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version 1.2, 2021/01/20 08:17:54

version 1.7, 2022/01/13 02:38:00

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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 1.6 2021/12/11 11:40:45 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

Line 19

@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

Line 38

@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

Line 123 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

Line 134 systems, Linear Algebra and its Applications (LAA), 42

Line 136 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})

Line 151 the parameter vector of the GKZ system.

Line 154 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

Line 168 the set of the cocycles standing for Av_1, Av_2, ...,

Line 171 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.

Line 225 Option @var{cg}. A constant matrix given by this optio

Line 231 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 ---

Line 298 stopped in step_up at line 342 in file "./mt_gkz/saito

Line 308 stopped in step_up at line 342 in file "./mt_gkz/saito

@findex mt_gkz.ff

@table @t

@item mt_gkz.ff(@var{Rvec0},@var{A},@var{Beta},@var{Ap})

@item mt_gkz.ff(@var{Rvec0},@var{A},@var{Ap},@var{Beta})

@item mt_gkz.ff1(@var{Rvec0},@var{A},@var{Beta},@var{Ap})

@item mt_gkz.ff2(@var{Rvec0},@var{A},@var{Beta},@var{Ap},@var{BF},@var{C})

:: @code{ff} returns a differential operator whose action to 1 gives

Line 556 obtains connection matrices.

Line 566 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

Line 697 Example:

Line 756 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

Line 713 Example:

Line 773 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

Line 725 Example:

Line 786 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

Line 789 the source code.

Line 851 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

Line 1194 Here, @math{n_{2_{1_{3_1}}}} and @math{d_{6_1}} are ar

Line 1262 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

Line 1208 The number of polynomial factors in the integrand. see

Line 1276 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 ---

Line 1227 Example:

Line 1297 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.

Line 1238 Example:

Line 1308 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

@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

Line 1271 A positive integer.

Line 1341 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.

Line 1293 Example:

Line 1363 Example:

[2927] N=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]]

Line 1316 intersection matrix is given by

Line 1386 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.

Line 1351 a list of variables of P.

Line 1421 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

Line 1360 If W is chose so that there are several initial terms,

Line 1430 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.

Line 1413 As for the definition of the standard pair, see Chapte

Line 1483 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 ---

Line 1425 Example: An example of a non-unimodular triangulation

Line 1495 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]]]

Line 1448 are 2,1 and 1. Moreover, the top-dimensional standard

Line 1518 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.

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