=================================================================== RCS file: /home/cvs/OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v retrieving revision 1.2 retrieving revision 1.6 diff -u -p -r1.2 -r1.6 --- OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi 2021/01/20 08:17:54 1.2 +++ OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi 2021/12/11 11:40:45 1.6 @@ -1,4 +1,4 @@ -%% $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 1.5 2021/10/27 06:13:24 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 @@ -38,7 +38,7 @@ @title GKZ hypergeometric system @subtitle Pfaffian system (Pfaff equation), contiguity relations, cohomology intersection @subtitle Version 1.0 -@subtitle January 20, 2021 +@subtitle December 21, 2021 @author by S-J. Matsubara-Heo, N.Takayama @page @@ -123,6 +123,8 @@ systems, Linear Algebra and its Applications (LAA), 42 * mt_gkz.ff1:: * mt_gkz.ff2:: * mt_gkz.rvec_to_fvec:: +* mt_gkz.fvec_to_conn_mat:: +* mt_gkz.contiguity:: @end menu @node Pfaff equation for given cocycles,,, Pfaff equation @@ -134,6 +136,7 @@ systems, Linear Algebra and its Applications (LAA), 42 @subsection @code{mt_gkz.pfaff_eq} @comment --- 索引用キーワード @findex mt_gkz.pfaff_eq +@findex mt_gkz.use_hilbert_driven @table @t @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX}) @@ -151,7 +154,7 @@ the parameter vector of the GKZ system. @item Ap See [MT2020]. @item Rvec -It is used to specify a basis of cocycles. See [MT2020] +It is used to specify a basis of cocycles as explained below. See also [MT2020]. @item DirX a list of dxi's. @end table @@ -168,6 +171,9 @@ the set of the cocycles standing for Av_1, Av_2, ..., (see [MT2020]) is supposed to be the basis to construct the Pfaffian system. +The exponents @math{(q_\ell, q)} of the integral representation +@math{\int \prod h_\ell^{-q_\ell} x^q {{dx} \over {x}}} +is shifted by Av_i@math{=:A_{v_i}} as @math{(q_\ell,q)+A_{v_i}}. Let a_1, a_2, ..., a_n be the column vectors of the matrix A and v be a column vector (x_1, x_2, ..., x_n)^T. Av is defined as a_1 x_1 + a_2 x_2 + ... + a_n x_n. @@ -225,6 +231,10 @@ Option @var{cg}. A constant matrix given by this optio for the Gauge transformation of the Pfaffian system. In other words, the basis of cocycles specified by @var{Rvec} 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 @comment --- @example〜@end example は実行例の表示 --- @@ -556,6 +566,55 @@ obtains connection matrices. @end table @comment おわり. +@comment --- contiguity +@comment --- section 名を正確に --- +@node mt_gkz.contiguity,,, Pfaff equation for given cocycles +@subsection @code{mt_gkz.contiguity} +@comment --- 索引用キーワード +@findex mt_gkz.contiguity + +@table @t +@item mt_gkz.contiguity(@var{A},@var{Beta},@var{Ap},@var{Rvec1},@var{Rvec2}) +:: It returns the coefficient matrix P that satisfies +@var{Rvec1} = P @var{Rvec2}. +@end table + +@comment --- 引数の簡単な説明 --- +@table @var +@item return +The coefficient matrix P that satisfies @var{Rvec1} = P @var{Rvec2}. +@item A Beta Ap Rvec1 Rvec2 +Same with @ref{mt_gkz.pfaff_eq}. +@end table + +@comment --- ここで関数の詳しい説明 --- +@comment --- @itemize〜@end itemize は箇条書き --- +@comment --- @bullet は黒点付き --- +@itemize @bullet +@item +It returns the contiguity relation between +@var{Rvec1} and @var{Rvec2} +@end itemize + +@comment --- @example〜@end example は実行例の表示 --- +Example: +@example +[1883] import("mt_gkz.rr"); +[3200] PP=mt_gkz.contiguity(A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]], + Beta=[-g1,-g2,-c], + Ap = [[1,1,0,0],[0,0,1,1],[0,0,0,0]], + Rvec1 = [[1,0,0,0],[0,0,1,0]], + Rvec2 = [[0,0,1,0],[1,0,0,0]]); +@end example + +@comment --- 参照(リンク)を書く --- +@table @t +@item Refer to +@ref{mt_gkz.pfaff_eq} +@ref{mt_gkz.fvec_to_conn_mat} +@end table +@comment おわり. + @comment ---------- New Chapter --------------- @node b function,,, Top @chapter b function @@ -697,6 +756,7 @@ Example: * mt_gkz.get_check_fvec:: * mt_gkz.get_bf_step_up:: * mt_gkz.mytoric_ideal:: +* mt_gkz.cbase_by_euler:: @end menu @node some utility functions,,, utilities @@ -713,6 +773,7 @@ Example: @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 +@node mt_gkz.cbase_by_euler,,, some utility functions @findex mt_gkz.reduce_by_toric @findex mt_gkz.tk_base_equal @@ -725,6 +786,7 @@ Example: @findex mt_gkz.get_check_fvec @findex mt_gkz.get_bf_step_up @findex mt_gkz.mytoric_ideal +@findex mt_gkz.cbase_by_euler @comment --- @example〜@end example は実行例の表示 --- We only show examples on these functions. As for details, please see @@ -789,6 +851,12 @@ the source code. [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. +[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 @@ -1194,7 +1262,7 @@ 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 @@ -1208,9 +1276,11 @@ 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 --- @@ -1227,7 +1297,7 @@ 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. @@ -1238,20 +1308,20 @@ 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 @@ -1271,7 +1341,7 @@ 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. @@ -1293,7 +1363,7 @@ 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]] @@ -1316,7 +1386,7 @@ 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. @@ -1351,7 +1421,7 @@ 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 @@ -1360,7 +1430,7 @@ 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. @@ -1413,9 +1483,9 @@ 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 --- @@ -1425,13 +1495,13 @@ 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]]] @@ -1448,7 +1518,7 @@ 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.