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version 1.2, 2021/01/20 08:17:54 version 1.5, 2021/10/27 06:13:24
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 %% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.1 2021/01/20 03:05:29 takayama Exp $  %% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.4 2021/03/29 05:08:01 takayama Exp $
 %% xetex mt_gkz-en.texi   (.texi までつける. )  %% xetex mt_gkz-en.texi   (.texi までつける. )
 %% @math{tex形式の数式}  %% @math{tex形式の数式}
 %% 参考: http://www.fan.gr.jp/~ring/doc/texinfo/texinfo-ja_14.html#SEC183  %% 参考: http://www.fan.gr.jp/~ring/doc/texinfo/texinfo-ja_14.html#SEC183
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Line 19 
 @comment --- おまじない終り ---  @comment --- おまじない終り ---
   
 @comment --- GNU info ファイルの名前 ---  @comment --- GNU info ファイルの名前 ---
 @setfilename mt_gkz_man  @setfilename asir-contrib-mt_gkz_man
   
 @comment --- タイトル ---  @comment --- タイトル ---
 @settitle GKZ hypergeometric system  @settitle GKZ hypergeometric system
Line 134  systems, Linear Algebra and its Applications (LAA), 42
Line 134  systems, Linear Algebra and its Applications (LAA), 42
 @subsection @code{mt_gkz.pfaff_eq}  @subsection @code{mt_gkz.pfaff_eq}
 @comment --- 索引用キーワード  @comment --- 索引用キーワード
 @findex mt_gkz.pfaff_eq  @findex mt_gkz.pfaff_eq
   @findex mt_gkz.use_hilbert_driven
   
 @table @t  @table @t
 @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})  @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
Line 225  Option @var{cg}. A constant matrix given by this optio
Line 226  Option @var{cg}. A constant matrix given by this optio
 for the Gauge transformation of the Pfaffian system.  for the Gauge transformation of the Pfaffian system.
 In other words, the basis of cocycles specified by @var{Rvec}  In other words, the basis of cocycles specified by @var{Rvec}
 is transformed by the constant matrix given by this option.  is transformed by the constant matrix given by this option.
   @item
   By mt_gkz.use_hilbert_driven(Rank), the rank of the GKZ system is assumed to be
   Rank. It makes the computation of Groebner basis by yang.rr faster.
   This option is disabled by mt_gkz.use_hilbert_driven(0);
 @end itemize  @end itemize
   
 @comment --- @example〜@end example は実行例の表示 ---  @comment --- @example〜@end example は実行例の表示 ---
Line 697  Example: 
Line 702  Example: 
 * mt_gkz.get_check_fvec::  * mt_gkz.get_check_fvec::
 * mt_gkz.get_bf_step_up::  * mt_gkz.get_bf_step_up::
 * mt_gkz.mytoric_ideal::  * mt_gkz.mytoric_ideal::
   * mt_gkz.cbase_by_euler::
 @end menu  @end menu
   
 @node some utility functions,,, utilities  @node some utility functions,,, utilities
Line 713  Example: 
Line 719  Example: 
 @node mt_gkz.get_check_fvec,,, some utility functions  @node mt_gkz.get_check_fvec,,, some utility functions
 @node mt_gkz.get_bf_step_up,,, some utility functions  @node mt_gkz.get_bf_step_up,,, some utility functions
 @node mt_gkz.mytoric_ideal,,, some utility functions  @node mt_gkz.mytoric_ideal,,, some utility functions
   @node mt_gkz.cbase_by_euler,,, some utility functions
   
 @findex mt_gkz.reduce_by_toric  @findex mt_gkz.reduce_by_toric
 @findex mt_gkz.tk_base_equal  @findex mt_gkz.tk_base_equal
Line 725  Example: 
Line 732  Example: 
 @findex mt_gkz.get_check_fvec  @findex mt_gkz.get_check_fvec
 @findex mt_gkz.get_bf_step_up  @findex mt_gkz.get_bf_step_up
 @findex mt_gkz.mytoric_ideal  @findex mt_gkz.mytoric_ideal
   @findex mt_gkz.cbase_by_euler
   
