%% $OpenXM: OpenXM/src/k097/Doc/complex.texi,v 1.5 2001/05/27 12:48:02 takayama Exp $

/*&ja

@node COMPLEX 函数,,, Top
@chapter COMPLEX 函数
@section 函数一覧
*/
/*&en

@node COMPLEX function,,, Top
@chapter COMPLEX function
@section A list of functions
*/
/*&C
@menu
* Res_solv::
* Res_solv2::
* Kernel::
* Kernel2::
* Gb::
* Gb_h::
* Res_shiftMatrix::
@end menu

*/

/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of Res_solv %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Res_solv::
@end menu
@node Res_solv,,, COMPLEX 函数
@subsection @code{Res_solv}
@findex Res_solv
@table @t
@item Res_solv(@var{m},@var{d})
::  一次不定方程式 u @var{m} =@var{d} の解をもとめる. 
@item Res_solv(@var{m},@var{d},@var{r})
::  一次不定方程式 u @var{m} =@var{d} の解をもとめる. @var{r} は ring.
@end table

*/
/*&en
@c %%%%%%%%%%%%%%%%%%%%  start of Res_solv %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Res_solv::
@end menu
@node Res_solv,,, COMPLEX function
@subsection @code{Res_solv}
@findex Res_solv
@table @t
@item Res_solv(@var{m},@var{d})
::  Find a solution u of the linear indefinite equation u @var{m} =@var{d}.
@item Res_solv(@var{m},@var{d},@var{r})
::  Find a solution u of the linear indefinite equation u @var{m} =@var{d}.
@var{r} is a ring object.
@end table

*/

/*&ja
@table @var
@item return
[c,r] が戻り値のとき c/r が解 u (横ベクトル).
@item m
行列またはベクトル
@item d
ベクトル またはスカラー
@end table

@itemize @bullet
@item  一次不定方程式 u @var{m} =@var{d} の解をもとめる.
@item  @var{m}, @var{d} の順に環の定義を検索し, その環と同じ変数集合を
もつ微分作用素環(graded reverse lexicographic order)で不定方程式を解く.
環 @var{r} があたえられているときは, @var{r} と同じ変数集合をもつ
微分作用素環(graded reverse lexicographic order)で不定方程式を解く.
@item @var{m}, @var{d} が定数成分のときは, 環 @var{r} を与える必要がある.
(@var{m}, @var{d} より環の情報をとりだせないため).
@end itemize

*/
/*&en
@table @var
@item return
When [c,r] is the return value,  c/r is the solution u.
@item m
Matrix or vector
@item d
Vector or scalar
@end table

@itemize @bullet
@item  Find a solution u of the linear indefinite equation u @var{m} =@var{d}.
@item It solves the linear indefinite equation in the ring of differential
operators (with graded reverse lexicographic order) of the same set 
of variables of the ring to which @var{m} or @var{d} belongs.
When the ring @var{r} is given, 
it solves the linear indefinite equation in the ring of differential
operators (with graded reverse lexicographic order) of the same set 
of variables of the ring @var{r}.
@item When @var{m} and @var{d} consist of constants, a ring @var{r}
should be given.
@end itemize

*/

/*&C
@example
In(16)= RingD("x,y");
In(17)= mm=[Dx,Dy,x];
In(18)= Res_solv(mm,1):
[    [    x , 0 , -Dx ]  , -1 ] 
@end example
*/
/*&ja
これは -x*Dx + 0*Dy+Dx*x = 1 であることを示す.
*/
/*&en
The output means that  -x*Dx + 0*Dy+Dx*x = 1.
*/

/*&C
@example
In(4)=RingD("x");
     m=[ [x*Dx+2, 0],[Dx+3,x^3],[3,x],[Dx*(x*Dx+3)-(x*Dx+2)*(x*Dx-4),0]];
     d=[1,0];
     Res_solv(m,d):

[    [    x^2*Dx-x*Dx-4*x-1 , 0 , 0 , x ]  , -2 ] 
@end example
*/
/*&ja
これは
-(1/2)*(x^2*Dx-x*Dx-4*x-1)*[x*Dx+2, 0]-(1/2)*[Dx*(x*Dx+3)-(x*Dx+2)*(x*Dx-4),0]
= [1,0]
であることを示す.
*/
/*&en
The output implies that
-(1/2)*(x^2*Dx-x*Dx-4*x-1)*[x*Dx+2, 0]-(1/2)*[Dx*(x*Dx+3)-(x*Dx+2)*(x*Dx-4),0]
= [1,0]
*/

