File: [local] / OpenXM / src / asir-contrib / packages / doc / mt_gkz / mt_gkz-en.texi (download)
Revision 1.8, Sun Feb 13 05:59:56 2022 UTC (2 years, 4 months ago) by takayama
Branch: MAIN
CVS Tags: HEAD
Changes since 1.7: +21 -4
lines
A new function contiguity_by_fvec.
[3366] Fvec411=mt_gkz.rvec_to_fvec(Rvec411=[[1,1,0]],
A=[[1,1,1],[1,0,1],[0,1,1]],
Ap=[[1,1,1],[0,0,0],[0,0,0]],
Beta=[eps,-eps*del,-eps*del])$
Fvec411d=[mt_gkz.dmul(dx1,Fvec411[0],[x1,x2,x3])];
[(dx1^2*dx2)/(eps^2-eps)]
[3367] mt_gkz.contiguity_by_fvec(A,Beta,Ap,Fvec411d,Fvec411);
1 .ooo
[ ((del+1)*eps-1)/(x1) ]
|
%% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.8 2022/02/13 05:59:56 takayama Exp $
%% xetex mt_gkz-en.texi (.texi までつける. )
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%% 参考: http://www.fan.gr.jp/~ring/doc/texinfo/texinfo-ja_14.html#SEC183
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%% 英語版, 以下コメントは @comment で始める. \input texinfo 以降は普通の tex 命令は使えない.
\input texinfo-ja
@iftex
@catcode`@#=6
@def@fref#1{@xrefX[#1,,@code{#1},,,]}
@def@b#1{{@bf #1}}
@catcode`@#=@other
@end iftex
@overfullrule=0pt
@documentlanguage en
@c -*-texinfo-*-
@comment --- おまじない終り ---
@comment --- GNU info ファイルの名前 ---
@setfilename asir-contrib-mt_gkz_man
@comment --- タイトル ---
@settitle GKZ hypergeometric system
@comment --- おまじない ---
@ifinfo
@macro fref{name}
@ref{\name\,,@code{\name\}}
@end macro
@end ifinfo
@titlepage
@comment --- おまじない終り ---
@comment --- タイトル, バージョン, 著者名, 著作権表示 ---
@title GKZ hypergeometric system
@subtitle Pfaffian system (Pfaff equation), contiguity relations, cohomology intersection
@subtitle Version 1.0
@subtitle February 13, 2022
@author by S-J. Matsubara-Heo, N.Takayama
@page
@vskip 0pt plus 1filll
Copyright @copyright{} Risa/Asir committers
2004--2020. All rights reserved.
@end titlepage
@comment --- おまじない ---
@synindex vr fn
@comment --- おまじない終り ---
@comment --- @node は GNU info, HTML 用 ---
@comment --- @node の引数は node-name, next, previous, up ---
@node Top,, (dir), (dir)
@comment --- @menu は GNU info, HTML 用 ---
@comment --- chapter 名を正確に並べる ---
@menu
* About this document::
* Pfaff equation::
* b function::
* Utilities::
* Index::
@end menu
@comment --- chapter の開始 ---
@comment --- 親 chapter 名を正確に. 親がない場合は Top ---
@node About this document,,, Top
@chapter About this document
This document explains Risa/Asir functions for GKZ hypergeometric system
(A-hypergeometric system). @* @comment 強制改行
Loading the package:
@example
import("mt_gkz.rr");
@end example
@noindent
References cited in this document.
@itemize @bullet
@item [MT2020]
Saiei-Jaeyeong Matsubara-Heo, Nobuki Takayama,
Algorithms for Pfaffian Systems and Cohomology Intersection Numbers of Hypergeometric Integrals,
Lecture Notes in Computer Science 12097 (2020), 73--84.
Errata is posted on @uref{http://arxiv.org/abs/???}.
E-attachments can be obtainable at
@uref{http://www.math.kobe-u.ac.jp/OpenXM/Math/intersection2}
@item [GM2020]
Yoshiaki Goto, Saiei-Jaeyeong Matsubara-Heo,
Homology and cohomology intersection numbers of GKZ systems, arXiv:2006.07848
@item [SST1999]
M.Saito, B.Sturmfels, N.Takayama, Hypergeometric polynomials
and integer programming, Compositio Mathematica, 155 (1999), 185--204
@item [SST2000]
M.Saito, B.Sturmfels, N.Takayama, Groebner Deformations of Hypergeometric
Differential Equations. Springer, 2000.
@end itemize
References for maple packages IntegrableConnections and OreMorphisms.
@itemize @bullet
@item [BCEW]
M.Barkatou, T.Cluzeau, C.El Bacha, J.-A.Weil,
IntegrableConnections – a maple package for computing closed form solutions of integrable connections
(2012). @uref{https://www.unilim.fr/pages perso/thomas.cluzeau/Packages/IntegrableConnections/PDS.html}
@item [CQ]
T.Cluzeau and A.Quadrat,
OreMorphisms: A homological algebraic package for factoring, reducing and decomposing linear functional systems (2009). @uref{https://who.rocq.inria.fr/Alban.Quadrat/OreMorphisms/index.html}
@item [CQ08]
T.Cluzeau, A.Quadrat, Factoring and decomposing a class of linear functional
systems, Linear Algebra and its Applications (LAA), 428(1): 324-381, 2008.
