Return to poly.texi CVS log

Up to [local] / OpenXM / src / asir-doc / parts / builtin

Imported OpenXM/src/asir-doc. 

@node $BB?9`<0$*$h$SM-M}<0$N1i;;(B,,, $BAH$_9~$_H!?t(B
@section $BB?9`<0(B, $BM-M}<0$N1i;;(B

@menu
* var::
* vars::
* uc::
* coef::
* deg mindeg::
* nmono::
* ord::
* sdiv sdivm srem sremm sqr sqrm::
* tdiv::
* %::
* subst psubst::
* diff::
* res::
* fctr sqfr::
* modfctr::
* ufctrhint::
* ptozp::
* prim cont::
* gcd gcdz::
* red::
@end menu

@node var,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{var}
@findex var

@table @t
@item var(@var{rat}) 
:: @var{rat} $B$N<gJQ?t(B. 
@end table

@table @var
@item return
$BITDj85(B
@item rat
$BM-M}<0(B
@end table

@itemize @bullet
@item
$B<gJQ?t$K4X$7$F$O(B, @xref{Asir $B$G;HMQ2DG=$J7?(B}.
@item 
$B%G%U%)%k%H$NJQ?t=g=x$O<!$N$h$&$K$J$C$F$$$k(B. 

@code{x}, @code{y}, @code{z}, @code{u}, @code{v}, @code{w}, @code{p}, @code{q}, @code{r}, @code{s}, @code{t}, @code{a}, @code{b}, @code{c}, @code{d}, @code{e},
@code{f}, @code{g}, @code{h}, @code{i}, @code{j}, @code{k}, @code{l}, @code{m}, @code{n}, @code{o},$B0J8e$OJQ?t$N8=$l$?=g(B. 
@end itemize

@example
[0] var(x^2+y^2+a^2);
x
[1] var(a*b*c*d*e);
a
[2] var(3/abc+2*xy/efg);
abc
@end example

@table @t
@item $B;2>H(B
@fref{ord}, @fref{vars}.
@end table

@node vars,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{vars}
@findex vars

@table @t
@item vars(@var{obj})
:: @var{obj} $B$K4^$^$l$kJQ?t$N%j%9%H(B. 
@end table

@table @var
@item return
$B%j%9%H(B
@item obj
$BG$0U(B
@end table

@itemize @bullet
@item
$BM?$($i$l$?<0$K4^$^$l$kJQ?t$N%j%9%H$rJV$9(B. 
@item
$BJQ?t=g=x$N9b$$$b$N$+$i=g$KJB$Y$k(B. 
@end itemize

@example
[0] vars(x^2+y^2+a^2);
[x,y,a]
[1] vars(3/abc+2*xy/efg);
[abc,xy,efg]
[2] vars([x,y,z]);
[x,y,z]
@end example

@table @t
@item $B;2>H(B
@fref{var}, @fref{uc}, @fref{ord}.
@end table

@node uc,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{uc}
@findex uc

@table @t
@item uc()
:: $B?7$?$JITDj85$r@8@.$9$k(B. 
@end table

@table @var
@item return
@code{vtype} $B$,(B 1 $B$NITDj85(B
@end table

@itemize @bullet
@item
@code{uc()} $B$r<B9T$9$k$?$S$K(B, @code{_0}, @code{_1}, @code{_2},... $B$H$$$&(B
$BITDj85$r@8@.$9$k(B. 
@item
@code{uc()} $B$G@8@.$5$l$?ITDj85$O(B, $BD>@\%-!<%\!<%I$+$iF~NO$9$k$3$H$,$G$-$J$$(B. 
$B$3$l$O(B, $B%W%m%0%i%`Cf$GL$Dj78?t$r<+F0@8@.$9$k>l9g(B, $BF~NO$J$I$K4^$^$l$k(B
$BITDj85$HF10l$N$b$N$,@8@.$5$l$k$3$H$rKI$0$?$a$G$"$k(B. 
@item
$BDL>o$NITDj85(B (@code{vtype} $B$,(B 0) $B$N<+F0@8@.$K$O(B @code{rtostr()}, 
@code{strtov()} $B$rMQ$$$k(B. 
@item
@code{uc()} $B$G@8@.$5$l$?ITDj85$NITDj85$H$7$F$N7?(B (@code{vtype}) $B$O(B 1 $B$G$"$k(B. 
(@xref{$BITDj85$N7?(B})
@end itemize

@example
[0] A=uc();
_0
[1] B=uc();
_1
[2] (uc()+uc())^2;
_2^2+2*_3*_2+_3^2
[3] (A+B)^2;
_0^2+2*_1*_0+_1^2
@end example

@table @t
@item $B;2>H(B
@fref{vtype}, @fref{rtostr}, @fref{strtov}.
@end table

@node coef,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{coef}
@findex coef

@table @t
@item coef(@var{poly},@var{deg}[,@var{var}])
:: @var{poly} $B$N(B @var{var} ($B>JN,;~$O<gJQ?t(B) $B$K4X$9$k(B @var{deg} $B<!$N78?t(B. 
@end table