 @comment --- @example〜@end example は実行例の表示 ---  @comment --- @example〜@end example は実行例の表示 ---
 We only show examples on these functions. As for details, please see  We only show examples on these functions. As for details, please see
Line 789  the source code.
Line 797  the source code.
 [3192] mt_gkz.mytoric_ideal(0 | use_4ti2=0);  [3192] mt_gkz.mytoric_ideal(0 | use_4ti2=0);
 // A slower method is used to obtain a generator set of the toric ideal  // A slower method is used to obtain a generator set of the toric ideal
 // defined by the matrix A.  4ti2 is not needed. Default.  // defined by the matrix A.  4ti2 is not needed. Default.
   [3193] mt_gkz.cbase_by_euler(A=[[1,1,1,1],[0,1,3,4]]);
   // Cohomology basis of the GKZ system defined by A for generic beta.
   // Basis is given by a set of Euler operators.
   // It is an implementation of the algorithm in http://dx.doi.org/10.1016/j.aim.2016.10.021
   // beta is set by random numbers. Option: no_prob=1
   
 @end example  @end example
   
   
Line 1194  Here, @math{n_{2_{1_{3_1}}}} and @math{d_{6_1}} are ar
Line 1208  Here, @math{n_{2_{1_{3_1}}}} and @math{d_{6_1}} are ar
 @comment --- 引数の簡単な説明 ---  @comment --- 引数の簡単な説明 ---
 @table @var  @table @var
 @item return  @item return
 a rational function which is the cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[{dx\over x}],[{dx\over x}]\rangle_{ch}} in terms of the regular triangulation T. Here, @math{n} is the number of integration variables and @math{dx\over x} is the volume form @math{{dx_1\over x_1}\wedge\cdots\wedge{dx_n\over x_n}} of the complex @math{n}-torus.  a rational function which is the cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[{dt\over t}],[{dt\over t}]\rangle_{ch}} in terms of the regular triangulation T. Here, @math{n} is the number of integration variables and @math{dt\over t} is the volume form @math{{dt_1\over t_1}\wedge\cdots\wedge{dt_n\over t_n}} of the complex @math{n}-torus.
 @item A,Beta  @item A,Beta
 see @code{pfaff_eq}.  see @code{pfaff_eq}.
 @item T  @item T
Line 1208  The number of polynomial factors in the integrand. see
Line 1222  The number of polynomial factors in the integrand. see
 @comment --- @bullet は黒点付き ---  @comment --- @bullet は黒点付き ---
 @itemize @bullet  @itemize @bullet
 @item  @item
 This function is useful when the basis of the cohomology group @math{\{\omega_i\}_{i=1}^r} is given so that @math{\omega_1=[{dx\over x}]}.  This function is useful when the basis of the cohomology group @math{\{\omega_i\}_{i=1}^r} is given so that @math{\omega_1=[{dt\over t}]}.
 @item  @item
 @code{leading_term} can be used more generally.  One can find a regular triangulation by using a function @code{mt_gkz.regular_triangulation}.
   @item
   @code{mt_gkz.leading_terms} can be used more generally.
 @end itemize  @end itemize
   
 @comment --- @example〜@end example は実行例の表示 ---  @comment --- @example〜@end example は実行例の表示 ---
Line 1227  Example:
Line 1243  Example:
 @comment --- 参照(リンク)を書く ---  @comment --- 参照(リンク)を書く ---
 @table @t  @table @t
 @item Refer to  @item Refer to
 @ref{mt_gkz.leading_term}.  @ref{mt_gkz.leading_terms}.
 @end table  @end table
 @comment おわり.  @comment おわり.
   
Line 1238  Example:
Line 1254  Example:
   
   
   
 @node mt_gkz.leading_term,,, Normalizing constant  @node mt_gkz.leading_terms,,, Normalizing constant
 @subsection @code{mt_gkz.leading_term}  @subsection @code{mt_gkz.leading_terms}
 @comment --- 索引用キーワード  @comment --- 索引用キーワード
 @findex mt_gkz.leading_terms  @findex mt_gkz.leading_terms
   
 @table @t  @table @t
 @item mt_gkz.leading_term(@var{A},@var{Beta},@var{W},@var{Q1},@var{Q2},@var{K},@var{N})  @item mt_gkz.leading_terms(@var{A},@var{Beta},@var{W},@var{Q1},@var{Q2},@var{K},@var{N})
 :: It returns the W-leading terms of a cohomology intersection number specified by Q1 and Q2 up to W-degree=(minimum W-degree)+N.  :: It returns the W-leading terms of a cohomology intersection number specified by Q1 and Q2 up to W-degree=(minimum W-degree)+N.
 @end table  @end table
   
