===================================================================
RCS file: /home/cvs/OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v
retrieving revision 1.1
retrieving revision 1.4
diff -u -p -r1.1 -r1.4
--- OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi	2021/01/20 03:05:29	1.1
+++ OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi	2021/03/29 05:08:01	1.4
@@ -1,4 +1,4 @@
-%% $OpenXM$
+%% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.3 2021/03/02 10:57:17 takayama Exp $
 %% xetex mt_gkz-en.texi   (.texi までつける. ) 
 %% @math{tex形式の数式} 
 %% 参考: http://www.fan.gr.jp/~ring/doc/texinfo/texinfo-ja_14.html#SEC183
@@ -19,7 +19,7 @@
 @comment --- おまじない終り ---
 
 @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 ---
 @settitle GKZ hypergeometric system
@@ -696,6 +696,7 @@ Example: 
 * mt_gkz.ord_xi::
 * mt_gkz.get_check_fvec::
 * mt_gkz.get_bf_step_up::
+* mt_gkz.mytoric_ideal::
 @end menu
 
 @node some utility functions,,, utilities
@@ -711,6 +712,7 @@ Example: 
 @node mt_gkz.ord_xi,,, 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.mytoric_ideal,,, some utility functions
 
 @findex mt_gkz.reduce_by_toric
 @findex mt_gkz.tk_base_equal
@@ -722,6 +724,7 @@ Example: 
 @findex mt_gkz.ord_xi
 @findex mt_gkz.get_check_fvec
 @findex mt_gkz.get_bf_step_up
+@findex mt_gkz.mytoric_ideal
 
 @comment --- @example〜@end example は実行例の表示 ---
 We only show examples on these functions. As for details, please see
@@ -780,6 +783,12 @@ the source code.
      -s_2^3+(3*s_1+1)*s_2^2-3*s_1*s_2 s_2^3-3*s_2^2+2*s_2 ],
   [ x3^3*dx4^2+ ---snip---
     3*x3^2*x4*dx4^2+ --- snip---]]
+[3191] mt_gkz.mytoric_ideal(0 | use_4ti2=1);
+// 4ti2 is used to obtain a generator set of the toric ideal 
+// defined by the matrix A
+[3192] mt_gkz.mytoric_ideal(0 | use_4ti2=0);
+// A slower method is used to obtain a generator set of the toric ideal 
+// defined by the matrix A.  4ti2 is not needed. Default.
 @end example
 
 
@@ -847,7 +856,7 @@ the source code.
 * mt_gkz.secondary_eq::
 * mt_gkz.generate_maple_file_IC::
 * mt_gkz.generate_maple_file_MR::
-* mt_gkz.normalizing_constant::
+* mt_gkz.principal_normalizing_constant::
 @end menu
 
 
@@ -873,7 +882,7 @@ the source code.
 @comment --- 引数の簡単な説明 --- 
 @table @var
 @item return
-a matrix which is equal to the Kronecker product of @var{A} and @var{B} (https://en.wikipedia.org/wiki/Kronecker_product).
+a matrix which is equal to the Kronecker product of @var{A} and @var{B} (@uref{https://en.wikipedia.org/wiki/Kronecker_product}).
 @item A,B
 list
 @end table
@@ -886,7 +895,7 @@ list
 [[a,b],[c,d]]
 [2645] B=[[e,f],[g,h]];
 [[e,f],[g,h]]
-[2646] kronecker_prd(A,B);
+[2646] mt_gkz.kronecker_prd(A,B);
 [ e*a f*a e*b f*b ]
 [ g*a h*a g*b h*b ]
 [ e*c f*c e*d f*d ]
@@ -941,7 +950,7 @@ Example:
 [2650] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
 [2651] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
 [2652] Xrule=[[x2,1],[x3,1]]$
-[2653] P=secondary_eq(A,Beta,Ap,Rvec,DirX|xrule=Xrule)$
+[2653] P=mt_gkz.secondary_eq(A,Beta,Ap,Rvec,DirX|xrule=Xrule)$
 --snip--
 [2654] length(P);
 2
@@ -1013,7 +1022,7 @@ Example: 
 [2684] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
 [2685] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
 [2687] Xrule=[[x2,1],[x3,1]]$
-[2688] generate_maple_file_IC(A,Beta,Ap,Rvec,DirX|xrule=Xrule,filename="Test.ml")$
+[2688] mt_gkz.generate_maple_file_IC(A,Beta,Ap,Rvec,DirX|xrule=Xrule,filename="Test.ml")$
 