@table @var
@item return
$BB?9`<0(B
@item poly
$BB?9`<0(B
@item var
$BITDj85(B
@item deg
$B<+A3?t(B
@end table

@itemize @bullet
@item
@var{poly} $B$N(B @var{var} $B$K4X$9$k(B @var{deg} $B<!$N78?t$r=PNO$9$k(B. 
@item
@var{var} $B$O(B, $B>JN,$9$k$H<gJQ?t(B @t{var}(@var{poly}) $B$@$H$_$J$5$l$k(B. 
@item
@var{var} $B$,<gJQ?t$G$J$$;~(B, @var{var} $B$,<gJQ?t$N>l9g$KHf3S$7$F(B
$B8zN($,Mn$A$k(B. 
@end itemize

@example
[0] A = (x+y+z)^3;
x^3+(3*y+3*z)*x^2+(3*y^2+6*z*y+3*z^2)*x+y^3+3*z*y^2+3*z^2*y+z^3
[1] coef(A,1,y);
3*x^2+6*z*x+3*z^2
[2] coef(A,0);
y^3+3*z*y^2+3*z^2*y+z^3
@end example

@table @t
@item $B;2>H(B
@fref{var}, @fref{deg mindeg}.
@end table

@node deg mindeg,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{deg}, @code{mindeg}
@findex deg
@findex mindeg

@table @t
@item deg(@var{poly},@var{var})
:: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B. 
@item mindeg(@var{poly},@var{var})
:: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:GDc<!?t(B. 
@end table

@table @var
@item return	
$B<+A3?t(B
@item poly
$BB?9`<0(B
@item var
$BITDj85(B
@end table

@itemize @bullet
@item
$BM?$($i$l$?B?9`<0$NJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B, $B:GDc<!?t$r=PNO$9$k(B. 
@item
$BJQ?t(B @var{var} $B$r>JN,$9$k$3$H$O=PMh$J$$(B. 
@end itemize

@example
[0] deg((x+y+z)^10,x);
10
[1] deg((x+y+z)^10,w);
0
[75] mindeg(x^2+3*x*y,x);
1
@end example

@node nmono,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{nmono}
@findex nmono

@table @t
@item nmono(@var{rat})
:: @var{rat} $B$NC19`<0$N9`?t(B. 
@end table

@table @var
@item return
$B<+A3?t(B
@item rat
$BM-M}<0(B
@end table

@itemize @bullet
@item
$BB?9`<0$rE83+$7$?>uBV$G$N(B 0 $B$G$J$$78?t$r;}$DC19`<0$N9`?t$r5a$a$k(B. 
@item
$BM-M}<0$N>l9g$O(B, $BJ,;R$HJ,Jl$N9`?t$NOB$,JV$5$l$k(B. 
@item
$BH!?t7A<0(B (@xref{$BITDj85$N7?(B}) $B$O(B, $B0z?t$,2?$G$"$C$F$bC19`$H$_$J$5$l$k(B. (1 $B8D$NITDj85$HF1$8(B. )
@end itemize

@example
[0] nmono((x+y)^10);
11
[1] nmono((x+y)^10/(x+z)^10);
22
[2] nmono(sin((x+y)^10));
1
@end example

@table @t
@item $B;2>H(B
@fref{vtype}.
@end table

@node ord,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{ord}
@findex ord

@table @t
@item ord([@var{varlist}])
:: $BJQ?t=g=x$N@_Dj(B
@end table

@table @var
@item return	
$BJQ?t$N%j%9%H(B
@item varlist
$BJQ?t$N%j%9%H(B
@end table

@itemize @bullet
@item
$B0z?t$,$"$k$H$-(B, $B0z?t$NJQ?t%j%9%H$r@hF,$K=P$7(B, $B;D$j$NJQ?t$,$=$N8e$K(B
$BB3$/$h$&$KJQ?t=g=x$r@_Dj$9$k(B. $B0z?t$N$"$k$J$7$K4X$o$i$:(B, @code{ord()}
$B$N=*N;;~$K$*$1$kJQ?t=g=x%j%9%H$rJV$9(B. 

@item
$B$3$NH!?t$K$h$kJQ?t=g=x$NJQ99$r9T$C$F$b(B, $B4{$K%W%m%0%i%`JQ?t$J$I$K(B
$BBeF~$5$l$F$$$k<0$NFbIt7A<0$O?7$7$$=g=x$K=>$C$F$OJQ99$5$l$J$$(B. 
$B=>$C$F(B, $B$3$NH!?t$K$h$k=g=x$NJQ99$O(B, @b{Asir} $B$N5/F0D>8e(B, 
$B$"$k$$$O(B, $B?7$?$JJQ?t$,8=$l$?;~E@$K9T$o$l$k(B
$B$Y$-$G$"$k(B. $B0[$J$kJQ?t=g=x$N$b$H$G@8@.$5$l$?<0$I$&$7$N1i;;(B
$B$,9T$o$l$?>l9g(B, $BM=4|$;$L7k2L$,@8$:$k$3$H$b$"$jF@$k(B. 
@end itemize

