OpenXM/src/asir-doc/parts/builtin/poly.texi - diff

Return to poly.texi CVS log

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

Diff for /OpenXM/src/asir-doc/parts/builtin/poly.texi between version 1.1.1.1 and 1.6

version 1.1.1.1, 1999/12/08 05:47:44

version 1.6, 2003/11/27 15:56:08

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/poly.texi,v 1.5 2003/04/20 08:01:29 noro Exp $

\BJP

@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

\BEG

@node Polynomials and rational expressions,,, Built-in Function

@section operations with polynomials and rational expressions

@menu

* var::

Line 24

Line 31

* red::

@end menu

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

\JP @node var,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node var,,, Polynomials and rational expressions

@subsection @code{var}

@findex var

@table @t

@item var(@var{rat})

:: @var{rat} $B$N<gJQ?t(B.

\JP :: @var{rat} $B$N<gJQ?t(B.

\EG :: Main variable (indeterminate) of @var{rat}.

@end table

@table @var

@item return

$BITDj85(B

\JP $BITDj85(B

\EG indeterminate

@item rat

$BM-M}<0(B

\JP $BM-M}<0(B

\EG rational expression

@end table

@itemize @bullet

\BJP

@item

$B<gJQ?t$K4X$7$F$O(B, @xref{Asir $B$G;HMQ2DG=$J7?(B}.

@item

Line 48

Line 60

@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.

\BEG

@item

See @ref{Types in Asir} for main variable.

@item

Indeterminates (variables) are ordered by default as follows.

@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}. The other variables will be ordered after the above noted variables

so that the first comer will be ordered prior to the followers.

@end itemize

@example

Line 60 abc

Line 85 abc

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{ord}, @fref{vars}.

@end table

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

\JP @node vars,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node vars,,, Polynomials and rational expressions

@subsection @code{vars}

@findex vars

@table @t

@item vars(@var{obj})

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

\JP :: @var{obj} $B$K4^$^$l$kJQ?t$N%j%9%H(B.

\EG :: A list of variables (indeterminates) in an expression @var{obj}.

@end table

@table @var

@item return

$B%j%9%H(B

\JP $B%j%9%H(B

\EG list

@item obj

$BG$0U(B

\JP $BG$0U(B

\EG arbitrary

@end table

@itemize @bullet

\BJP

@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.

\BEG

@item

Returns a list of variables (indeterminates) contained in a given expression.

@item

Lists variables according to the variable ordering.

@end itemize

@example

Line 97 abc

Line 135 abc

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{var}, @fref{uc}, @fref{ord}.

@end table

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

\JP @node uc,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node uc,,, Polynomials and rational expressions

@subsection @code{uc}

@findex uc

@table @t

@item uc()

:: $B?7$?$JITDj85$r@8@.$9$k(B.

\JP :: $BL$Dj78?tK!$N$?$a$NITDj85$r@8@.$9$k(B.

\EG :: Create a new indeterminate for an undermined coeficient.

@end table

@table @var

@item return

@code{vtype} $B$,(B 1 $B$NITDj85(B

\JP @code{vtype} $B$,(B 1 $B$NITDj85(B

\EG indeterminate with its @code{vtype} 1.

@end table

@itemize @bullet

\BJP

@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.

Line 128 abc

Line 171 abc

@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})

(@xref{$BITDj85$N7?(B}.)

\BEG

@item

At every evaluation of command @code{uc()}, a new indeterminate in

the sequence of indeterminates @code{_0}, @code{_1}, @code{_2}, @dots{}

is created successively.

@item

Indeterminates created by @code{uc()} cannot be input on the keyboard.

By this property, you are free, no matter how many indeterminates you

will create dynamically by a program, from collision of created names

with indeterminates input from the keyboard or from program files.

@item

Functions, @code{rtostr()} and @code{strtov()}, are used to create

ordinary indeterminates (indeterminates having 0 for their @code{vtype}).

@item

Kernel sub-type of indeterminates created by @code{uc()} is 1.

(@code{vtype(uc())}=1)

@end itemize

@example

Line 143 _0^2+2*_1*_0+_1^2

Line 204 _0^2+2*_1*_0+_1^2

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{vtype}, @fref{rtostr}, @fref{strtov}.

@end table

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

\JP @node coef,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node coef,,, Polynomials and rational expressions

@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.