/*&C

@example

In(4)= r=RingD("x,y");
In(5)= Res_solv([[1,2],[3,4]],[5,0],r):
[    [    10 , -5 ]  , -1 ] 

@end example


*/


/*&ja

@table @t
@item 参照
    @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 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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/

/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of Res_solv2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Res_solv2::
@end menu
@node Res_solv2,,, COMPLEX 函数
@subsection @code{Res_solv2}
@findex Res_solv2
@table @t
@item Res_solv2(@var{m},@var{v},@var{j})
::  一次不定方程式 u @var{m} =@var{v} mod @var{j} の解をもとめる. 
@item Res_solv2(@var{m},@var{v},@var{j},@var{r})
::  一次不定方程式 u @var{m} =@var{v} mod @var{j} の解をもとめる. 
@var{r} は ring.
@end table

*/
/*&en
@c %%%%%%%%%%%%%%%%%%%%  start of Res_solv2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Res_solv2::
@end menu
@node Res_solv2,,, COMPLEX function
@subsection @code{Res_solv2}
@findex Res_solv2
@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}.
@var{r} is a ring object.
@end table

*/

/*&ja
@table @var
@item return
[c,r] が戻り値のとき c/r が解 u (横ベクトル).
@item m
行列またはベクトル
@item v  j
ベクトル またはスカラー
@end table

@itemize @bullet
@item  一次不定方程式 u @var{m} =@var{v} mod @var{j} の解をもとめる.
@item これは, @var{m} を
 @var{m} :  D^p ---> D^q/@var{j} 
なる左 D homomorphism とみなすとき,
@var{m}^(-1)(@var{v}) を求めることに相当する.
@item  @var{m}, @var{v} の順に環の定義を検索し, その環と同じ変数集合を
もつ微分作用素環(graded reverse lexicographic order)で不定方程式を解く.
環 @var{r} があたえられているときは, @var{r} と同じ変数集合をもつ
微分作用素環(graded reverse lexicographic order)で不定方程式を解く.
@item @var{m}, @var{v} が定数成分のときは, 環 @var{r} を与える必要がある.
(@var{m}, @var{v} より環の情報をとりだせないため).
@end itemize

*/
/*&en
@table @var
@item return
When [c,r] is the return value,  c/r is the solution u.
@item m
Matrix or vector
@item v j
Vector or scalar
@end table

@itemize @bullet
@item  Find a solution u of the linear indefinite equation u @var{m} =@var{v}
mod @var{j}.
@item Let  @var{m} be the left D-homomorphism
 @var{m} :  D^p ---> D^q/@var{j}.
The function returns an element in 
@var{m}^(-1)(@var{v}).
@item It solves the linear indefinite equation in the ring of differential
operators (with graded reverse lexicographic order) of the same set 
of variables of the ring to which @var{m} or @var{v} belongs.
When the ring @var{r} is given, 
it solves the linear indefinite equation in the ring of differential
operators (with graded reverse lexicographic order) of the same set 
of variables of the ring @var{r}.
@item When @var{m} and @var{v} consist of constants, a ring @var{r}
should be given.
@end itemize

*/

/*&C
@example
In(28)= r=RingD("x,y");
In(29)= Res_solv2([x,y],[x^2+y^2],[x]):
[    [    0 , y ]  , 1 ] 

@end example
*/
/*&ja
これは 0*x + y*y = x^2+y^2 mod x であることを示す.
*/
/*&en
The output means that  0*x + y*y = x^2+y^2 mod x
*/

/*&C
@example

In(32)= Res_solv2([x,y],[x^2+y^2],[],r):
[    [    x , y ]  , 1 ] 


@end example
*/
/*&ja
これは
  x*x + y*y = x^2+y^2
であることを示す.
*/
/*&en
The output implies that
  x*x + y*y = x^2+y^2.
*/



/*&ja

@table @t
@item 参照
    @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 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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/