@example
[0] ord();
[x,y,z,u,v,w,p,q,r,s,t,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,_z,_u,_v,_w,_p,
_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),(_x)^(_y),
log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),cosh(_x),sinh(_x),
tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]
[1] ord([dx,dy,dz,a,b,c]);
[dx,dy,dz,a,b,c,x,y,z,u,v,w,p,q,r,s,t,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,_z,_u,_v,
_w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),
(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),
cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]
@end example

@node sdiv sdivm srem sremm sqr sqrm,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{sdiv}, @code{sdivm}, @code{srem}, @code{sremm}, @code{sqr}, @code{sqrm}
@findex sdiv
@findex sdivm
@findex srem
@findex sremm
@findex sqr
@findex sqrm

@table @t
@item sdiv(@var{poly1},@var{poly2}[,@var{v}])
@itemx sdivm(@var{poly1},@var{poly2},@var{mod}[,@var{v}])
:: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&$r5a$a$k(B. 
@item srem(@var{poly1},@var{poly2}[,@var{v}])
@item sremm(@var{poly1},@var{poly2},@var{mod}[,@var{v}])
:: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>jM>$r5a$a$k(B. 
@item sqr(@var{poly1},@var{poly2}[,@var{v}])
@item sqrm(@var{poly1},@var{poly2},@var{mod}[,@var{v}])
:: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&(B, $B>jM>$r(B
$B5a$a$k(B. 
@end table

@table @var
@item return
@code{sdiv()}, @code{sdivm()}, @code{srem()}, @code{sremm()} : $BB?9`<0(B, @code{sqr()}, @code{sqrm()} : @code{[$B>&(B,$B>jM>(B]} $B$J$k%j%9%H(B
@item poly1 poly2
$BB?9`<0(B
@item v
$BITDj85(B
@item mod
$BAG?t(B
@end table

@itemize @bullet
@item
@var{poly1} $B$r(B @var{poly2} $B$N<gJQ?t(B @t{var}(@var{poly2}) 
( $B0z?t(B @var{v} $B$,$"$k>l9g$K$O(B @var{v}) $B$K4X$9$kB?9`<0$H8+$F(B, 
@var{poly2} $B$G(B, $B3d$j;;$r9T$&(B.
@item
@code{sdivm()}, @code{sremm()}, @code{sqrm()} $B$O(B GF(@var{mod}) $B>e$G7W;;$9$k(B. 
@item
$BB?9`<0$N=|;;$O(B, $B<g78?t$I$&$7$N3d;;$K$h$jF@$i$l$?>&$H(B, $B<gJQ?t$NE,Ev$JQQ$N(B
$B@Q$r(B @var{poly2} $B$K3]$1$F(B, @var{poly1} $B$+$i0z$/$H$$$&A`:n$r(B
@var{poly1} $B$N<!?t$,(B @var{poly2} $B$N<!?t$h$j>.$5$/$J$k$^$G7+$jJV$7$F(B
$B9T$&(B. $B$3$NA`:n$,(B, $BB?9`<0$NHO0OFb$G9T$o$l$k$?$a$K$O(B, $B3F%9%F%C%W$K$*$$$F(B
$B<g78?t$I$&$7$N=|;;$,(B, $BB?9`<0$H$7$F$N@0=|$G$"$kI,MW$,$"$k(B. $B$3$l$,(B, $B!V=|;;(B
$B$,:G8e$^$G<B9T$G$-$k!W$3$H$N0UL#$G$"$k(B. 
@item
$BE57?E*$J>l9g$H$7$F(B, @var{poly2} $B$N<g78?t$,(B, $BM-M}?t$G$"$k>l9g(B, $B$"$k$$$O(B, 
@var{poly2} $B$,(B @var{poly1} $B$N0x;R$G$"$k$3$H$,$o$+$C$F$$$k>l9g$J$I(B
$B$,$"$k(B. 
@item
@code{sqr()} $B$O>&$H>jM>$rF1;~$K5a$a$?$$;~$KMQ$$$k(B. 
@item
$B@0?t=|;;$N>&(B, $B>jM>$O(B @code{idiv}, @code{irem} $B$rMQ$$$k(B. 
@item
$B78?t$KBP$9$k>jM>1i;;$O(B @code{%} $B$rMQ$$$k(B. 
@end itemize

@example
[0] sdiv((x+y+z)^3,x^2+y+a);    
x+3*y+3*z
[1] srem((x+y+z)^2,x^2+y+a);
(2*y+2*z)*x+y^2+(2*z-1)*y+z^2-a
[2] X=(x+y+z)*(x-y-z)^2;
x^3+(-y-z)*x^2+(-y^2-2*z*y-z^2)*x+y^3+3*z*y^2+3*z^2*y+z^3
[3] Y=(x+y+z)^2*(x-y-z);  
x^3+(y+z)*x^2+(-y^2-2*z*y-z^2)*x-y^3-3*z*y^2-3*z^2*y-z^3
[4] G=gcd(X,Y);
x^2-y^2-2*z*y-z^2
[5] sqr(X,G);
[x-y-z,0]
[6] sqr(Y,G);
[x+y+z,0]
[7] sdiv(y*x^3+x+1,y*x+1);  
divsp: cannot happen
return to toplevel
@end example