\JP :: @var{poly} $B$N(B @var{var} ($B>JN,;~$O<gJQ?t(B) $B$K4X$9$k(B @var{deg} $B<!$N78?t(B.

\BEG

:: The coefficient of a polynomial @var{poly} at degree @var{deg}

with respect to the variable @var{var} (main variable if unspecified).

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item var

$BITDj85(B

\JP $BITDj85(B

\EG indeterminate

@item deg

$B<+A3?t(B

\JP $B<+A3?t(B

\EG non-negative integer

@end table

@itemize @bullet

\BJP

@item

@var{poly} $B$N(B @var{var} $B$K4X$9$k(B @var{deg} $B<!$N78?t$r=PNO$9$k(B.

@item

Line 175 _0^2+2*_1*_0+_1^2

Line 247 _0^2+2*_1*_0+_1^2

@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.

\BEG

@item

The coefficient of a polynomial @var{poly} at degree @var{deg}

with respect to the variable @var{var}.

@item

The default value for @var{var} is the main variable, i.e.,

@t{var(@var{poly})}.

@item

For multi-variate polynomials, access to coefficients depends on

the specified indeterminates. For example, taking coef for the main

variable is much faster than for other variables.

@end itemize

@example

Line 187 y^3+3*z*y^2+3*z^2*y+z^3

Line 272 y^3+3*z*y^2+3*z^2*y+z^3

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{var}, @fref{deg mindeg}.

@end table

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

\JP @node deg mindeg,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node deg mindeg,,, Polynomials and rational expressions

@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.

\JP :: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:G9b<!?t(B.

\EG :: The degree of a polynomial @var{poly} with respect to variable.

@item mindeg(@var{poly},@var{var})

:: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:GDc<!?t(B.

\JP :: @var{poly} $B$N(B, $BJQ?t(B @var{var} $B$K4X$9$k:GDc<!?t(B.

\BEG

:: The least exponent of the terms with non-zero coefficients in

a polynomial @var{poly} with respect to the variable @var{var}.

In this manual, this quantity is sometimes referred to the minimum

degree of a polynomial for short.

@end table

@table @var

@item return

$B<+A3?t(B

\JP $B<+A3?t(B

\EG non-negative integer

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item var

$BITDj85(B

\JP $BITDj85(B

\EG indeterminate

@end table

@itemize @bullet

\BJP

@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.

\BEG

@item

The least exponent of the terms with non-zero coefficients in

a polynomial @var{poly} with respect to the variable @var{var}.

In this manual, this quantity is sometimes referred to the minimum

degree of a polynomial for short.

@item

Variable @var{var} must be specified.

@end itemize

@example

Line 228 y^3+3*z*y^2+3*z^2*y+z^3

Line 336 y^3+3*z*y^2+3*z^2*y+z^3

@end example

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

\JP @node nmono,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node nmono,,,Polynomials and rational expressions

@subsection @code{nmono}

@findex nmono

@table @t

@item nmono(@var{rat})

:: @var{rat} $B$NC19`<0$N9`?t(B.

\JP :: @var{rat} $B$NC19`<0$N9`?t(B.

\EG :: Number of monomials in rational expression @var{rat}.

@end table

@table @var

@item return

$B<+A3?t(B

\JP $B<+A3?t(B

\EG non-negative integer

@item rat

$BM-M}<0(B

\JP $BM-M}<0(B

\EG rational expression

@end table

@itemize @bullet

\BJP

@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. )

$BH!?t7A<0(B (@ref{$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. )

\BEG

@item

Number of monomials with non-zero number coefficients in the full

expanded form of the given polynomial.

@item

For a rational expression, the sum of the numbers of monomials

of the numerator and denominator.

@item

A function form is regarded as a single indeterminate no matter how

complex arguments it has.

@end itemize

@example

Line 263 y^3+3*z*y^2+3*z^2*y+z^3

Line 388 y^3+3*z*y^2+3*z^2*y+z^3

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{vtype}.

@end table

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

\JP @node ord,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node ord,,, Polynomials and rational expressions

@subsection @code{ord}

@findex ord

@table @t

@item ord([@var{varlist}])

:: $BJQ?t=g=x$N@_Dj(B

\JP :: $BJQ?t=g=x$N@_Dj(B

\EG :: It sets the ordering of indeterminates (variables).