 
 //A file named Test.ml is automatically generated as follows:
@@ -1115,7 +1124,7 @@ Example:
 [2672] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
 [2673] Xvar=[x1,x4]$
 [2674] Xrule=[[x2,1],[x3,1]]$
-[2675] generate_maple_file_MR(A,Beta,Ap,Rvec,DirX,2,2|xrule=Xrule)$
+[2675] mt_gkz.generate_maple_file_MR(A,Beta,Ap,Rvec,DirX,2,2|xrule=Xrule)$
 
 
 //A file "auto-generated-MR.ml" is automatically generated as follows:
@@ -1185,7 +1194,7 @@ Here, @math{n_{2_{1_{3_1}}}} and @math{d_{6_1}} are ar
 @comment --- 引数の簡単な説明 --- 
 @table @var
 @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
 see @code{pfaff_eq}.
 @item T
@@ -1199,9 +1208,11 @@ The number of polynomial factors in the integrand. see
 @comment --- @bullet は黒点付き ---
 @itemize @bullet
 @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
-@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
 
 @comment --- @example〜@end example は実行例の表示 ---
@@ -1211,14 +1222,14 @@ Example:
 [2677] Beta=[b1,b2,b3]$
 [2678] K=2$
 [2679] T=[[1,2,3],[2,3,4]]$
-[2680] normalizing_constant(A,T,Beta,K);
+[2680] mt_gkz.principal_normalizing_constant(A,T,Beta,K);
 (-b1-b2)/(b3*b1+b3*b2-b3^2)
 @end example
 
 @comment --- 参照(リンク)を書く ---
 @table @t
 @item Refer to
-@ref{mt_gkz.leading_term}.
+@ref{mt_gkz.leading_terms}.
 @end table
 @comment おわり. 
 
@@ -1229,20 +1240,20 @@ Example:
 
 
 
-@node mt_gkz.leading_term,,, Normalizing constant
-@subsection @code{mt_gkz.leading_term}
+@node mt_gkz.leading_terms,,, Normalizing constant
+@subsection @code{mt_gkz.leading_terms}
 @comment --- 索引用キーワード
 @findex mt_gkz.leading_terms
 
 @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.
 @end table
 
 @comment --- 引数の簡単な説明 --- 
 @table @var
 @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
 see @code{pfaff_eq}.
 @item W
@@ -1262,7 +1273,7 @@ A positive integer.
 @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}.
 @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
 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.
@@ -1284,7 +1295,7 @@ Example:
 2
 [2927] N=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--
 [2929] NC;
 [[(-3)/(x1),-5],[0,-4],[0,-3]]
@@ -1307,7 +1318,7 @@ intersection matrix is given by
 @comment --- 参照(リンク)を書く ---
 @table @t
 @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
 @comment おわり. 
 
@@ -1342,7 +1353,7 @@ a list of variables of P.
 @comment --- @bullet は黒点付き ---
 @itemize @bullet
 @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
 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
@@ -1351,7 +1362,7 @@ If W is chose so that there are several initial terms,
 @comment --- 参照(リンク)を書く ---
 @table @t
 @item Refer to
-@ref{mt_gkz.leading_term}.
+@ref{mt_gkz.leading_terms}.
 @end table
 @comment おわり. 
 
@@ -1404,9 +1415,9 @@ As for the definition of the standard pair, see Chapte
 @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}}.
 @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
-These functions are utilized in @code{leading_term}.
+These functions are utilized in @code{leading_terms}.
 @end itemize
 
 @comment --- @example〜@end example は実行例の表示 ---
@@ -1416,13 +1427,13 @@ Example: An example of a non-unimodular triangulation 
 [[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]]
 [3257] W=[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--
 [[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--
 [[2,4,5],[2,3,5],[1,2,3]]
-[3260] top_standard_pairs(A,W);
+[3260] mt_gkz.top_standard_pairs(A,W);
 --snip--
 [[[[0,0],[0,1]],[2,4,5]],[[[0,0]],[2,3,5]],[[[0,0]],[1,2,3]]]
 
@@ -1439,7 +1450,7 @@ are 2,1 and 1. Moreover, the top-dimensional standard 
 @comment --- 参照(リンク)を書く ---
 @table @t
 @item Refer to
-@ref{mt_gkz.leading_term}.
+@ref{mt_gkz.leading_terms}.
 @end table
 @comment おわり.