@table @t
@item $B;2>H(B
@fref{idiv irem}, @fref{%}.
@end table

@node tdiv,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{tdiv}
@findex tdiv

@table @t
@item tdiv(@var{poly1},@var{poly2})
:: @var{poly1} $B$,(B @var{poly2} $B$G3d$j@Z$l$k$+$I$&$+D4$Y$k(B. 
@end table

@table @var
@item return	
$B3d$j@Z$l$k$J$i$P>&(B, $B3d$j@Z$l$J$1$l$P(B 0
@item poly1 poly2
$BB?9`<0(B
@end table

@itemize @bullet
@item
@var{poly2} $B$,(B @var{poly1} $B$rB?9`<0$H$7$F3d$j@Z$k$+$I$&$+D4$Y$k(B. 
@item
$B$"$kB?9`<0$,4{Ls0x;R$G$"$k$3$H$O$o$+$C$F$$$k$,(B, $B$=$N=EJ#EY$,$o$+$i$J$$(B
$B>l9g$K(B, @code{tdiv()} $B$r7+$jJV$78F$V$3$H$K$h$j=EJ#EY$,$o$+$k(B. 
@end itemize

@example
[11] Y=(x+y+z)^5*(x-y-z)^3;  
x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6+(-6*y^3-18*z*y^2-18*z^2*y-6*z^3)*x^5
+(6*y^5+30*z*y^4+60*z^2*y^3+60*z^3*y^2+30*z^4*y+6*z^5)*x^3+(2*y^6+12*z*y^5
+30*z^2*y^4+40*z^3*y^3+30*z^4*y^2+12*z^5*y+2*z^6)*x^2+(-2*y^7-14*z*y^6
-42*z^2*y^5-70*z^3*y^4-70*z^4*y^3-42*z^5*y^2-14*z^6*y-2*z^7)*x-y^8-8*z*y^7
-28*z^2*y^6-56*z^3*y^5-70*z^4*y^4-56*z^5*y^3-28*z^6*y^2-8*z^7*y-z^8
[12] for(I=0,F=x+y+z,T=Y; T=tdiv(T,F); I++); 
[13] I;
5
@end example

@table @t
@item $B;2>H(B
@fref{sdiv sdivm srem sremm sqr sqrm}.
@end table

@node %,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{%}
@findex %

@table @t
@item @var{poly} % @var{m}
:: $B@0?t$K$h$k>jM>(B
@end table

@table @var
@item return	
$B@0?t$^$?$OB?9`<0(B
@item poly
$B@0?t$^$?$O@0?t78?tB?9`<0(B
@item m
$B@0?t(B
@end table

@itemize @bullet
@item
@var{poly} $B$N3F78?t$r(B @var{m} $B$G3d$C$?>jM>$GCV$-49$($?B?9`<0$rJV$9(B. 
@item
$B7k2L$N78?t$OA4$F@5$N@0?t$H$J$k(B. 
@item
@var{poly} $B$O@0?t$G$b$h$$(B. $B$3$N>l9g(B, $B7k2L$,@5$K@55,2=$5$l$k$3$H$r=|$1$P(B
@code{irem()} $B$HF1MM$KMQ$$$k$3$H$,$G$-$k(B. 
@item
@var{poly} $B$N78?t(B, @var{m} $B$H$b@0?t$G$"$kI,MW$,$"$k$,(B, $B%A%'%C%/$O9T$J$o$l$J$$(B. 
@end itemize

@example
[0] (x+2)^5 % 3;
x^5+x^4+x^3+2*x^2+2*x+2
[1] (x-2)^5 % 3;
x^5+2*x^4+x^3+x^2+2*x+1
[2] (-5) % 4;
3
[3] irem(-5,4);
-1
@end example

@table @t
@item $B;2>H(B
@fref{idiv irem}.
@end table

@node subst psubst,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{subst}, @code{psubst}
@findex subst
@findex psubst

@table @t
@item subst(@var{rat}[,@var{varn},@var{ratn}]*)
@item psubst(@var{rat}[,@var{var},@var{rat}]*)
:: @var{rat} $B$N(B @var{varn} $B$K(B @var{ratn} $B$rBeF~(B
(@var{n=1,2},... $B$G:8$+$i1&$K=g<!BeF~$9$k(B). 
@end table

@table @var
@item return
$BM-M}<0(B
@item rat,ratn
$BM-M}<0(B
@item varn
$BITDj85(B
@end table