@end table

@table @var

@item return

$BJQ?t$N%j%9%H(B

\JP $BJQ?t$N%j%9%H(B

\EG list of indeterminates

@item varlist

$BJQ?t$N%j%9%H(B

\JP $BJQ?t$N%j%9%H(B

\EG list of indeterminates

@end table

@itemize @bullet

\BJP

@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()}

Line 296 y^3+3*z*y^2+3*z^2*y+z^3

Line 427 y^3+3*z*y^2+3*z^2*y+z^3

$B$"$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.

\BEG

@item

When an argument is given,

this function rearranges the ordering of variables (indeterminates)

so that the indeterminates in the argument @var{varlist} precede

and the other indeterminates follow in the system's variable ordering.

Regardless of the existence of an argument, it always returns the

final variable ordering.

@item

Note that no change will be made to the variable ordering of internal

forms of objects which already exists in the system, no matter what

reordering you specify. Therefore, the reordering should be limited to

the time just after starting @b{Asir}, or to the time when one has

decided himself to start a totally new computation which has no relation

with the previous results.

Note that unexpected results may be obtained from operations between

objects which are created under different variable ordering.

@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,

[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,

_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),(_x)^(_y),

_w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,

log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),cosh(_x),sinh(_x),

exp(_x),(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),

tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]

(-_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,

[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,

_w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,exp(_x),

_z,_u,_v,_w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,

(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),(-_x^2+1)^(-1/2),

_o,exp(_x),(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),

cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]

(-_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

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

\EG @node sdiv sdivm srem sremm sqr sqrm,,, Polynomials and rational expressions

@subsection @code{sdiv}, @code{sdivm}, @code{srem}, @code{sremm}, @code{sqr}, @code{sqrm}

@findex sdiv

@findex sdivm

Line 323 cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-

Line 477 cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-

@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.

\JP :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>&$r5a$a$k(B.

\BEG

:: Quotient of @var{poly1} divided by @var{poly2} provided that the

division can be performed within polynomial arithmetic over the

rationals.

@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.

\JP :: @var{poly1} $B$r(B @var{poly2} $B$G3d$k=|;;$,:G8e$^$G<B9T$G$-$k>l9g$K>jM>$r5a$a$k(B.

\BEG

:: Remainder of @var{poly1} divided by @var{poly2} provided that the

division can be performed within polynomial arithmetic over the

rationals.

@item sqr(@var{poly1},@var{poly2}[,@var{v}])

@item sqrm(@var{poly1},@var{poly2},@var{mod}[,@var{v}])

\BJP

:: @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.

\BEG

:: Quotient and remainder of @var{poly1} divided by @var{poly2} provided

that the division can be performed within polynomial arithmetic over

the rationals.

@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

\JP @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

\EG @code{sdiv()}, @code{sdivm()}, @code{srem()}, @code{sremm()} : polynomial @code{sqr()}, @code{sqrm()} : a list @code{[quotient,remainder]}

@item poly1 poly2

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item v

$BITDj85(B

\JP $BITDj85(B

\EG indeterminate

@item mod

$BAG?t(B

\JP $BAG?t(B

\EG prime

@end table

@itemize @bullet

\BJP

@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,

Line 368 cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-

Line 544 cosh(_x),sinh(_x),tanh(_x),(_x^2+1)^(-1/2),(_x^2-1)^(-

$B@0?t=|;;$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.

\BEG

@item

Regarding @var{poly1} as an uni-variate polynomial in the main variable

of @var{poly2},

i.e. @t{var(@var{poly2})} (@var{v} if specified), @code{sdiv()} and

@code{srem()} compute

the polynomial quotient and remainder of @var{poly1} divided by @var{poly2}.

@item @code{sdivm()}, @code{sremm()}, @code{sqrm()} execute the same

operation over GF(@var{mod}).

@item

Division operation of polynomials is performed by the following steps:

(1) obtain the quotient of leading coefficients; let it be Q;

(2) remove the leading term of @var{poly1} by subtracting, from

@var{poly1}, the product of Q with some powers of main variable

and @var{poly2}; obtain a new @var{poly1};

(3) repeat the above step until the degree of @var{poly1} become smaller

than that of @var{poly2}.

For fulfillment, by operating in polynomials, of this procedure, the

divisions at step (1) in every repetition must be an exact division of

polynomials. This is the true meaning of what we say

``division can be performed within polynomial arithmetic

over the rationals.''