 @comment --- 引数の簡単な説明 ---  @comment --- 引数の簡単な説明 ---
 @table @var  @table @var
 @item return  @item return
 a list [[C1,DEG1],[C2,DEG2],...]. Each CI is a rational function depending on Beta times a monomial @math{x^m} in x-variables. DEGI is the W-degree of @math{x^m}. The cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[h^{-q_1^\prime}x^{q_1^{\prime\prime}}{dx\over x}],[h^{-q_2^\prime}x^{q_2^{\prime\prime}}{dx\over x}]\rangle_{ch}} has a Laurent expansion of the form C1+C2+....  a list [[C1,DEG1],[C2,DEG2],...]. Each CI is a rational function depending on Beta times a monomial @math{x^m} in x-variables. DEGI is the W-degree of @math{x^m}. The cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[h^{-q_1^\prime}t^{q_1^{\prime\prime}}{dt\over t}],[h^{-q_2^\prime}t^{q_2^{\prime\prime}}{dt\over t}]\rangle_{ch}} has a Laurent expansion of the form C1+C2+....
 @item A,Beta  @item A,Beta
 see @code{pfaff_eq}.  see @code{pfaff_eq}.
 @item W  @item W
Line 1271  A positive integer.
Line 1287  A positive integer.
 @item  @item
 For a monomial @math{x^m=x_1^{m_1}\cdots x_n^{m_n}} and a weight vector @math{W=(w_1,\dots,w_n)}, the W-degree of @math{x^m} is given by the dot product @math{m\cdot W=m_1w_1+\cdots +m_nw_n}.  For a monomial @math{x^m=x_1^{m_1}\cdots x_n^{m_n}} and a weight vector @math{W=(w_1,\dots,w_n)}, the W-degree of @math{x^m} is given by the dot product @math{m\cdot W=m_1w_1+\cdots +m_nw_n}.
 @item  @item
 The W-leading terms of the cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[h^{-q_1^\prime}x^{q_1^{\prime\prime}}{dx\over x}],[h^{-q_2^\prime}x^{q_2^{\prime\prime}}{dx\over x}]\rangle_{ch}} can be computed by means of Theorem 2.6 of [GM2020]. See also Theorem 3.4.2 of [SST2000].  The W-leading terms of the cohomology intersection number @math{{1\over (2\pi\sqrt{-1})^n} \langle[h^{-q_1^\prime}t^{q_1^{\prime\prime}}{dt\over t}],[h^{-q_2^\prime}t^{q_2^{\prime\prime}}{dt\over t}]\rangle_{ch}} can be computed by means of Theorem 2.6 of [GM2020]. See also Theorem 3.4.2 of [SST2000].
 @item  @item
 If the weight vector is not generic, you will receive an error message such as "WARNING(initial_mon): The weight may not be generic". In this case, the output may be wrong and you should retake a suitable W. To be more precise, W should be chosen from an open cone of the Groebner fan.  If the weight vector is not generic, you will receive an error message such as "WARNING(initial_mon): The weight may not be generic". In this case, the output may be wrong and you should retake a suitable W. To be more precise, W should be chosen from an open cone of the Groebner fan.
 @item Option xrule.  @item Option xrule.
Line 1293  Example:
Line 1309  Example:
 2  2
 [2927] N=2;  [2927] N=2;
 2  2
 [2928] NC=leading_term(A,Beta,W,Q[0],Q[1],K,N|xrule=[[x2,1],[x3,1],[x4,1]])$  [2928] NC=mt_gkz.leading_terms(A,Beta,W,Q[0],Q[1],K,N|xrule=[[x2,1],[x3,1],[x4,1]])$
 --snip--  --snip--
 [2929] NC;  [2929] NC;
 [[(-3)/(x1),-5],[0,-4],[0,-3]]  [[(-3)/(x1),-5],[0,-4],[0,-3]]
Line 1316  intersection matrix is given by
Line 1332  intersection matrix is given by
 @comment --- 参照(リンク)を書く ---  @comment --- 参照(リンク)を書く ---
 @table @t  @table @t
 @item Refer to  @item Refer to
 @ref{mt_gkz.leading_term}, @ref{mt_gkz.generate_maple_file_IC}, @ref{mt_gkz.generate_maple_file_MR}.  @ref{mt_gkz.leading_terms}, @ref{mt_gkz.generate_maple_file_IC}, @ref{mt_gkz.generate_maple_file_MR}.
 @end table  @end table
 @comment おわり.  @comment おわり.
   