/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of Kernel %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@c Kernel は minimal.k にあるが complex.k などに移動すべき.
@menu
* Kernel::
@end menu
@node Kernel,,, COMPLEX 函数
@subsection @code{Kernel}
@findex Kernel
@table @t
@item Kernel(@var{m})
::  一次不定方程式 u @var{m} =0 の解空間の基底を求める.
@item Kernel(@var{m},@var{r})
::  一次不定方程式 u @var{m} =0 の解空間の基底を求める. @var{r} は ring.
@end table

*/
/*&en
@c %%%%%%%%%%%%%%%%%%%%  start of Kernel %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Kernel::
@end menu
@node Kernel,,, COMPLEX function
@subsection @code{Kernel}
@findex Kernel
@table @t
@item Kernel(@var{m})
::  Find solution basis of the linear indefinite equation u @var{m} =0.
@item Kernel(@var{m},@var{r})
::  Find solution basis of the linear indefinite equation u @var{m} =0.
@var{r} is a ring object.
@end table

*/

/*&ja
@table @var
@item return
リスト
@item m
行列またはベクトル
@end table

@itemize @bullet
@item  一次不定方程式 u @var{m} =0 の解空間の基底を求める.
@item 戻り値を k とするとき k[0] が 解空間の基底の集合.
k[1] は [gb, backward transformation, syzygy without dehomogenization].
@item  @var{m} より環の定義を検索し, その環でカーネルを計算する.
環 @var{r} があたえられているときは, @var{r} でカーネルを計算する.
@item @var{m} が定数成分のときは, 環 @var{r} を与える必要がある.
(@var{m} より環の情報をとりだせないため).
@item BUG:  Kernel および Res_solv (syz, res-solv) のみが, 環引数に
対応してる. (2000, 12/29 現在).
@end itemize

*/
/*&en
@table @var
@item return
List
@item m
Matrix or vector
@end table

@itemize @bullet
@item  Find solution basis of the linear indefinite equation u @var{m} =0.
@item When the return value is  k, k[0] is a set of generators of the kernel.
k[1] is [gb, backward transformation, syzygy without dehomogenization].
@item It finds the kernel in the ring 
to which @var{m} belongs.
When the ring @var{r} is given, 
it finds the kernel in the ring @var{r}.
@item When @var{m} consists of constants, a ring @var{r}
should be given.
@end itemize

*/

/*&C
@example
In(16)= RingD("x,y");
In(17)= mm=[[Dx],[Dy],[x]];
In(18)= Pmat(Kernel(mm));
 [ 
  [ 
    [    -x*Dx-2 , 0 , Dx^2 ] 
    [    -x*Dy , -1 , Dx*Dy ] 
    [    -x^2 , 0 , x*Dx-1 ] 
  ]
  [ 
   [ 
     [    -1 ] 
   ]
   [ 
     [    x , 0 , -Dx ] 
   ]
   [ 
     [    -x*Dx-2 , 0 , Dx^2 ] 
     [    -x*Dy , -1 , Dx*Dy ] 
     [    -x^2 , 0 , x*Dx-1 ] 
   ]
  ]
 ]

@end example
*/


/*&C

@example

In(4)= r=RingD("x,y");
In(5)= k=Kernel([[1,2],[2,4]],r); k[0]:
[    [    2 , -1 ]  ] 
@end example


*/


/*&ja

@table @t
@item 参照
    @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 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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/
/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of Kernel2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Kernel2::
@end menu
@node Kernel2,,, COMPLEX 函数
@subsection @code{Kernel2}
@findex Kernel2
@table @t
@item Kernel2(@var{m},@var{j})
::  @var{m} : D^p ---> D^q/@var{j} の Kernel を求める. 
@item Kernel2(@var{m},@var{j},@var{r})
::  @var{m} : D^p ---> D^q/@var{j} の Kernel を求める. @var{r} は ring.
@end table

*/
/*&en
@c %%%%%%%%%%%%%%%%%%%%  start of Kernel2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Kernel2::
@end menu
@node Kernel2,,, COMPLEX function
@subsection @code{Kernel2}
@findex Kernel2
@table @t
@item Kernel2(@var{m})
::  Get the kernel of @var{m} : D^p ---> D^q/@var{j}.
@item Kernel2(@var{m},@var{r})
::   Get the kernel of @var{m} : D^p ---> D^q/@var{j}.
@var{r} is a ring object.
@end table

