(B@code{adjoint1}(@var{F}) $B$O$3$NB?9`<0$rJV$9!#(B
@item $B:G=i$K%Q%i%a!<%?!<$N%j%9%H$H!"$=$ND9$5$,I=<($5$l$k!#(B
@item @var{F}$B$O=EJ#0x;R$r;}$C$F$$$F$O$$$1$J$$!#(B
@end itemize

@example
[1] adjoint2(x^6+3*y^2*x^4+(3*y^4-4*z^2*y^2)*x^2+y^6);
[c2,c3,c4,c6,c7] 5
(c2-c4)*x^4+c3*y*x^3+(c2*y^2+c6*z*y)*x^2+(c3*y^3+c7*z*y^2)*x+c4*y^4
[2] adjoint1(F);
[c1,c7,c11,c12,c13,c15,c16,c17,c18,c19,c20] 11
(c1*y+(c11-c15+c18-c20)*z)*x^4+(c13*y^2+c7*z*y+c11*z^2)*x^3+(c17*z*y^2+c12*z^2*y
+c15*z^3)*x^2+(c13*z^2*y^2+c16*z^3*y+c18*z^4)*x+c17*z^3*y^2+c19*z^4*y+c20*z^5
@end example

@table @t
@item $B;2>H(B
@ref{restriction}
@end table


@node intpt,,, $B<g$J4X?t(B
@subsection @code{intpt}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex intpt

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item intpt(@var{F}) 
:: $BFs<!6J@~(B@var{F=0} $B>e$N@0?tE@(B@code{[x,y,z]} $B$r$R$H$D8+$D$1$FJV$9!#@0?tE@$,B8:_$7$J$1$l$P!"J8;zNs(B@code{no integer solution}$B$rJV$9!#(B
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
$B%j%9%H!"$"$k$$$OJ8;zNs(B@code{no integer solution}.
@item F
$BJQ?t(Bx,y,z $B$NFs<!$N@F<!B?9`<0(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item $BFs<!6J@~(B@var{F=0} $B>e$K@0?tE@(B(affine$B$G$$$&M-M}E@(B)$B$,(B
$B$"$l$P!"$=$N:BI8(B@code{[x,y,z]}$B$rJV$9!#(B@code{x},@code{y},
@code{z} $B$O$9$Y$F@0?t$G$"$k!#@0?tE@$,B8:_$7$J$$$H$-$O(B
$BJ8;zNs(B@code{no integer solution} $B$rJV$9!#(B 
@item $B;085Fs<!7A<0$N@0?t2r$r5a$a$k8EE5E*$J(BLegendre$B$NJ}K!$rMQ$$$F$$$k!#%5%V%k!<%A%s$GFs<!$N9gF1J}Dx<0$r2r$/:]!"C1$KAmEv$jK!$rMQ$$$F$$$k$@$1$N$G!"(B@var{F} $B$N78?t$,Bg$-$/$J$k$HHs>o$K;~4V$,$+$+$k!#(B
@end itemize

@example
[1] intpt(22*x^2-10*y^2+z^2+5*x*y+13*y*x-z*x);
[71,-121,473]
[2] intpt(22*x^2-10*y^2+z^2+5*x*y+12*y*x-z*x);
no integer solution
@end example


@node parametrize,,, $B<g$J4X?t(B
@subsection @code{parametrize}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex parametrize