@item

There are typical cases where the division is possible:

leading coefficient of @var{poly2} is a rational number;

@var{poly2} is a factor of @var{poly1}.

@item

Use @code{sqr()} to get both the quotient and remainder at once.

@item

Use @code{idiv()}, @code{irem()} for integer quotient.

@item

For remainder operation on all integer coefficients, use @code{%}.

@end itemize

@example

Line 391 return to toplevel

Line 601 return to toplevel

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{idiv irem}, @fref{%}.

@end table

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

\JP @node tdiv,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node tdiv,,, Polynomials and rational expressions

@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.

\JP :: @var{poly1} $B$,(B @var{poly2} $B$G3d$j@Z$l$k$+$I$&$+D4$Y$k(B.

\EG :: Tests whether @var{poly2} divides @var{poly1}.

@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

\JP $B3d$j@Z$l$k$J$i$P>&(B, $B3d$j@Z$l$J$1$l$P(B 0

\EG Quotient if @var{poly2} divides @var{poly1}, 0 otherwise.

@item poly1 poly2

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@end table

@itemize @bullet

\BJP

@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.

\BEG

@item

Tests whether @var{poly2} divides @var{poly1} in polynomial ring.

@item

One application of this function: Consider the case where a polynomial

is certainly an irreducible factor of the other polynomial, but

the multiplicity of the factor is unknown. Application of @code{tdiv()}

to the polynomials repeatedly yields the multiplicity.

@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

x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6

+(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

+(-6*y^3-18*z*y^2-18*z^2*y-6*z^3)*x^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

+(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

-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

+(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

-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

+(-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;

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{sdiv sdivm srem sremm sqr sqrm}.

@end table

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

\JP @node %,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node %,,, Polynomials and rational expressions

@subsection @code{%}

@findex %

@table @t

@item @var{poly} % @var{m}

:: $B@0?t$K$h$k>jM>(B

\JP :: $B@0?t$K$h$k>jM>(B

\EG :: integer remainder to all integer coefficients of the polynomial.

@end table

@table @var

@item return

$B@0?t$^$?$OB?9`<0(B

\JP $B@0?t$^$?$OB?9`<0(B

\EG integer or polynomial

@item poly

$B@0?t$^$?$O@0?t78?tB?9`<0(B

\JP $B@0?t$^$?$O@0?t78?tB?9`<0(B

\EG integer or polynomial with integer coefficients

@item m

$B@0?t(B

\JP $B@0?t(B

\EG intger

@end table

@itemize @bullet

\BJP

@item

@var{poly} $B$N3F78?t$r(B @var{m} $B$G3d$C$?>jM>$GCV$-49$($?B?9`<0$rJV$9(B.

@item

Line 464 x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6+(-6*y^3-18*

Line 699 x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6+(-6*y^3-18*

@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.

\BEG

@item

Returns a polynomial whose coefficients are remainders of the

coefficients of the input polynomial divided by @var{m}.

@item

The resulting coefficients are all normalized to non-negative integers.

@item

An integer is allowed for @var{poly}. This can be used for an

alternative for @code{irem()} except that the result is normalized to

a non-negative integer.

@item

Coefficients of @var{poly} and @var{m} must all be integers, though the

type checking is not done.

@end itemize

@example

Line 478 x^5+2*x^4+x^3+x^2+2*x+1

Line 728 x^5+2*x^4+x^3+x^2+2*x+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{idiv irem}.

@end table

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

\JP @node subst psubst,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node subst psubst,,, Polynomials and rational expressions

@subsection @code{subst}, @code{psubst}

@findex subst

@findex psubst

Line 490 x^5+2*x^4+x^3+x^2+2*x+1

Line 742 x^5+2*x^4+x^3+x^2+2*x+1

@table @t

@item subst(@var{rat}[,@var{varn},@var{ratn}]*)

@item psubst(@var{rat}[,@var{var},@var{rat}]*)

\BJP

:: @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).

(@var{n}=1,2,... $B$G:8$+$i1&$K=g<!BeF~$9$k(B).

\BEG

:: Substitute @var{ratn} for @var{varn} in expression @var{rat}.

(@var{n}=1,2,@dots{}.

Substitution will be done successively from left to right

if arguments are repeated.)