*/

/*&ja
@table @var
@item return
リスト
@item m  j
行列またはベクトル
@end table

@itemize @bullet
@item @var{m} : D^p ---> D^q/@var{j} の Kernel を求める. 
@item D^p は横ベクトルであり, u が D^p の元のとき,
    u @var{m} で写像を定義する.
@item  @var{m} より環の定義を検索し, その環でカーネルを計算する.
環 @var{r} があたえられているときは, @var{r} でカーネルを計算する.
@item @var{m} が定数成分のときは, 環 @var{r} を与える必要がある.
(@var{m} より環の情報をとりだせないため).
@end itemize

*/
/*&en
@table @var
@item return
List
@item m  j
Matrix or vector
@end table

@itemize @bullet
@item Get a set of generators of the the kernel 
of @var{m} : D^p ---> D^q/@var{j}.
@item D^p is a set of row vectors. When u is an element of D^p,
define the map from D^p to D^q/@var{j} by u @var{m}.
@item It finds the kernel in the ring 
to which @var{m} belongs.
When the ring @var{r} is given, 
it finds the kernel in the ring @var{r}.
@item When @var{m} consists of constants, a ring @var{r}
should be given.
@end itemize

*/

/*&C
@example
In(27)= r=RingD("x,y");
In(28)= Kernel2([[x,y],[x^2,x*y]],[]):
[    [    -x , 1 ]  ] 
In(29)=Kernel2([[x,y],[x^2,x*y]],[[x,y]]):
[    [    1 , 0 ]  , [    0 , 1 ]  ] 

In(41)=Kernel2([0],[0],r):
[    [    1 ]  , [    0 ]  ] 
In(42)=Kernel2([[0,0],[0,0]],[[0,0]],r):
[    [    1 , 0 ]  , [    0 , 1 ]  , [    0 , 0 ]  ] 
In(43)=Kernel2([[0,0,0],[0,0,0]],[],r):
[    [    1 , 0 ]  , [    0 , 1 ]  ] 

@end example
*/


/*&ja

@table @t
@item 参照
    @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 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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/

/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of Gb %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Gb::
@end menu
@node Gb,,, COMPLEX 函数
@node Gb_h,,, COMPLEX 函数
@subsection @code{Gb}
@findex Gb
@findex Gb_h
@table @t
@item Gb(@var{f})
::  @var{f} のグレブナ基底をもとめる.
@item Gb(@var{f},@var{r})
::  @var{f} のグレブナ基底をもとめる. @var{r} は ring.
@item Gb_h(@var{f})
::  @var{f} のグレブナ基底をもとめる.
@item Gb_h(@var{f},@var{r})
::  @var{f} のグレブナ基底をもとめる. @var{r} は ring.
@end table

*/
/*&en
@c %%%%%%%%%%%%%%%%%%%%  start of Gb %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Gb::
@end menu
@node Gb,,, COMPLEX function
@node Gb_h,,, COMPLEX function
@subsection @code{Gb}
@findex Gb
@table @t
@item Gb(@var{f})
::  It computes the Grobner basis of @var{f}.
@item Gb(@var{m},@var{r})
::  It computes the Grobner basis of @var{f}.
@var{r} is a ring object.
@item Gb_h(@var{f})
::  It computes the Grobner basis of @var{f}.
@item Gb_h(@var{m},@var{r})
::  It computes the Grobner basis of @var{f}.
@var{r} is a ring object.
@end table

*/

/*&ja
@table @var
@item return
リスト
@item f
行列またはベクトル
@end table

@itemize @bullet
@item  @var{f} のグレブナ基底をもとめる.
@item _h 付きの場合は, 同次ワイル代数で計算をおこなう.
@item 戻り値を k とするとき k[0] が グレブナ基底.
環が weight vector 付きで定義されたときは,
k[1] は initial ideal または initial module.
@item  @var{m} より環の定義を検索し, その環でグレブナ基底を計算する.
環 @var{r} があたえられているときは, @var{r} でグレブナ基底を計算する.
@item @var{m} が定数成分のときは, 環 @var{r} を与える必要がある.
(@var{m} より環の情報をとりだせないため).
@end itemize