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item parametrize(@var{F}) 
:: $BM-M}6J@~(B@var{F=0} $B$r%Q%i%a%H%i%$%:$9$kB?9`<0$NAH$rJV$9!#(B
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
$B%j%9%H(B
@item F
$BM-M}6J@~$NDj5AB?9`<0!JJQ?t(Bx,y,z $B$N@F<!B?9`<0!K(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item $BM-M}6J@~(B@var{F=0}$B!J<o?t$,(B0$B$N6J@~!K$O!"JQ?t(Bt $B$NB?9`<0(BP(t),Q(t),R(t) $B$*$h$S(Bx,y,z$B$N@F<!B?9`<0(BS(x,y,z),T(x,y,z)$B$rMQ$$$F(B(x:y:z)=(P(t):Q(t):R(t)), t=T(x,y,z)/S(x,y,z) $B$H%Q%i%a!<%?!<I=<($5$l$k!#(B@code{parametrize}(@var{F}) $B$O$3$l$i$NB?9`<0$+$i$J$k%j%9%H(B@code{[P(t),Q(t),R(t),T(x,y,z)/S(x,y,z)]} $B$rJV$9!J(BGCD(@code{P(t)},@code{Q(t)},@code{R(t)})=1 $B$G$"$k!K!#0lHL$K$O(BP(t),Q(t),R(t) $B$O78?t$KM-M}?t$NJ?J}:,$r4^$`B?9`<0$H$J$k$,!"M-M}?t78?t$NB?9`<0$G6J@~$r%Q%i%a%H%i%$%:$G$-$k>l9g$O!"(B@b{$B>o$K(B}$BM-M}?t78?t$NB?9`<0$NAH$rJV$9!JNc$($P6J@~$N<!?t$,4q?t$N>l9g!K!#(B
@item @var{F} $B$O(B@tex $\overline{Q}[x,y,z]$ @end tex
$B$K$*$$$F4{Ls$G!"$+$D<o?t$,(B0$B$G$J$1$l$P$J$i$J$$$,!"$3$l$i$N>r7o$,K~$?$5$l$F$$$k$+$I$&$+$N%A%'%C%/$O$J$5$l$J$$!#(B
@end itemize

@example
[1] parametrize(x^4+(2*y^2-z^2)*x^2+y^4+z^2*y^2);
[-t^3-t,t^3-t,t^4+1,(-x^2-y^2)/(z*x+z*y)]
[2] parametrize((x^2+y^2)^3-4*x^2*y^2*z^2);
heuristic2 failed...
heuristic3 succeed
[32256*t^6-133120*t^5-129024*t^4+1064960*t^3-516096*t^2
-2129920*t+2064384,-127008*t^6+1048320*t^5-2671232*t^4
+10684928*t^2-16773120*t+8128512,274625*t^6-3194100*t^5
+15678780*t^4-41555808*t^3+62715120*t^2-51105600*t+17576000,
(-126*x^4+1040*y*x^3-382*y^2*x^2+1040*y^3*x-256*y^4)
/(-65*x^4+520*y*x^3+(-65*y^2-32*z*y)*x^2+(520*y^3+256*z*y^2)*x)]
[3] parametrize(22*x^2-10*y^2+z^2+5*x*y+12*y*x-z*x);
[(220*#6-10)*t^2+(-22*#6+1),(374*#6-17)*t^2+(-22*#6-43)*t,
(220*#6+210)*t^2+(-374*#6+17)*t+22,(-y)/((22*#6-1)*x+z)]
@end example

@table @t
@item $B;2>H(B
@ref{genus}
@end table


@node $B$=$NB>$N4X?t(B,,, $B4X?t4J0W%^%K%e%"%k(B
@section $B$=$NB>$N4X?t(B

@menu
* tdeg::
* homzation::
* random_line::
* multia::
* irr_conic::
* lissajou::
* restriction::
@end menu

@node tdeg,,, $B$=$NB>$N4X?t(B
@subsection @code{tdeg}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex tdeg

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item tdeg(Poly) 
:: $BB?9`<0(B@var{Poly}$B$NA4<!?t$rJV$9!#(B
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
0$B0J>e$N@0?t(B
@item Poly
$BB?9`<0(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item $BB?9`<0(B@var{Poly}$B$NA4<!?t$rJV$9!#(B
@end itemize

@comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
@example
[1] tdeg(u^3+v^3-x*y*z*w);
4
[956] tdeg((x^3+y^2+z)*(a^2+b+1));
5
@end example


@node homzation,,, $B$=$NB>$N4X?t(B
@subsection @code{homzation}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex homzation

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item homzation(AF) 
:: $BJQ?t(Bx,y$B$NB?9`<0$r@F<!2=$7$F(Bx,y,z$B$N@F<!B?9`<0$K$9$k!#(B
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
$BJQ?t(Bx,y,z$B$N@F<!B?9`<0(B
@item F
$BJQ?t(Bx,y$B$NB?9`<0(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item $BJQ?t(Bx,y$B$NB?9`<0$r@F<!2=$7$F(Bx,y,z$B$N@F<!B?9`<0$K$9$k!#F~NO$9$kB?9`<0$NJQ?t$O(Bx,y$B$G$J$1$l$P$J$i$J$$!#(B
@end itemize

@comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
@example
[1] homzation((x^2+4*x^3+6*x^4)-4*x^4*y
+(-2*x-4*x^2-2*x^3)*y^2+y^4);
(-4*y+6*z)*x^4+(-2*y^2+4*z^2)*x^3
+(-4*z*y^2+z^3)*x^2-2*z^2*y^2*x+z*y^4
[958] homzation(u*v+1);
Input must be polynomial of variable x,y
@end example


@node random_line,,, $B$=$NB>$N4X?t(B
@subsection @code{random_line}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex random_line

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item random_line(@var{Pt},B[,@var{Seed}]) 
:: $BE@(B@var{Pt}(=@code{[x,y,z]})$B$rDL$kD>@~$r$R$H$D%i%s%@%`$K(B
$BJV$9!#(B
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
$BJQ?t(Bx,y,z$B$N0l<!<0(B
@item Pt
$BE@$rI=$9%j%9%H(B
@item B
$B<+A3?t(B
@item Seed
$B<+A3?t(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item $BE@(B@var{Pt}(=@code{[x,y,z]})$B$rDL$kD>@~$NJ}Dx<0$G(B
$B3F78?t$NCM$,(B-B$B0J>e(BB$BL$K~$N$b$N$r!"$R$H$D%i%s%@%`$KJV$9!#(B
@item Seed$B$O%5%V%k!<%A%s$G(Brandom([Seed])$B$rMQ$$$k:]$K;HMQ(B
$B$5$l$k!#(B
@end itemize

@comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
@example
[1] random_line([0,0,1],1);
x-8*y
@end example


@node multia,,, $B$=$NB>$N4X?t(B
@subsection @code{multia}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex multia

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item multia(F,Pt) 
:: $B6J@~(B@var{F=0} $B$NE@(B@var{Pt}(=@code{[x,y,z]})$B$K$*$1$k(B
$B=EJ#EY$rJV$9!#(B
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
0$B0J>e$N<+A3?t(B
@item F
$BJQ?t(Bx,y,z $B$N@F<!B?9`<0(B
@item Pt
$BE@$rI=$9%j%9%H(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item $B6J@~(B@var{F=0} $B$NE@(B@var{Pt}(=@code{[x,y,z]})$B$K$*$1$k(B
$B=EJ#EY$rJV$9!#(BF$B$r(BN $B3,JPHyJ,$7$FF@$i$l$kB?9`<0$,=i$a$FE@(BPt$B$G(B
0$B$K$J$i$J$$$H$-!"@0?t(BN$B$r6J@~(B@var{F=0}$B$NE@(BPt$B$K$*$1$k=EJ#EY(B
$B$H$$$&!#(B
@end itemize

@comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
@example
[1] multia((4*y^2+4*z^2)*x^4+8*z^3*x^3+8*z^2*y^2*x^2-8*z^5*x+
4*z^4*y^2-4*z^6,[0,0,1]);
0
[2] multia((4*y^2+4*z^2)*x^4+8*z^3*x^3+8*z^2*y^2*x^2-8*z^5*x+
4*z^4*y^2-4*z^6,[0,1,0]);
4
[3] multia((4*y^2+4*z^2)*x^4+8*z^3*x^3+8*z^2*y^2*x^2-8*z^5*x+
4*z^4*y^2-4*z^6,[1,0,0]);
2
@end example

@comment --- $B;2>H(B($B%j%s%/(B)$B$r=q$/(B ---
@table @t
@item $B;2>H(B
@ref{sing}
@ref{nbh}
@end table


@node irr_conic,,, $B$=$NB>$N4X?t(B
@subsection @code{irr_conic}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex irr_conic

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item irr_conic(@var{F}) 
:: $B;085Fs<!7A<0(B@var{F}$B$,(B
@tex
$\overline{Q}[x,y,z]$
@end tex
$B$G4{Ls$+$I$&$+$rH=Dj$9$k!#(B
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
$BJ8;zNs(B
@item F
$BJQ?t(Bx,y,z $B$NFs<!$N@F<!B?9`<0(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item $B;085Fs<!7A<0(B@var{F}$B$,(B
@tex
$\overline{Q}[x,y,z]$
@end tex
$B$G4{Ls$J$i$P(B@code{irreducible}$B$r!"2DLs$J$i$P(B@code{reducible}
$B$rJV$9!#(B
@end itemize

@comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
@example
[1] irr_conic(x^2+y^2+z^2-x*y-y*z-z*x);
reducible
[2] fctr(x^2+y^2+z^2-x*y-y*z-z*x);
[[1,1],[x^2+(-y-z)*x+y^2-z*y+z^2,1]]
@end example