@end itemize
@node Pfaff equation,,, Top
@chapter Pfaff equation
@menu
* mt_gkz.pfaff_eq::
* mt_gkz.ff::
* mt_gkz.ff1::
* mt_gkz.ff2::
* mt_gkz.rvec_to_fvec::
* mt_gkz.fvec_to_conn_mat::
* mt_gkz.contiguity::
* mt_gkz.contiguity_by_fvec::
@end menu
@node Pfaff equation for given cocycles,,, Pfaff equation
@section Pfaff equation for given cocycles
@comment **********************************************************
@comment --- 関数 pfaff_eq
@node mt_gkz.pfaff_eq,,, Pfaff equation for given cocycles
@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})
:: It returns the Pfaff equation for the GKZ system defined by @var{A} and @var{Beta} with respect to cocycles defined by @var{Rvec}.
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
a list of coefficients of the Pfaff equation with respect to the direction @var{DirX}
@item A
the matrix A of the GKZ system.
@item Beta
the parameter vector of the GKZ system.
@item Ap
See [MT2020].
@item Rvec
It is used to specify a basis of cocycles as explained below. See also [MT2020].
@item DirX
a list of dxi's.
@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
The independent variables are x1, x2, x3, ...
@item
When @var{Rvec}=[v_1, v_2, ..., v_r] where r is the rank of the GKZ system,
the set of the cocycles standing for Av_1, Av_2, ..., Av_r
(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.
@item
When the columns of @var{A} are expressed as
@math{e_i \otimes \alpha_{i_j}},
the columns of @var{Ap} is
@math{e_i \otimes 0} where $e_i$ is the i-th unit vector.
See [MT2020] on the definition of @var{Ap}.
Here are some examples.
When @var{A} is
@verbatim
[[1,1,0,0],
[0,0,1,1],
[0,1,0,1]]
@end verbatim
@var{Ap} is
@verbatim
[[1,1,0,0],
[0,0,1,1],
[0,0,0,0]] <-- zero row
@end verbatim
When @var{A} is
@verbatim
[[1,1,1,0,0,0],
[0,0,0,1,1,1],
[0,1,0,0,1,0],
[0,0,1,0,0,1]
]
@end verbatim
@var{Ap} is
@verbatim
[[1,1,1,0,0,0],
[0,0,0,1,1,1],
[0,0,0,0,0,0], <-- zero row
[0,0,0,0,0,0] <-- zero row
]
@end verbatim
See also page 223 of [SST2000].
@item
Option @var{xrule}. When the option @var{xrule} is given,
the x variables specified by this option are specialized to numbers.
@item
Option @var{shift}. When the matrix @var{A} is not normal
(the associated toric ideal is not normal), a proper shift vector
must be given to obtain an element of the b-ideal. Or, use the option
@var{b_ideal} below. See [SST1999] on the theory.
@item
Option @var{b_ideal}. When the matrix @var{A} is not normal,
the option @code{b_ideal=1} obtains b-ideals and the first element
of each b-ideal is used as the b-function. The option @var{shift}
is ignored.
@item
Option @var{cg}. A constant matrix given by this option is used
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 ---
Example: Gauss hypergeometric system, see [GM2020] example ??.
@example
[1883] import("mt_gkz.rr");
[2657] PP=mt_gkz.pfaff_eq(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]],
Rvec = [[1,0,0,0],[0,0,1,0]],
DirX=[dx4,dx3] | xrule=[[x1,1],[x2,1]],
cg=matrix_list_to_matrix([[1,0],[-1,1]]))$
Bfunctions=[s_1*s_2-s_1*s_3+s_1^2,s_1*s_3,s_2^2+(-s_3+s_1)*s_2,s_3*s_2]
-- snip --
[2658] PP[0];
[ (g2*x3-g2)/(x4-x3) (g2*x3)/(x4-x3) ]
[ ((-g2*x3-c+g2)*x4+(c-g1)*x3+g1)/(x4^2-x3*x4)
((-g2*x3-c)*x4+(c-g1)*x3)/(x4^2-x3*x4) ]
[2659] PP[1];
[ (-g2*x4+g2)/(x4-x3) (-g2*x4)/(x4-x3) ]
[ ((g2*x3+c-g2-1)*x4+(-c+g1+1)*x3-g1)/(x3*x4-x3^2)
((g2*x3+c-g2-1)*x4+(-c+g1+g2+1)*x3)/(x3*x4-x3^2) ]
@end example
@*
Example: The role of shift.
When the toric ideal is not normal, a proper shift vector
must be given with the option @code{shift} to find an element of the b-ideal.