@itemize @bullet
@item
$BM-M}<0$NFCDj$NITDj85$K(B, $BDj?t$"$k$$$OB?9`<0(B, $BM-M}<0$J$I$rBeF~$9$k$N$KMQ$$$k(B. 
@item
@t{subst}(@var{rat},@var{var1},@var{rat1},@var{var2},@var{rat2},...) $B$O(B, 
@t{subst}(@t{subst}(@var{rat},@var{var1},@var{rat1}),@var{var2},@var{rat2},...) 
$B$HF1$80UL#$G$"$k(B. 
@item
$BF~NO$N:8B&$+$i=g$KBeF~$r7+$jJV$9$?$a$K(B, $BF~NO$N=g$K$h$C$F7k2L$,JQ$o$k$3$H$,$"$k(B. 
@item
@code{subst()} $B$O(B, @code{sin()} $B$J$I$NH!?t$N0z?t$KBP$7$F$bBeF~$r9T$&(B. 
@code{psubst()} $B$O(B, $B$3$N$h$&$JH!?t$r0l$D$NFHN)$7$?ITDj85$H8+$J$7$F(B, $B$=(B
$B$N0z?t$K$OBeF~$O9T$o$J$$(B. (partial substitution $B$N$D$b$j(B)
@item
@b{Asir} $B$G$O(B, $BM-M}<0$NLsJ,$O<+F0E*$K$O9T$o$J$$$?$a(B, 
$BM-M}<0$NBeF~$O(B, $B;W$o$L7W;;;~4V$NA}Bg$r0z$-5/$3$9>l9g$,$"$k(B. 
$BM-M}<0$rBeF~$9$k>l9g$K$O(B, $BLdBj$K1~$8$?FH<+$NH!?t$r=q$$$F(B, 
$B$J$k$Y$/J,Jl(B, $BJ,;R$,Bg$-$/$J$i$J$$$h$&$KG[N8$9$k$3$H$b$7$P$7$PI,MW$H$J$k(B. 
@item
$BJ,?t$rBeF~$9$k>l9g$bF1MM$G$"$k(B. 
@end itemize

@example
[0] subst(x^3-3*y*x^2+3*y^2*x-y^3,y,2);
x^3-6*x^2+12*x-8
[1] subst(@@@@,x,-1);
-27
[2] subst(x^3-3*y*x^2+3*y^2*x-y^3,y,2,x,-1);
-27
[3] subst(x*y^3,x,y,y,x);  
x^4
[4] subst(x*y^3,y,x,x,y);    
y^4
[5] subst(x*y^3,x,t,y,x,t,y);
y*x^3
[6] subst(x*sin(x),x,t);
sint(t)*t
[7] psubst(x*sin(x),x,t);
sin(x)*t
@end example

@node diff,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{diff}
@findex diff

@table @t
@item diff(@var{rat}[,@var{varn}]*)
@item diff(@var{rat},@var{varlist})
:: @var{rat} $B$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G=g<!HyJ,$9$k(B. 
@end table

@table @var
@item return
$B<0(B
@item rat
$BM-M}<0(B ($B=iEyH!?t$r4^$s$G$b$h$$(B)
@item varn
$BITDj85(B
@item varlist
$BITDj85$N%j%9%H(B
@end table

@itemize @bullet
@item
$BM?$($i$l$?=iEyH!?t$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G(B
$B=g<!HyJ,$9$k(B. 
@item
$B:8B&$NITDj85$h$j(B, $B=g$KHyJ,$7$F$$$/(B. $B$D$^$j(B, @t{diff}(@var{rat},@t{x,y}) $B$O(B, 
@t{diff}(@t{diff}(@var{rat},@t{x}),@t{y}) $B$HF1$8$G$"$k(B. 
@end itemize

@example
[0] diff((x+2*y)^2,x);  
2*x+4*y
[1] diff((x+2*y)^2,x,y);
4
[2] diff(x/sin(log(x)+1),x);
(sin(log(x)+1)-cos(log(x)+1))/(sin(log(x)+1)^2)
[3] diff(sin(x),[x,x,x,x]);
sin(x)
@end example

@node res,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{res}
@findex res

@table @t
@item res(@var{var},@var{poly1},@var{poly2}[,@var{mod}])
:: @var{var} $B$K4X$9$k(B @var{poly1} $B$H(B @var{poly2} $B$N=*7k<0(B. 
@end table

@table @var
@item return
$BB?9`<0(B
@item var
$BITDj85(B
@item poly1,poly2
$BB?9`<0(B
@item mod
$BAG?t(B
@end table

@itemize @bullet
@item
$BFs$D$NB?9`<0(B @var{poly1} $B$H(B @var{poly2} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k(B
$B=*7k<0$r5a$a$k(B. 
@item
$BItJ,=*7k<0%"%k%4%j%:%`$K$h$k(B. 
@item
$B0z?t(B @var{mod} $B$,$"$k;~(B, GF(@var{mod}) $B>e$G$N7W;;$r9T$&(B. 
@end itemize

@example
[0] res(t,(t^3+1)*x+1,(t^3+1)*y+t);
-x^3-x^2-y^3
@end example

@node fctr sqfr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{fctr}, @code{sqfr}
@findex fctr
@findex sqfr