@end table

@table @var

@item return

$BM-M}<0(B

\JP $BM-M}<0(B

@item rat,ratn

\EG rational expression

$BM-M}<0(B

@item rat ratn

\JP $BM-M}<0(B

\EG rational expression

@item varn

$BITDj85(B

\JP $BITDj85(B

\EG indeterminate

@end table

@itemize @bullet

\BJP

@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

Line 523 x^5+2*x^4+x^3+x^2+2*x+1

Line 787 x^5+2*x^4+x^3+x^2+2*x+1

$B$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.

@item

@code{subst}$B$N0z?t(B@var{rat}$B$,%j%9%H(B,$BG[Ns(B,$B9TNs(B,$B$"$k$$$OJ,;6I=8=B?9`<0$G(B

$B$"$C$?>l9g$K$O(B, $B$=$l$>$l$NMWAG$^$?$O78?t$KBP$7$F:F5"E*$K(B@code{subst}$B$r(B

$B9T$&(B.

\BEG

@item

Substitutes rational expressions for specified kernels in a rational

expression.

@item

@t{subst}(@var{r},@var{v1},@var{r1},@var{v2},@var{r2},@dots{})

has the same effect as

@t{subst}(@t{subst}(@var{r},@var{v1},@var{r1}),@var{v2},@var{r2},@dots{}).

@item

Note that repeated substitution is done from left to right successively.

You may get different result by changing the specification order.

@item

Ordinary @code{subst()} performs

substitution at all levels of a scalar algebraic expression creeping

into arguments of function forms recursively.

Function @code{psubst()} regards such a function form as an independent

indeterminate, and does not attempt to apply substitution to its

arguments. (The name comes after Partial SUBSTitution.)

@item

Since @b{Asir} does not reduce common divisors of a rational expression

automatically, substitution of a rational expression to an expression

may cause unexpected increase of computation time.

Thus, it is often necessary to write a special function to meet the

individual problem so that the denominator and the numerator do not

become too large.

@item

The same applies to substitution by rational numbers.

@end itemize

@example

Line 544 sint(t)*t

Line 841 sint(t)*t

sin(x)*t

@end example

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

\JP @node diff,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node diff,,, Polynomials and rational expressions

@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.

\JP :: @var{rat} $B$r(B @var{varn} $B$"$k$$$O(B @var{varlist} $B$NCf$NJQ?t$G=g<!HyJ,$9$k(B.

\BEG

:: Differentiate @var{rat} successively by @var{var}'s for the first

form, or by variables in @var{varlist} for the second form.

@end table

@table @var

@item return

$B<0(B

\JP $B<0(B

\EG expression

@item rat

$BM-M}<0(B ($B=iEyH!?t$r4^$s$G$b$h$$(B)

\JP $BM-M}<0(B ($B=iEyH!?t$r4^$s$G$b$h$$(B)

\EG rational expression which contains elementary functions.

@item varn

$BITDj85(B

\JP $BITDj85(B

\EG indeterminate

@item varlist

$BITDj85$N%j%9%H(B

\JP $BITDj85$N%j%9%H(B

\EG list of indeterminates

@end table

@itemize @bullet

\BJP

@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.

\BEG

@item

Differentiate @var{rat} successively by @var{var}'s for the first

form, or by variables in @var{varlist} for the second form.

@item

differentiation is performed by the specified indeterminates (variables)

from left to right.

@t{diff}(@var{rat},@t{x,y}) is the same as

@t{diff}(@t{diff}(@var{rat},@t{x}),@t{y}).

@end itemize

@example

Line 585 sin(x)*t

Line 903 sin(x)*t

sin(x)

@end example

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

\JP @node res,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node res,,, Polynomials and rational expressions

@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.

\JP :: @var{var} $B$K4X$9$k(B @var{poly1} $B$H(B @var{poly2} $B$N=*7k<0(B.

\EG :: Resultant of @var{poly1} and @var{poly2} with respect to @var{var}.

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item var

$BITDj85(B

\JP $BITDj85(B

@item poly1,poly2

\EG indeterminate

$BB?9`<0(B

@item poly1 poly2

\JP $BB?9`<0(B

\EG polynomial

@item mod

$BAG?t(B

\JP $BAG?t(B

\EG prime

@end table

@itemize @bullet

\BJP

@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.

Line 613 sin(x)

Line 938 sin(x)

$BItJ,=*7k<0%"%k%4%j%:%`$K$h$k(B.

