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version 1.1, 2001/01/04 12:29:32

version 1.5, 2001/05/27 12:48:02

Line 1

%% $OpenXM$

%% $OpenXM: OpenXM/src/k097/Doc/complex.texi,v 1.4 2001/01/13 01:17:36 takayama Exp $

/*&C

\input texinfo

@iftex

@catcode`@#=6

@def@b#1{{@bf@gt #1}}

@catcode`@#=@other

@end iftex

@overfullrule=0pt

@c -*-texinfo-*-

@comment %**start of header

@setfilename complex

@settitle complex

@comment %**end of header

@comment %@setchapternewpage odd

@iftex

@comment @finalout

@end iftex

@titlepage

//&C @title Kan/k0 complex

//&ja @subtitle Kan/k0 complex $B%Q%C%1!<%8(B User's Manual ($BF|K\8lHG(B)

//&en @subtitle Kan/k0 complex Package User's Manual

/*&C

@subtitle Edition 1.1.3 for OpenXM/kan/k0

@subtitle December 31, 2000

@author by Nobuki Takayama

@end titlepage

@synindex vr fn

@comment node-name, next, previous, up

@node Top,, (dir), (dir)

/*&ja

@menu

* COMPLEX $BH!?t(B::

* $B4pK\(B $BH!?t(B::

* $B:w0z(B::

@end menu

/*&en

@menu

* COMPLEX function::

* Primitive function::

* index::

@end menu

/*&ja

@node COMPLEX $BH!?t(B,,, Top

@chapter COMPLEX $BH!?t(B

@section $BH!?t0lMw(B

Line 92

Line 35

@findex Res_solv

@table @t

@item Res_solv(@var{m},@var{d})

:: $B0l<!ITDjJ}Dx<0(B u @var{m} =@var{d} $B$N2r$r$b$H$a$k(B.

@item Res_solv(@var{m},@var{d},@var{r})

:: $B0l<!ITDjJ}Dx<0(B u @var{m} =@var{d} $B$N2r$r$b$H$a$k(B. @var{r} $B$O(B ring.

@end table

Line 218 In(5)= Res_solv([[1,2],[3,4]],[5,0],r):

Line 161 In(5)= Res_solv([[1,2],[3,4]],[5,0],r):

@table @t

@item $B;2>H(B

@code{Res_solv_h}, @code{Kernel}, @code{GetRing}, @code{SetRing}.

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Res_solv %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&en

@table @t

@item Reference

@item See also

@code{Res_solv_h}, @code{Kernel}, @code{GetRing}, @code{SetRing}.

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Res_solv %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Line 240 In(5)= Res_solv([[1,2],[3,4]],[5,0],r):

Line 187 In(5)= Res_solv([[1,2],[3,4]],[5,0],r):

@findex Res_solv2

@table @t

@item Res_solv2(@var{m},@var{v},@var{j})

:: $B0l<!ITDjJ}Dx<0(B u @var{m} =@var{v} mod @var{j} $B$N2r$r$b$H$a$k(B.

@item Res_solv2(@var{m},@var{v},@var{j},@var{r})

:: $B0l<!ITDjJ}Dx<0(B u @var{m} =@var{v} mod @var{j} $B$N2r$r$b$H$a$k(B.

@var{r} $B$O(B ring.

Line 258 In(5)= Res_solv([[1,2],[3,4]],[5,0],r):

Line 205 In(5)= Res_solv([[1,2],[3,4]],[5,0],r):

@table @t

@item Res_solv2(@var{m},@var{v},@var{j})

:: Find a solution u of the linear indefinite equation u @var{m} =@var{v}

mod @var{j}.

@item Res_solv2(@var{m},@var{v},@var{j},@var{r})

:: Find a solution u of the linear indefinite equation u @var{m} =@var{v}

mod @var{j}.

Line 363 The output implies that

Line 310 The output implies that

@table @t

@item $B;2>H(B

@code{Res_solv2_h}, @code{Kernel2}, @code{GetRing}, @code{SetRing}.

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Res_solv2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&en

@table @t

@item Reference

@item See also

@code{Res_solv2_h}, @code{Kernel2}, @code{GetRing}, @code{SetRing}.

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Res_solv2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Line 502 In(5)= k=Kernel([[1,2],[2,4]],r); k[0]:

Line 453 In(5)= k=Kernel([[1,2],[2,4]],r); k[0]:

@table @t

@item $B;2>H(B

@code{Kernel_h}, @code{Res_solv}, @code{GetRing}, @code{SetRing}.

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Kernel %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&en

@table @t

@item Reference

@item See also

@code{Kernel_h}, @code{Res_solv}, @code{GetRing}, @code{SetRing}.

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Kernel %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Line 614 In(43)=Kernel2([[0,0,0],[0,0,0]],[],r):

Line 569 In(43)=Kernel2([[0,0,0],[0,0,0]],[],r):

@item $B;2>H(B

@code{Kernel2_h}, @code{Res_solv2}, @code{GetRing}, @code{SetRing},

@code{Kernel}.