Line 1351  a list of variables of P.
Line 1367  a list of variables of P.
 @comment --- @bullet は黒点付き ---  @comment --- @bullet は黒点付き ---
 @itemize @bullet  @itemize @bullet
 @item  @item
 This function is supposed to be combined with @code{leading_term} to compute the leading term of a cohomology intersection number.  This function is supposed to be combined with @code{leading_terms} to compute the leading term of a cohomology intersection number.
 @item  @item
 If W is chose so that there are several initial terms, you will receive an error message "WARNING(leading_term_rat):The weight vector may not be generic."  If W is chose so that there are several initial terms, you will receive an error message "WARNING(leading_term_rat):The weight vector may not be generic."
 @end itemize  @end itemize
Line 1360  If W is chose so that there are several initial terms,
Line 1376  If W is chose so that there are several initial terms,
 @comment --- 参照(リンク)を書く ---  @comment --- 参照(リンク)を書く ---
 @table @t  @table @t
 @item Refer to  @item Refer to
 @ref{mt_gkz.leading_term}.  @ref{mt_gkz.leading_terms}.
 @end table  @end table
 @comment おわり.  @comment おわり.
   
Line 1413  As for the definition of the standard pair, see Chapte
Line 1429  As for the definition of the standard pair, see Chapte
 @item  @item
 We set n=length(A) and set BS1:=@math{\{ 1,2,...,n\}\setminus S1}. Then, each L1[I] is an exponent @math{\bf k} of a top-dimensional standard pair @math{(\partial^{\bf k}_{BS1},S1)}. Here, @math{\bf k} is a list of length n-length(S1) and @math{\partial_{BS1}=(\partial_J)_{J\in BS1}}.  We set n=length(A) and set BS1:=@math{\{ 1,2,...,n\}\setminus S1}. Then, each L1[I] is an exponent @math{\bf k} of a top-dimensional standard pair @math{(\partial^{\bf k}_{BS1},S1)}. Here, @math{\bf k} is a list of length n-length(S1) and @math{\partial_{BS1}=(\partial_J)_{J\in BS1}}.
 @item  @item
 If the weight vector is not generic, you will receive an error message such as "WARNING(initial_mon): The weight may not be generic". See also @code{leading_term}.  If the weight vector is not generic, you will receive an error message such as "WARNING(initial_mon): The weight may not be generic". See also @code{leading_terms}.
 @item  @item
 These functions are utilized in @code{leading_term}.  These functions are utilized in @code{leading_terms}.
 @end itemize  @end itemize
   
 @comment --- @example〜@end example は実行例の表示 ---  @comment --- @example〜@end example は実行例の表示 ---
Line 1425  Example: An example of a non-unimodular triangulation 
Line 1441  Example: An example of a non-unimodular triangulation 
 [[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]]  [[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]]
 [3257] W=[2,0,1,2,2];  [3257] W=[2,0,1,2,2];
 [2,0,1,2,2]  [2,0,1,2,2]
 [3258] toric_gen_initial(A,W);  [3258] mt_gkz.toric_gen_initial(A,W);
 --snip--  --snip--
 [[x1*x5,x1*x4,x3^2*x4],[x1,x2,x3,x4,x5]]  [[x1*x5,x1*x4,x3^2*x4],[x1,x2,x3,x4,x5]]
 [3259] regular_triangulation(A,W);  [3259] mt_gkz.regular_triangulation(A,W);
 --snip--  --snip--
 [[2,4,5],[2,3,5],[1,2,3]]  [[2,4,5],[2,3,5],[1,2,3]]
 [3260] top_standard_pairs(A,W);  [3260] mt_gkz.top_standard_pairs(A,W);
 --snip--  --snip--
 [[[[0,0],[0,1]],[2,4,5]],[[[0,0]],[2,3,5]],[[[0,0]],[1,2,3]]]  [[[[0,0],[0,1]],[2,4,5]],[[[0,0]],[2,3,5]],[[[0,0]],[1,2,3]]]
   
Line 1448  are 2,1 and 1. Moreover, the top-dimensional standard 
Line 1464  are 2,1 and 1. Moreover, the top-dimensional standard 
 @comment --- 参照(リンク)を書く ---  @comment --- 参照(リンク)を書く ---
 @table @t  @table @t
 @item Refer to  @item Refer to
 @ref{mt_gkz.leading_term}.  @ref{mt_gkz.leading_terms}.
 @end table  @end table
 @comment おわり.  @comment おわり.
   
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