@item

$B0z?t(B @var{mod} $B$,$"$k;~(B, GF(@var{mod}) $B>e$G$N7W;;$r9T$&(B.

\BEG

@item

Resultant of two polynomials @var{poly1} and @var{poly2}

with respect to @var{var}.

@item

Sub-resultant algorithm is used to compute the resultant.

@item

The computation is done over GF(@var{mod}) if @var{mod} is specified.

@end itemize

@example

Line 620 sin(x)

Line 955 sin(x)

-x^3-x^2-y^3

@end example

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

\JP @node fctr sqfr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node fctr sqfr,,, Polynomials and rational expressions

@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.

\JP :: @var{poly} $B$r4{Ls0x;R$KJ,2r$9$k(B.

\EG :: Factorize polynomial @var{poly} over the rationals.

@item sqfr(@var{poly})

:: @var{poly} $B$rL5J?J}J,2r$9$k(B.

\JP :: @var{poly} $B$rL5J?J}J,2r$9$k(B.

\EG :: Gets a square-free factorization of polynomial @var{poly}.

@end table

@table @var

@item return

$B%j%9%H(B

\JP $B%j%9%H(B

\EG list

@item poly

$BM-M}?t78?t$NB?9`<0(B

\JP $BM-M}?t78?t$NB?9`<0(B

\EG polynomial with rational coefficients

@end table

@itemize @bullet

\BJP

@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.

Line 650 sin(x)

Line 991 sin(x)

@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)

\BEG

@item

Factorizes polynomial @var{poly} over the rationals.

@code{fctr()} for irreducible factorization;

@code{sqfr()} for square-free factorization.

@item

The result is represented by a list, whose elements are a pair

represented as

[[@b{num},1],[@b{factor},@b{multiplicity}],...].

@item

Products of all @b{factor}^@b{multiplicity} and @b{num} is equal to

@var{poly}.

@item

The number @b{num} is determined so that (@var{poly}/@b{num}) is an

integral polynomial and its content (GCD of all coefficients) is 1.

(@xref{ptozp}.)

@end itemize

@example

Line 670 x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4

Line 1030 x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{ufctrhint}.

@end table

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

\JP @node ufctrhint,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node ufctrhint,,, Polynomials and rational expressions

@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

\JP :: $B<!?t>pJs$rMQ$$$?(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B

\BEG

:: Factorizes uni-variate polynomial @var{poly} over the rational number

field when the degrees of its factors are known to be some integer

multiples of @var{hint}.

@end table

@table @var

@item return

$B%j%9%H(B

\JP $B%j%9%H(B

\EG list

@item poly

$BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B

\JP $BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B

\EG uni-variate polynomial with rational coefficients

@item hint

$B<+A3?t(B

\JP $B<+A3?t(B

\EG non-negative integer

@end table

@itemize @bullet

\BJP

@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,

$B$"$kB?9`<0$N%N%k%`(B (@ref{$BBe?tE*?t$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.

\BEG

@item

By any reason, if the degree of all the irreducible factors of @var{poly}

is known to be some multiples of @var{hint}, factors can be computed

more efficiently by the knowledge than @code{fctr()}.

@item

When @var{hint} is 1, @code{ufctrhint()} is the same as @code{fctr()} for

uni-variate polynomials.

An typical application where @code{ufctrhint()} is effective:

Consider the case where @var{poly} is a norm (@ref{Algebraic numbers})

of a certain polynomial over an extension field with its extension

degree @var{d}, and it is square free; Then, every irreducible factor

has a degree that is a multiple of @var{d}.

@end itemize

@example

Line 707 x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4

Line 1093 x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4

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

-x^81+1215*x^78-567405*x^75+139519665*x^72-19360343142*x^69

-88249977024390*x^63-4856095669551930*x^60+1999385245240571421*x^57

+1720634125410*x^66-88249977024390*x^63-4856095669551930*x^60

-15579689952590251515*x^54+15956967531741971462865*x^51

+1999385245240571421*x^57-15579689952590251515*x^54

+15956967531741971462865*x^51

...

+140395588720353973535526123612661444550659875*x^6

+10122324287343155430042768923500799484375*x^3

Line 727 t^9-15*t^6-87*t^3-125

Line 1114 t^9-15*t^6-87*t^3-125

[[-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^18-324*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^18-324*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

\JP @item $B;2>H(B

\EG @item References

@fref{fctr sqfr}.