@table @t
@item fctr(@var{poly})
:: @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B. 
@item sqfr(@var{poly})
:: @var{poly} $B$rL5J?J}J,2r$9$k(B. 
@end table

@table @var
@item return
$B%j%9%H(B
@item poly
$BM-M}?t78?t$NB?9`<0(B
@end table

@itemize @bullet
@item
$BM-M}?t78?t$NB?9`<0(B @var{poly} $B$r0x?tJ,2r$9$k(B. @code{fctr()} $B$O4{Ls0x;RJ,2r(B, 
@code{sqfr()} $B$OL5J?J}0x;RJ,2r(B. 
@item
$B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B. 
@item
@b{$B?t78?t(B} $B$H(B $BA4$F$N(B @b{$B0x;R(B}^@b{$B=EJ#EY(B} $B$N@Q$,(B @var{poly} $B$HEy$7$$(B. 
@item
@b{$B?t78?t(B} $B$O(B, (@var{poly}/@b{$B?t78?t(B}) $B$,(B, $B@0?t78?t$G(B, $B78?t$N(B GCD $B$,(B 1 $B$H$J$k(B
$B$h$&$JB?9`<0$K$J$k$h$&$KA*$P$l$F$$$k(B. (@code{ptozp()} $B;2>H(B)
@end itemize

@example
[0] fctr(x^10-1);
[[1,1],[x-1,1],[x+1,1],[x^4+x^3+x^2+x+1,1],[x^4-x^3+x^2-x+1,1]]
[1] fctr(x^3+y^3+(z/3)^3-x*y*z);
[[1/27,1],[9*x^2+(-9*y-3*z)*x+9*y^2-3*z*y+z^2,1],[3*x+3*y+z,1]]
[2] A=(a+b+c+d)^2;
a^2+(2*b+2*c+2*d)*a+b^2+(2*c+2*d)*b+c^2+2*d*c+d^2
[3] fctr(A);
[[1,1],[a+b+c+d,2]]
[4] A=(x+1)*(x^2-y^2)^2; 
x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4
[5] sqfr(A);
[[1,1],[x+1,1],[-x^2+y^2,2]]
[6] fctr(A);
[[1,1],[x+1,1],[-x-y,2],[x-y,2]]
@end example

@table @t
@item $B;2>H(B
@fref{ufctrhint}.
@end table

@node ufctrhint,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{ufctrhint}
@findex ufctrhint

@table @t
@item ufctrhint(@var{poly},@var{hint})
:: $B<!?t>pJs$rMQ$$$?(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B
@end table

@table @var
@item return
$B%j%9%H(B
@item poly
$BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B
@item hint
$B<+A3?t(B
@end table

@itemize @bullet
@item
$B3F4{Ls0x;R$N<!?t$,(B @var{hint} $B$NG\?t$G$"$k$3$H$,$o$+$C$F$$$k>l9g$K(B
@var{poly} $B$N4{Ls0x;RJ,2r$r(B @code{fctr()} $B$h$j8zN(NI$/9T$&(B. 
@var{poly} $B$,(B, @var{d} $B<!$N3HBgBN>e$K$*$1$k(B
$B$"$kB?9`<0$N%N%k%`(B (@xref{$BBe?tE*?t$K4X$9$k1i;;(B}) $B$GL5J?J}$G$"$k>l9g(B, 
$B3F4{Ls0x;R$N<!?t$O(B @var{d} $B$NG\?t$H$J$k(B. $B$3$N$h$&$J>l9g$K(B
$BMQ$$$i$l$k(B. 
@end itemize

@example
[10] A=t^9-15*t^6-87*t^3-125;               
t^9-15*t^6-87*t^3-125
0msec
[11] N=res(t,subst(A,t,x-2*t),A);           
-x^81+1215*x^78-567405*x^75+139519665*x^72-19360343142*x^69+1720634125410*x^66
-88249977024390*x^63-4856095669551930*x^60+1999385245240571421*x^57
-15579689952590251515*x^54+15956967531741971462865*x^51
...
+140395588720353973535526123612661444550659875*x^6
+10122324287343155430042768923500799484375*x^3
+139262743444407310133459021182733314453125
980msec + gc : 250msec
[12] sqfr(N);
[[-1,1],[x^81-1215*x^78+567405*x^75-139519665*x^72+19360343142*x^69
-1720634125410*x^66+88249977024390*x^63+4856095669551930*x^60
-1999385245240571421*x^57+15579689952590251515*x^54
...
-10122324287343155430042768923500799484375*x^3
-139262743444407310133459021182733314453125,1]]
20msec
[13] fctr(N);                               
[[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],
[x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3
+296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],
[x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3+31524548679,1],
[x^18+10773*x^12+2784051*x^6+307546875,1]]
167.050sec + gc : 1.890sec
[14] ufctrhint(N,9);
[[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],
[x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3
+296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],
[x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3+31524548679,1],
[x^18+10773*x^12+2784051*x^6+307546875,1]]
119.340sec + gc : 1.300sec
@end example

@table @t
@item $B;2>H(B
@fref{fctr sqfr}.
@end table

@node modfctr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{modfctr}
@findex modfctr