*/
/*&en
@table @var
@item return
List
@item f
Matrix or vector
@end table

@itemize @bullet
@item  Compute the Grobner basis of @var{f}.
@item Functions with _h  computes Grobner bases in the homogenized Weyl
algebra.
@item When the return value is  k, k[0] is the Grobner basis.
k[1] is the initial ideal or the initial module of @var{f},
when the ring is defined with a weight vector.
@item It computes the Grobner basis in the ring 
to which @var{f} belongs.
When the ring @var{r} is given, 
it computes the Grobner basis in the ring @var{r}.
@item When @var{f} consists of constants, a ring @var{r}
should be given.
@end itemize

*/

/*&C
@example
In(5)= r=RingD("x,y");
In(6)= m=[[x^2+y^2-1],[x*y-1]];
In(7)= Gb(m):
[    [    [    x^2+y^2-1 ]  , [    x*y-1 ]  , [    y^3+x-y ]  ]  , 
[    [    x^2+y^2-1 ]  , [    x*y-1 ]  , [    y^3+x-y ]  ]  ] 

In(11)= RingD("x,y",[["x",1]]);
In(12)= r=RingD("x,y",[["x",1]]);
In(13)= Gb(m,r):
[    [    [    x+y^3-y ]  , [    -y^4+y^2-1 ]  ]  , 
[    [    x ]  , [    -y^4+y^2-1 ]  ]  ] 

@end example
*/


/*&ja

@table @t
@item 参照
    @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 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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/

/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of Res_shiftMatrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Res_shiftMatrix::
@end menu
@node Res_shiftMatrix,,, COMPLEX 函数
@subsection @code{Res_shiftMatrix}
@findex Res_shiftMatrix
@table @t
@item Res_shiftMatrix(@var{m},@var{v})
::  Degree shift ベクトル @var{m} に対応する行列を作る.
@item Res_shiftMatrix(@var{f},@var{v},@var{r})
::  Degree shift ベクトル @var{m} に対応する行列を作る. @var{r} は ring.
@end table

*/
/*&en
@c %%%%%%%%%%%%%%%%%%%%  start of Res_shiftMatrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Res_shiftMatrix::
@end menu
@node Res_shiftMatrix,,, COMPLEX function
@subsection @code{Res_shiftMatrix}
@findex Res_shiftMatrix
@table @t
@item Res_shiftMatrix(@var{m},@var{v})
::  Generate a matrix associated to a degree shift vector @var{m}
@item Res_shiftMatrix(@var{m},@var{v},@var{r})
::  Generate a matrix associated to a degree shift vector @var{m}
@var{r} is a ring object.
@end table

*/

/*&ja
@table @var
@item return
行列.
@item m
ベクトル
@item v
多項式変数または文字列
@end table

@itemize @bullet
@item diag(@var{v}^(@var{m}1), ..., @var{v}^(@var{m}n))
なる n × n 行列を戻す.
@end itemize

*/
/*&en
@table @var
@item return
Matrix
@item m
Vector
@item v
多項式変数または文字列
@end table

@itemize @bullet
@item Returns n by n matrix
diag(@var{v}^(@var{m}1), ..., @var{v}^(@var{m}n))
@end itemize

*/

/*&C
@example
In(5)= r=RingD("x,y");
In(6)= Res_shiftMatrix([-1,0,3],x):
[    [    x^(-1) , 0 , 0 ]  , [    0 , 1 , 0 ]  , [    0 , 0 , x^3 ]  ] 

@end example
*/

/*&C
@example
In(9)=  rrr = RingD("t,x,y",[["t",1,"x",-1,"y",-1,"Dx",1,"Dy",1]]);
In(10)=  m=[Dx-(x*Dx+y*Dy+2),Dy-(x*Dx+y*Dy+2)];
In(12)=  m=Gb(m);
In(13)=  k = Kernel_h(m[0]);
In(14)=  Pmat(k[0]);
 [ 
   [    -Dy+3*h , Dx-3*h , 1 ] 
   [    -x*Dx+x*Dy-y*Dy-3*x*h , y*Dy+3*x*h , h-x ] 
 ]