@end table

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

\JP @node modfctr,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node modfctr,,, Polynomials and rational expressions

@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

\JP :: $BM-8BBN>e$G$NB?9`<0$N0x?tJ,2r(B

\EG :: Factorizer over small finite fields

@end table

@table @var

@item return

$B%j%9%H(B

\JP $B%j%9%H(B

\EG list

@item poly

$B@0?t78?t$N(B 1 $BJQ?tB?9`<0(B

\JP $B@0?t78?t$NB?9`<0(B

\EG Polynomial with integer coefficients

@item mod

$B<+A3?t(B

\JP $B<+A3?t(B

\EG non-negative integer

@end table

@itemize @bullet

\BJP

@item

2^31 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$G0lJQ?tB?9`<0(B

2^29 $BL$K~$N<+A3?t(B @var{mod} $B$rI8?t$H$9$kAGBN>e$GB?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.

@item

$BBg$-$J0L?t$r;}$DM-8BBN>e$N0x?tJ,2r$K$O(B @code{fctr_ff} $B$rMQ$$$k(B.

(@ref{$BM-8BBN$K4X$9$k1i;;(B},@pxref{fctr_ff}).

\BEG

@item

This function factorizes a polynomial @var{poly} over

the finite prime field of characteristic @var{mod}, where

@var{mod} must be smaller than 2^29.

@item

The result is represented by a list, whose elements are a pair

represented as

[[@b{num},1],[@b{factor},@b{multiplicity}],...].

@item

Products of all @b{factor}^@b{multiplicity} and @b{num} is equal to

@var{poly}.

@item

To factorize polynomials over large finite fields, use

@code{fctr_ff} (@pxref{Finite fields},@ref{fctr_ff}).

@end itemize

@example

Line 777 t^9-15*t^6-87*t^3-125

Line 1194 t^9-15*t^6-87*t^3-125

[[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]]

[1] modfctr(2*x^6+(y^2+z*y)*x^4+2*z*y^3*x^2+(2*z^2*y^2+z^3*y)*x+z^4,3);

[[2,1],[2*x^3+z*y*x+z^2,1],[2*x^3+y^2*x+2*z^2,1]]

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{fctr sqfr}.

@end table

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

\JP @node ptozp,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node ptozp,,, Polynomials and rational expressions

@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.

\JP :: @var{poly} $B$rM-M}?tG\$7$F@0?t78?tB?9`<0$K$9$k(B.

\BEG

:: Converts a polynomial @var{poly} with rational coefficients into

an integral polynomial such that GCD of all its coefficients is 1.

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@end table

@itemize @bullet

\BJP

@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.

Line 812 t^9-15*t^6-87*t^3-125

Line 1240 t^9-15*t^6-87*t^3-125

$BJ,;RB?9`<0$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.

\BEG

@item

Converts the given polynomial by multiplying some rational number

into an integral polynomial such that GCD of all its coefficients is 1.

@item

In general, operations on polynomials can be

performed faster for integer coefficients than for rational number

coefficients. Therefore, this function is conveniently used to improve

efficiency.

@item

Function @code{red} does not convert rational coefficients of the

numerator.

You cannot obtain an integral polynomial by direct use of the function

@code{nm()}. The function @code{nm()} returns the numerator of its

argument, and a polynomial with rational coefficients is

the numerator of itself and will be returned as it is.

@end itemize

@example

Line 822 t^9-15*t^6-87*t^3-125

Line 1268 t^9-15*t^6-87*t^3-125

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{nm dn}.

@end table

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

\JP @node prim cont,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node prim cont,,, Polynomials and rational expressions

@subsection @code{prim}, @code{cont}

@findex prim

@table @t

@item prim(@var{poly}[,@var{v}])

:: @var{poly} $B$N86;OE*ItJ,(B (primitive part).

\JP :: @var{poly} $B$N86;OE*ItJ,(B (primitive part).

\EG :: Primitive part of @var{poly}.

@item cont(@var{poly}[,@var{v}])

:: @var{poly} $B$NMFNL(B (content).

\JP :: @var{poly} $B$NMFNL(B (content).

\EG :: Content of @var{poly}.

@end table

@table @var

@item return poly

$BM-M}?t78?tB?9`<0(B

\JP $BM-M}?t78?tB?9`<0(B

\EG polynomial over the rationals

@item v

$BITDj85(B

\JP $BITDj85(B

\EG indeterminate

@end table

@itemize @bullet

\BJP

@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.