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Kernel2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&en

@table @t

@item Reference

@item See also

@code{Kernel2_h}, @code{Res_solv2}, @code{GetRing}, @code{SetRing},

@code{Kernel}

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Kernel2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Line 743 In(13)= Gb(m,r):

Line 702 In(13)= Gb(m,r):

@item $B;2>H(B

@code{Gb_h}, @code{Kernel}, @code{Res_solv}, @code{RingD},

@code{GetRing}, @code{SetRing}.

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Gb %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&en

@table @t

@item Reference

@item See also

@code{Gb_h}, @code{Kernel}, @code{Res_solv}, @code{RingD},

@code{GetRing}, @code{SetRing}.

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Gb %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Line 865 In(19)=Pmat(Substitute(k2[0],t,1));

Line 828 In(19)=Pmat(Substitute(k2[0],t,1));

@table @t

@item $B;2>H(B

@code{Gb}, (m,(u,v))-$B%0%l%V%J4pDl(B

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Res_shiftMatrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&en

@table @t

@item Reference

@item See also

@code{Gb}, (m,(u,v))-Grobner basis

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Res_shiftMatrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Line 894 In(19)=Pmat(Substitute(k2[0],t,1));

Line 861 In(19)=Pmat(Substitute(k2[0],t,1));

/*&C

@menu

* changeRing::

* ChangeRing::

* GetRing::

* SetRing::

* Intersection::

* Getxvars::

* Firstn::

Line 904 In(19)=Pmat(Substitute(k2[0],t,1));

Line 869 In(19)=Pmat(Substitute(k2[0],t,1));

/*&ja

@c %%%%%%%%%%%%%%%%%%%% start changeRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@c %%%%%%%%%%%%%%%%%%%% start ChangeRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@node changeRing,,, $B4pK\(B $BH!?t(B

@node ChangeRing,,, $B4pK\(B $BH!?t(B

@subsection @code{changeRing}

@subsection @code{ChangeRing}

@findex changeRing

@findex ChangeRing

@table @t

@item changeRing(@var{f})

@item ChangeRing(@var{f})

:: changeRing $B$O(B @var{f} $B$NMWAG$,B0$9$k4D$r(B current ring $B$K$9$k(B.

:: ChangeRing $B$O(B @var{f} $B$NMWAG$,B0$9$k4D$r(B current ring $B$K$9$k(B.

@end table

@table @var

Line 923 true $B$+(B false

Line 888 true $B$+(B false

RingD("x,y");

f=[x+y,0];

RingD("p,q,r");

changeRing(f);

ChangeRing(f);

@end example

@c %%%%%%%%%%%%%%%%%%%% end of changeRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&ja

@c %%%%%%%%%%%%%%%%%%%% start of GetRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@menu

* GetRing::

@end menu

@node GetRing,,, $B4pK\(B $BH!?t(B

@subsection @code{GetRing}

@findex GetRing

@table @t

@item GetRing(@var{f})

@item Files

:: @var{f} $B$NMWAG$NB0$9$k4D$rLa$9(B.

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of ChangeRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@table @var

@item return

$B4D(B

@item f

$B%*%V%8%'%/%H(B

@end table

@itemize @bullet

@item @var{f} $B$NMWAG$^$?$O(B @var{f} $B<+?H$NB0$9$k4D$rLa$9(B.

$B4D$,$J$$$H$-$O(B null $B$rLa$9(B.

@end itemize

@example

In(16)= RingD("x,y");

In(17)= r = GetRing([x^2,0,1]);

@end example

@table @t

@item $B;2>H(B

@code{SetRing}, @code{ChangeRing}.

@end table

@c %%%%%%%%%%%%%%%%%%%% end of GetRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&ja

@c %%%%%%%%%%%%%%%%%%%% start of SetRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@menu

* SetRing::

@end menu

@node SetRing,,, $B4pK\(B $BH!?t(B

@subsection @code{SetRing}

@findex SetRing

@table @t

@item SetRing(@var{r})

:: Current ring $B$r(B @var{r} $B$KJQ99$9$k(B.

@end table

@table @var

@item return

$B$J$7(B

@item r

$B4D(B

@end table

@itemize @bullet

@item Current ring $B$r(B @var{r} $B$KJQ99$9$k(B.

@end itemize

@example

In(16)= RingD("x,y");

In(17)= f=[x^2,0,1]; r = GetRing(f);

In(19)= RingD("p,q");

In(20)= SetRing(r);

In(21)= f = DC(f,"polynomial");

@end example

@table @t

@item $B;2>H(B

@code{GetRing}, @code{ChangeRing}, @code{DC}.