@table @t
@item modfctr(@var{poly},@var{mod})
:: $BM-8BBN>e$G$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B
@end table

@table @var
@item return
$B%j%9%H(B
@item poly
$B@0?t78?t$N(B 1 $BJQ?tB?9`<0(B
@item mod
$B<+A3?t(B
@end table

@itemize @bullet
@item
2^31 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$G0lJQ?tB?9`<0(B
@var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B. 
@item
$B7k2L$O(B [[@b{$B?t78?t(B},1],[@b{$B0x;R(B},@b{$B=EJ#EY(B}],...] $B$J$k%j%9%H(B. 
@item
@b{$B?t78?t(B} $B$H(B $BA4$F$N(B @b{$B0x;R(B}^@b{$B=EJ#EY(B} $B$N@Q$,(B @var{poly} $B$HEy$7$$(B. 
@end itemize

@example
[0] modfctr(x^10+x^2+1,2147483647);
[[1,1],[x+1513477736,1],[x+2055628767,1],[x+91854880,1],
[x+634005911,1],[x+1513477735,1],[x+634005912,1],
[x^4+1759639395*x^2+2045307031,1]]
@end example

@table @t
@item $B;2>H(B
@fref{fctr sqfr}.
@end table

@node ptozp,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{ptozp}
@findex ptozp

@table @t
@item ptozp(@var{poly})
:: @var{poly} $B$rM-M}?tG\$7$F@0?t78?tB?9`<0$K$9$k(B. 
@end table

@table @var
@item return
$BB?9`<0(B
@item poly
$BB?9`<0(B
@end table

@itemize @bullet
@item
$BM?$($i$l$?B?9`<0(B @var{poly} $B$KE,Ev$JM-M}?t$r3]$1$F(B, $B@0?t78?t$+$D(B
$B78?t$N(B GCD $B$,(B 1 $B$K$J$k$h$&$K$9$k(B. 
@item
$BJ,?t$N;MB'1i;;$O(B, $B@0?t$N1i;;$KHf3S$7$FCY$$$?$a(B, $B<o!9$NB?9`<01i;;(B
$B$NA0$K(B, $BB?9`<0$r@0?t78?t$K$7$F$*$/$3$H$,K>$^$7$$(B. 
@item
$BM-M}<0$rLsJ,$9$k(B @code{red()} $B$GJ,?t78?tM-M}<0$rLsJ,$7$F$b(B, 
$BJ,;RB?9`<0$N78?t$OM-M}?t$N$^$^$G$"$j(B, $BM-M}<0$NJ,;R$r5a$a$k(B
@code{nm()} $B$G$O(B, $BJ,?t78?tB?9`<0$O(B, $BJ,?t78?t$N$^$^$N7A$G=PNO$5$l$k$?$a(B, 
$BD>$A$K@0?t78?tB?9`<0$rF@$k;v$O=PMh$J$$(B. 
@end itemize

@example
[0] ptozp(2*x+5/3);
6*x+5
[1] nm(2*x+5/3);   
2*x+5/3
@end example

@table @t
@item $B;2>H(B
@fref{nm dn}.
@end table

@node prim cont,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{prim}, @code{cont}
@findex prim

@table @t
@item prim(@var{poly}[,@var{v}])
:: @var{poly} $B$N86;OE*ItJ,(B (primitive part). 
@item cont(@var{poly}[,@var{v}])
:: @var{poly} $B$NMFNL(B (content). 
@end table

@table @var
@item return poly
$BM-M}?t78?tB?9`<0(B
@item v
$BITDj85(B
@end table

@itemize @bullet
@item
@var{poly} $B$N<gJQ?t(B ($B0z?t(B @var{v} $B$,$"$k>l9g$K$O(B @var{v}) 
$B$K4X$9$k86;OE*ItJ,(B, $BMFNL$r5a$a$k(B. 
@end itemize

@example
[0] E=(y-z)*(x+y)*(x-z)*(2*x-y);
(2*y-2*z)*x^3+(y^2-3*z*y+2*z^2)*x^2+(-y^3+z^2*y)*x+z*y^3-z^2*y^2
[1] prim(E);
2*x^3+(y-2*z)*x^2+(-y^2-z*y)*x+z*y^2
[2] cont(E);
y-z
[3] prim(E,z);
(y-z)*x-z*y+z^2
@end example

@table @t
@item $B;2>H(B
@fref{var}, @fref{ord}.
@end table

@node gcd gcdz,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{gcd}, @code{gcdz}
@findex gcd

@table @t
@item gcd(@var{poly1},@var{poly2}[,@var{mod}])
@item gcdz(@var{poly1},@var{poly2})
:: @var{poly1} $B$H(B @var{poly2} $B$N(B gcd. 
@end table

@table @var
@item return
$BB?9`<0(B
@item poly1,poly2
$BB?9`<0(B
@item mod
$BAG?t(B
@end table