In(15)=Pmat(m[0]);
  [    Dx*h-x*Dx-y*Dy-2*h^2 , Dy*h-x*Dx-y*Dy-2*h^2 , 
       x*Dx^2-x*Dx*Dy+y*Dx*Dy-y*Dy^2 ] 

In(18)=k2 = Gb_h(k[0]*Res_shiftMatrix([1,1,1],t));
In(19)=Pmat(Substitute(k2[0],t,1));
 [ 
   [    -Dy+3*h , Dx-3*h , 1 ] 
   [    -x*Dx+x*Dy-y*Dy-3*x*h , y*Dy+3*x*h , h-x ] 
 ]


@end example
*/


/*&ja

@table @t
@item 参照
    @code{Gb}, (m,(u,v))-グレブナ基底
@item Files
    @code{lib/restriction/complex.k}
@end table
@c  %%%%%%%%%%%%%%%%%%%%  end of Res_shiftMatrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/
/*&en

@table @t
@item See also
   @code{Gb}, (m,(u,v))-Grobner basis
@item Files
    @code{lib/restriction/complex.k}
@end table
@c  %%%%%%%%%%%%%%%%%%%%  end of Res_shiftMatrix %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/


@c --------------  Primitive 基本 ----------------------

/*&ja

@node 基本 函数,,, Top
@chapter 基本 函数
@section 函数一覧
*/
/*&en

@node Primitive function,,, Top
@chapter Primitive function
@section A list of functions
*/
/*&C
@menu
* ChangeRing::
* Intersection::
* Getxvars::
* Firstn::
@end menu
*/

/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start ChangeRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@node ChangeRing,,, 基本 函数
@subsection @code{ChangeRing}
@findex ChangeRing
@table @t
@item ChangeRing(@var{f})
::  ChangeRing は @var{f} の要素が属する環を current ring にする.
@end table
 
@table @var
@item return
true か false
@item f  リスト
@end table

@example
   RingD("x,y");
   f=[x+y,0];
   RingD("p,q,r");
   ChangeRing(f);
@end example

@table @t
@item Files
    @code{lib/restriction/complex.k}
@end table
@c  %%%%%%%%%%%%%%%%%%%%  end of ChangeRing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/


/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of Intersection %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Intersection::
@end menu
@node Intersection,,, 基本 函数
@subsection @code{Intersection}
@findex Intersection
@table @t
@item Intersection(@var{i},@var{j})
::  イデアル @var{i} と @var{j} の交わりを求める.
@item Intersection(@var{i},@var{j},@var{r})
::  イデアル @var{i} と @var{j} の交わりを求める. 計算を環 @var{r} 
でおこなう.
@end table

@table @var
@item return
リストであたえられたイデアルまたは自由加群の部分加群
@item i  j
イデアルまたは自由加群の部分加群
@item r
環
@end table

@itemize @bullet
@item  ::  イデアル @var{i} と @var{j} の交わりを求める.
@end itemize

@example
In(16)= RingD("x,y");
In(17)= mm=[ [x,0],[0,y] ]; nn = [ [x^2,0],[0,y^3]];
In(19)= Intersection(mm,nn):
In(33)=Intersection(mm,nn):
[    [    -x^2 , 0 ]  , [    0 , -y^3 ]  ] 
@end example

@table @t
@item 参照
@item Files
    @code{lib/restriction/complex.k}
@end table
@c  %%%%%%%%%%%%%%%%%%%%  end of Intersection %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/

/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of Getxvars %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Getxvars::
@end menu
@node Getxvars,,, 基本 函数
@subsection @code{Getxvars}
@findex Getxvars
@table @t
@item Getxvars()
::  x 変数を戻す
@end table

*/
/*&en
@c %%%%%%%%%%%%%%%%%%%%  start of Getxvars %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Getxvars::
@end menu
@node Getxvars,,, Primitive function
@subsection @code{Getxvars}
@findex Getxvars
@table @t
@item Getxvars()
::  Return x variables
@end table

*/

/*&ja
@table @var
@item return
[x_list, x_str]  x_list は x 変数のリスト, x_str は x 変数を , で区切った文字列.
@end table

*/
/*&en
@table @var
@item return
[x_list, x_str] x_list is a list of x variables, x_str is a string consisting
of x variables separated by commas.
@end table