@end table

@c %%%%%%%%%%%%%%%%%%%% end of SetRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&ja

@c %%%%%%%%%%%%%%%%%%%% start of Intersection %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@menu

Line 1043 In(33)=Intersection(mm,nn):

Line 938 In(33)=Intersection(mm,nn):

@table @t

@item $B;2>H(B

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Intersection %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Line 1108 In(5)=Getxvars():

Line 1004 In(5)=Getxvars():

@table @t

@item $B;2>H(B

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Getxvars %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&en

@table @t

@item Reference

@item See also

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Getxvars %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Line 1197 In(17)= Firstn(mm,2):

Line 1095 In(17)= Firstn(mm,2):

@table @t

@item $B;2>H(B

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Firstn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&en

@table @t

@item Reference

@item See also

@item Files

@code{lib/restriction/complex.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Firstn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&ja

@node $B:w0z(B,,, Top

@c %%%%%%%%%%%%%%%%%%%% start of GKZ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@unnumbered $B:w0z(B

@menu

* GKZ::

@end menu

@node GKZ,,, $B4pK\(B $BH!?t(B

@subsection @code{GKZ}

@findex GKZ

@table @t

@item GKZ(@var{a},@var{b})

:: $B9TNs(B @var{a} $B$H(B $B%Y%/%H%k(B @var{b} $B$GDj$^$k(B, GKZ hypergeometric system $B$rLa$9(B.

@end table

@table @var

@item return

D $B$N%$%G%"%k$N@8@.85(B

@item a

$B@0?t9TNs(B

@item b

$B@0?t%Y%/%H%k(B

@end table

@itemize @bullet

@item $B9TNs(B @var{a} $B$H(B $B%Y%/%H%k(B @var{b} $B$GDj$^$k(B, GKZ hypergeometric system $B$rLa$9(B.

@item Gelfand, Kapranov, Zelevinski $B$ND64v2?HyJ,J}Dx<0$rLa$9(B.

@end itemize

@example

In(3)= GKZ([[1,1,1,1],[0,1,2,3]],[-1,-2]):

[ x1*Dx1+x2*Dx2+x3*Dx3+x4*Dx4+1 , x2*Dx2+2*x3*Dx3+3*x4*Dx4+2 , Dx2^2-Dx1*Dx3 , -Dx3^2+Dx2*Dx4 , Dx2*Dx3-Dx1*Dx4 ]

@end example

@table @t

@item $B;2>H(B

@item Files

@code{lib/restriction/demo2.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of GKZ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&en

@node index,,, Top

/*&ja

@unnumbered index

@c %%%%%%%%%%%%%%%%%%%% start of Slope %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@menu

* Slope::

@end menu

@node Slope,,, $B4pK\(B $BH!?t(B

@subsection @code{Slope}

@findex Slope

@table @t

@item Slope(@var{i},@var{v},@var{f},@var{v})

:: $BHyJ,J}Dx<07O(B @var{i} $B$N(B slope $B$N=89g$r$b$H$a$k(B.

@end table

@table @var

@item return

Slope $B$N=89g$H$=$l$rM?$($k(B, weight vectors.

@item i

D $B$N%$%G%"%k$N@8@.85(B

@item v

$B6u4VJQ?t(B

@item f

F-filtration $B$rM?$($k(B weight vector (0,...,0,1,...,1).

@item v

V-filtration $B$rM?$($k(B weight vector.

$B$?$H$($P(B (0,...,0,-1,0,...,0,1).

@end table

@itemize @bullet

@item @var{i} $B$N(B x_i = 0 $B$K$=$C$?(B, $B86E@$G$N(B slope $B$NA4BN$r$b$H$a$k(B.

@item $B7W;;$K$O(B Assi, Castro, Granger $B$N%"%k%4%j%:%`$rMQ$$$k(B.

@item Geometric slope $B$r7W;;$9$k$?$a(B, radical $B$N7W;;$r$*$3$J$C$F$$$k(B.

$B$3$l$K$O(B ox_asir $B$rMQ$$$F$$$k(B.

@end itemize

@example

In(13)= a=GKZ([[1,3,7]],[-30]);

In(14)= a:

[ x1*Dx1+3*x2*Dx2+7*x3*Dx3+30 , -Dx1^3+Dx2 , -Dx1*Dx2^2+Dx3 , -Dx2^3+Dx1^2*Dx3 ]

In(15)= Slope(a,[x1,x2,x3],[0,0,0,1,1,1],[0,0,-1,0,0,1]):

[ [ (3)/(4) , [ 0 , 0 , -4 , 3 , 3 , 7 ] ] ]

@end example

@table @t

@item $B;2>H(B

@item Files

@code{lib/restriction/demo2.k}

@end table

@c %%%%%%%%%%%%%%%%%%%% end of Slope %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/*&C

@printindex fn

@printindex cp

@iftex

@vfill @eject

@end iftex

@summarycontents

@contents

@bye

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