@itemize @bullet
@item
$BFs$D$NB?9`<0$N:GBg8xLs<0(B (GCD) $B$r5a$a$k(B. 
@item
@code{gcd()} $B$OM-M}?tBN>e$NB?9`<0$H$7$F$N(B GCD $B$rJV$9(B. 
$B$9$J$o$A(B, $B7k2L$O@0?t78?t$G(B, $B$+$D78?t$N(B GCD
$B$,(B 1 $B$K$J$k$h$&$JB?9`<0(B, $B$^$?$O(B, $B8_$$$KAG$N>l9g$O(B 1 $B$rJV$9(B.
@item
@code{gcdz()} $B$O(B @var{poly1}, @var{poly2} $B$H$b$K@0?t78?t$N>l9g$K(B, 
$B@0?t4D>e$NB?9`<0$H$7$F$N(B GCD $B$rJV$9(B.
$B$9$J$o$A(B, @code{gcd()} $B$NCM$K(B, $B78?tA4BN$N@0?t(B GCD$B$NCM$r3]$1$?$b$N$rJV$9(B. 
@item
$B0z?t(B @var{mod} $B$,$"$k;~(B, @code{gcd()} $B$O(B GF(@var{mod}) $B>e$G$N(B GCD $B$rJV$9(B. 
@item
@code{gcd()}, @code{gcdz()} Extended Zassenhaus $B%"%k%4%j%:%`$K$h$k(B. 
$BM-8BBN>e$N(B GCD $B$O(B PRS $B%"%k%4%j%:%`$K$h$C$F$$$k$?$a(B, $BBg$-$JLdBj(B, 
GCD $B$,(B 1 $B$N>l9g$J$I$K$*$$$F8zN($,0-$$(B. 
@end itemize

@example
[0] gcd(12*(x^2+2*x+1)^2,18*(x^2+(y+1)*x+y)^3);
x^3+3*x^2+3*x+1
[1] gcdz(12*(x^2+2*x+1)^2,18*(x^2+(y+1)*x+y)^3);
6*x^3+18*x^2+18*x+6
[2] gcd((x+y)*(x-y)^2,(x+y)^2*(x-y));
x^2-y^2
[3] gcd((x+y)*(x-y)^2,(x+y)^2*(x-y),2);
x^3+y*x^2+y^2*x+y^3
@end example

@table @t
@item $B;2>H(B
@fref{igcd igcdcntl}.
@end table

@node red,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B
@subsection @code{red}
@findex red

@table @t
@item red(@var{rat})
:: @var{rat} $B$rLsJ,$7$?$b$N(B. 
@end table

@table @var
@item return
$BM-M}<0(B
@item rat
$BM-M}<0(B
@end table

@itemize @bullet
@item
@b{Asir} $B$OM-M}?t$NLsJ,$r>o$K<+F0E*$K9T$&(B. 
$B$7$+$7(B, $BM-M}<0$K$D$$$F$ODLJ,$O9T$&$,(B, 
$BLsJ,$O%f!<%6!<$,;XDj$7$J$$8B$j9T$o$J$$(B. 
$B$3$NLsJ,$r9T$&%3%^%s%I$,(B @t{red} $B$G$"$k(B. 
@item
EZGCD $B$K$h$j(B @var{rat} $B$NJ,;R(B, $BJ,Jl$rLsJ,$9$k(B. 
@item
$B=PNO$5$l$kM-M}<0$NJ,Jl$NB?9`<0$O(B, $B3F78?t$N(B GCD $B$,(B 1 $B$N(B
$B@0?t78?tB?9`<0$G$"$k(B. 
$BJ,;R$K$D$$$F$O@0?t78?tB?9`<0$H$J$k$H$O8B$i$J$$(B. 
@item
GCD $B$OBgJQ=E$$1i;;$J$N$G(B, $BB>$NJ}K!$G=|$1$k6&DL0x;R$O2DG=$J8B$j=|$/$N$,(B
$BK>$^$7$$(B. $B$^$?(B, $BJ,Jl(B, $BJ,;R$,Bg$-$/$J$C$F$+$i$N$3$NH!?t$N8F$S=P$7$O(B, 
$BHs>o$K;~4V$,3]$+$k>l9g$,B?$$(B. $BM-M}<01i;;$r9T$&>l9g$O(B, $B$"$kDxEY(B
$BIQHK$K(B, $BLsJ,$r9T$&I,MW$,$"$k(B. 
@end itemize

@example
[0] (x^3-1)/(x-1);
(x^3-1)/(x-1)
[1] red((x^3-1)/(x-1));
x^2+x+1
[2] red((x^3+y^3+z^3-3*x*y*z)/(x+y+z));
x^2+(-y-z)*x+y^2-z*y+z^2
[3] red((3*x*y)/(12*x^2+21*y^3*x));
(y)/(4*x+7*y^3)
[4] red((3/4*x^2+5/6*x)/(2*y*x+4/3*x));
(9/8*x+5/4)/(3*y+2)
@end example

@table @t
@item $B;2>H(B
@fref{nm dn}, @fref{gcd gcdz}, @fref{ptozp}.
@end table