@node lissajou,,, $B$=$NB>$N4X?t(B
@subsection @code{lissajou}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex lissajou

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item lissajou(M,N) 
:: @tex
$x=\sin(M\theta),y=\cos(N\theta)$
@end tex
$B$K$h$C$FDj5A$5$l$k%j%5!<%8%e6J@~$N1"4X?tI=<((B
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
$BJQ?t(Bx,y,z$B$N@F<!B?9`<0(B
@item M N
$B8_$$$KAG$J<+A3?t(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item @tex
$x=\sin(M\theta),y=\cos(N\theta)$
@end tex
$B$K$h$C$FDj5A$5$l$k%j%5!<%8%e6J@~$N1"4X?tI=<(!JJQ?t(Bx,y,z$B$N(B
$B@F<!B?9`<0!K$rJV$9!#(B
@end itemize

@comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
@example
[984] lissajou(3,4);
64*x^8-128*z^2*x^6+80*z^4*x^4-16*z^6*x^2+16*z^2*y^6
-24*z^4*y^4+9*z^6*y^2
[985] lissajou(2,7);
4096*x^14-14336*z^2*x^12+19712*z^4*x^10-13440*z^6*x^8
+4704*z^8*x^6-784*z^10*x^4+49*z^12*x^2+4*z^10*y^4-4*z^12*y^2
@end example


@node restriction,,, $B$=$NB>$N4X?t(B
@subsection @code{restriction}
@comment --- $B:w0zMQ%-!<%o!<%I(B---
@findex restriction

@comment --- $B4X?t$N4JC1$J@bL@(B ---
@table @t
@item restriction(@var{A},@var{List}) 
:: $BFCDj$NE@$rDL$k?oH<6J@~$NDj5AB?9`<0$r7W;;$7$?$$$H$-$KMQ$$$k!#(B
@end table

@comment --- $B0z?t$N4JC1$J@bL@(B ---
@table @var
@item return
$B@~7A$N%Q%i%a!<%?!<$r4^$`(Bx,y,z$B$N@F<!B?9`<0(B
@item A
@code{adjoint1,adjoint2}$B$+$iJV$5$l$k7A$HF1MM$N!"@~7A%Q%i%a!<%?!<(B
$B$D$-$NJQ?t(Bx,y,z$B$N@F<!B?9`<0(B
@item List
$BE@(B@code{[x,y,z]}$B$+$i$J$k%j%9%H(B
@end table

@comment --- $B$3$3$G4X?t$N>\$7$$@bL@(B ---
@itemize @bullet
@item @code{adjoint1,adjoint2}$B$+$iJV$5$l$k@~7A%Q%i%a!<%?!<IU$N(B
$B@F<!B?9`<0$,!"(B@var{List}$B$K4^$^$l$k3FE@$rNmE@$K$b$D$?$a$K$O!"(B
$B@~7A%Q%i%a!<%?!<$N4V$K$$$/$D$+$N!J(BQ$B>e$N!K0l<!4X78<0$,@.$jN)$F(B
$B$P$h$$!#$3$N>r7o$r2CL#$7$F!"?7$?$J@~7A%Q%i%a!<%?!<IU$N@F<!(B
$BB?9`<0$r:n$k!#(B
@item @var{List}$B$K4^$^$l$kE@$O!"(B@code{intersect}$B$d(B@code{sing}
$B$+$iJV$5$l$kE@$r;H$&$3$H$rA[Dj$7$F$$$k!#(B
@end itemize

@comment --- @example$B!A(B@end example $B$O<B9TNc$NI=<((B ---
@example
@end example

@comment --- $B;2>H(B($B%j%s%/(B)$B$r=q$/(B ---
@table @t
@item $B;2>H(B
@ref{adjoint1,adjoint2}
@end table


@comment --- $B"~"~"~"~(B  $B0J2<B>$N4X?t$K$D$$$F??;w$7$F5-=R$9$k(B. $B"~"~"~"~(B 


@comment --- $B$*$^$8$J$$(B ---
@node Index,,, Top
@unnumbered Index
@printindex fn
@printindex cp
@iftex
@vfill @eject
@end iftex
@summarycontents
@contents
@bye
@comment --- $B$*$^$8$J$$=*$j(B ---