\BEG

@item

The primitive part and the content of a polynomial @var{poly}

with respect to its main variable (@var{v} if specified).

@end itemize

@example

Line 862 y-z

Line 1321 y-z

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{var}, @fref{ord}.

@end table

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

\JP @node gcd gcdz,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node gcd gcdz,,, Polynomials and rational expressions

@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.

\JP :: @var{poly1} $B$H(B @var{poly2} $B$N(B gcd.

\EG :: The polynomial greatest common divisor of @var{poly1} and @var{poly2}.

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

@item poly1,poly2

\EG polynomial

$BB?9`<0(B

@item poly1 poly2

\JP $BB?9`<0(B

\EG polynomial

@item mod

$BAG?t(B

\JP $BAG?t(B

\EG prime

@end table

@itemize @bullet

\BJP

@item

$BFs$D$NB?9`<0$N:GBg8xLs<0(B (GCD) $B$r5a$a$k(B.

@item

Line 902 y-z

Line 1368 y-z

@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.

\BEG

@item

Functions @code{gcd()} and @code{gcdz()} return the greatest common divisor

(GCD) of the given two polynomials.

@item

Function @code{gcd()} returns an integral polynomial GCD over the

rational number field. The coefficients are normalized such that

their GCD is 1. It returns 1 in case that the given polynomials are

mutually prime.

@item

Function @code{gcdz()} works for arguments of integral polynomials,

and returns a polynomial GCD over the integer ring, that is,

it returns @code{gcd()} multiplied by the contents of all coefficients

of the two input polynomials.

@item

@code{gcd()} computes the GCD over GF(@var{mod}) if @var{mod} is specified.

@item

Polynomial GCD is computed by an improved algorithm based

on Extended Zassenhaus algorithm.

@item

GCD over a finite field is computed by PRS algorithm and it may not be

efficient for large inputs and co-prime inputs.

@end itemize

@example

Line 916 x^3+y*x^2+y^2*x+y^3

Line 1406 x^3+y*x^2+y^2*x+y^3

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{igcd igcdcntl}.

@end table

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

\JP @node red,,, $BB?9`<0$*$h$SM-M}<0$N1i;;(B

\EG @node red,,, Polynomials and rational expressions

@subsection @code{red}

@findex red

@table @t

@item red(@var{rat})

:: @var{rat} $B$rLsJ,$7$?$b$N(B.

\JP :: @var{rat} $B$rLsJ,$7$?$b$N(B.

\EG :: Reduced form of @var{rat} by canceling common divisors.

@end table

@table @var

@item return

$BM-M}<0(B

\JP $BM-M}<0(B

\EG rational expression

@item rat

$BM-M}<0(B

\JP $BM-M}<0(B

\EG rational expression

@end table

@itemize @bullet

\BJP

@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,

Line 953 GCD $B$OBgJQ=E$$1i;;$J$N$G(B, $BB>$NJ}K!$G=|$1$k6&D

Line 1449 GCD $B$OBgJQ=E$$1i;;$J$N$G(B, $BB>$NJ}K!$G=|$1$k6&D

$BK>$^$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.

\BEG

@item

@b{Asir} automatically performs cancellation of common divisors of rational numb

ers.

But, without an explicit command, it does not cancel common polynomial divisors

of rational expressions.

(Reduction of rational expressions to a common denominator will be always done.)

Use command @t{red()} to perform this cancellation.

@item

Cancel the common divisors of the numerator and the denominator of

a rational expression @var{rat} by computing their GCD.

@item

The denominator polynomial of the result is an integral polynomial

which has no common divisors in its coefficients,

while the numerator may have rational coefficients.

@item

Since GCD computation is a very hard operation, it is desirable to

detect and remove by any means common divisors as far as possible.

Furthermore, a call to this function after swelling of the denominator

and the numerator shall usually take a very long time. Therefore,

often, to some extent, reduction of common divisors is inevitable for

operations of rational expressions.

@end itemize

@example

Line 969 x^2+(-y-z)*x+y^2-z*y+z^2

Line 1489 x^2+(-y-z)*x+y^2-z*y+z^2

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item References

@fref{nm dn}, @fref{gcd gcdz}, @fref{ptozp}.

@end table

FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>