*/

/*&C
@example
In(4)=RingD("x,y");
In(5)=Getxvars():
[    [    y , x ]  , y,x, ] 
@end example

*/


/*&ja

@table @t
@item 参照
@item Files
    @code{lib/restriction/complex.k}
@end table
@c  %%%%%%%%%%%%%%%%%%%%  end of Getxvars %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/
/*&en

@table @t
@item See also
@item Files
    @code{lib/restriction/complex.k}
@end table
@c  %%%%%%%%%%%%%%%%%%%%  end of Getxvars %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/

/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of Firstn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Firstn::
@end menu
@node Firstn,,, 基本 函数
@subsection @code{Firstn}
@findex Firstn
@table @t
@item Firstn(@var{m},@var{n})
::  @var{m} の最初の @var{n} 個をとりだす.
@end table

*/
/*&en
@c %%%%%%%%%%%%%%%%%%%%  start of Firstn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Firstn::
@end menu
@node Firstn,,, Primitive function
@subsection @code{Firstn}
@findex Firstn
@table @t
@item Firstn(@var{m},@var{n})
::  Return the first @var{n} elements of @var{m}.
@end table

*/

/*&ja
@table @var
@item return
行列またはベクトル
@item m
行列またはベクトル
@item n
数
@end table

@itemize @bullet
@item  m の最初の n 個.  とくに m が行列のときは, 各行より最初の n 個をとりだした
もので作れらた行列を戻す.
@end itemize

*/
/*&en
@table @var
@item return
Matrix or vector
@item m
Matrix or vector
@item n
Number
@end table

@itemize @bullet
@item  The first n elements of m.  When m is a matrix, it returns the matrix
consisting of first n elements of rows of m.
@end itemize

*/

/*&C
@example
In(16)= mm = [[1,2,3],[4,5,6]];
In(17)= Firstn(mm,2):
    [[1,2], 
     [4,5]]
@end example
*/


/*&ja

@table @t
@item 参照
@item Files
    @code{lib/restriction/complex.k}
@end table
@c  %%%%%%%%%%%%%%%%%%%%  end of Firstn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/
/*&en

@table @t
@item See also
@item Files
    @code{lib/restriction/complex.k}


@end table
@c  %%%%%%%%%%%%%%%%%%%%  end of Firstn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/

/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of GKZ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* GKZ::
@end menu
@node GKZ,,, 基本 函数
@subsection @code{GKZ}
@findex GKZ
@table @t
@item GKZ(@var{a},@var{b})
::  行列 @var{a} と ベクトル @var{b} で定まる, GKZ hypergeometric system を戻す.
@end table

@table @var
@item return
D のイデアルの生成元
@item a
整数行列
@item b
整数ベクトル
@end table

@itemize @bullet
@item   行列 @var{a} と ベクトル @var{b} で定まる, GKZ hypergeometric system を戻す.
@item Gelfand, Kapranov, Zelevinski の超幾何微分方程式を戻す. 
@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 参照
@item Files
    @code{lib/restriction/demo2.k}
@end table
@c  %%%%%%%%%%%%%%%%%%%%  end of GKZ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/

/*&ja
@c %%%%%%%%%%%%%%%%%%%%  start of Slope %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@menu
* Slope::
@end menu
@node Slope,,, 基本 函数
@subsection @code{Slope}
@findex Slope
@table @t
@item Slope(@var{i},@var{v},@var{f},@var{v})
::  微分方程式系 @var{i} の slope の集合をもとめる.
@end table

@table @var
@item return
Slope の集合とそれを与える, weight vectors.
@item i
D のイデアルの生成元
@item v
空間変数
@item f
F-filtration を与える weight vector (0,...,0,1,...,1).
@item v
V-filtration を与える weight vector.
たとえば (0,...,0,-1,0,...,0,1).
@end table

@itemize @bullet
@item @var{i} の x_i = 0 にそった, 原点での slope の全体をもとめる.
@item 計算には Assi, Castro, Granger のアルゴリズムを用いる.
@item Geometric slope を計算するため, radical の計算をおこなっている.
これには ox_asir を用いている.
@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 参照
@item Files
    @code{lib/restriction/demo2.k}
@end table
@c  %%%%%%%%%%%%%%%%%%%%  end of Slope %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*/