@example
[1882] load("mt_gkz.rr");
[1883] A=[[1,1,1,1],[0,1,3,4]];
[[1,1,1,1],[0,1,3,4]]
[1884] Ap=[[1,1,1,1],[0,0,0,0]];
[[1,1,1,1],[0,0,0,0]]
[1885] Rvec=[[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
[[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
[2674] P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
| xrule=[[x1,1],[x2,2],[x3,4]] )$
dx remains
stopped in step_up at line 342 in file "./mt_gkz/saito-b.rr"
342 if (type(dn(Ans)) > 1) error("dx remains");
(debug) quit
// Since the toric ideal for A is not normal, it stops with the error.
[2675] P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
| shift=[1,0],xrule=[[x1,1],[x2,2],[x3,4]])$
// It works.
@end example
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.ff1}
@ref{mt_gkz.ff2}
@ref{mt_gkz.ff}
@ref{mt_gkz.rvec_to_fvec}
@end table
@comment mt_gkz.pfaff_eq の説明おわり. あとはこれの繰り返し.
@comment --- 個々の関数の説明 ---
@comment --- section 名を正確に ---
@node mt_gkz.ff2,,, Pfaff equation for given cocycles
@node mt_gkz.ff1,,, Pfaff equation for given cocycles
@node mt_gkz.ff,,, Pfaff equation for given cocycles
@subsection @code{mt_gkz.ff2}, @code{mt_gkz.ff1}, @code{mt_gkz.ff}
@comment --- 索引用キーワード
@findex mt_gkz.ff2
@findex mt_gkz.ff1
@findex mt_gkz.ff
@table @t
@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
the cocycle defined by @var{Rvec0}
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
@code{ff} returns a differential operator whose action to 1 of @math{M_A(\beta)}
gives the cocycle defined by @var{Rvec0}.
@item return
@code{ff1} returns a composite of step-down operators for the positive part
of @var{Rvec0}
@item return
@code{ff2} returns a composite of step-up operators for the positive part
of @var{Rvec0}
@item Rvec0
An element of @var{Rvec} explained in @ref{mt_gkz.pfaff_eq}.
@item BF
the list of b-functions to all directions.
@item C
the list of the step up operators for all a_1, a_2, ..., a_n.
@end table
Other arguments are same with those of @code{pfaff_eq}.
@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
The function @code{ff} generates the list of b-functions and the list of
step up operators and store them in the cache variable.
They can be obtained by calling as @code{S=mt_gkz.get_bf_step_up()}
where S[0] is the list of b-functions and S[1] is the list of step up
operators.
Step up operators are obtained by the algorithm given in [SST1999].
@item
Option nf. When nf=1, the output operator is reduced to the normal form
with respect to the Groebner basis of the GKZ system of the graded reverse
lexicographic order.
@item
Option shift. See @ref{mt_gkz.pfaff_eq}.
@item
Internal info: The function @code{mt_gkz.bb} gives the constant so that
the step up and step down operators (contiguity operators) give
contiguity relations for the integral representation in [MT2020].
Note that @code{mt_gkz.ff1} and @code{mt_gkz.ff2} give contiguity
relations which are constant multiple of those for hypergeometric
polynomials.
@item
Internal info: @code{mt_gkz.step_up} generates step up operators
of [SST1999] from b-functions by utilizing @code{mt_gkz.bf2euler}
and @code{mt_gkz.toric}.
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example: Step up operators compatible with the integral representation in [MT2020].
The function hgpoly_res defined in @code{check-by-hgpoly.rr} returns
a multiple of the hypergeometric polynomial which agrees with
the residue times a power of @math{2\pi \sqrt{-1}}
of the integral representation.
See [SST1999].
@example
[1883] import("mt_gkz.rr")$
[3175] load("mt_gkz/check-by-hgpoly.rr")$
[3176] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
[3177] B=newvect(3,[5,4,7])$ Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
[3179] Beta=[b1,b2,b3]$ R=[0,0,-1,0]$
[3180] F2=hgpoly_res(A,B,2); // HG polynomial. 2 is the number of e_i's.
10*x1^2*x2^3*x4^4+20*x1*x2^4*x3*x4^3+6*x2^5*x3^2*x4^2
[3182] mt_gkz.ff(R,A,Ap,Beta); // the operator standing for R
(x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
[3184] S=mt_gkz.get_bf_step_up(A); // b-function and non-reduced step up op's
[[ s_1*s_2-s_1*s_3+s_1^2 s_1*s_3 s_2^2+(-s_3+s_1)*s_2 s_3*s_2 ],
[ x2*x3*dx4+x1*x3*dx3+x1*x2*dx2+x1^2*dx1+x1
x2*x4*dx4+x1*x4*dx3+x2^2*dx2+x1*x2*dx1+x2
x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3
x4^2*dx4+x3*x4*dx3+x2*x4*dx2+x2*x3*dx1+x4 ]]
[3185] Fvec=mt_gkz.ff2(R,A,Beta,Ap,S[0],S[1]);
(x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
[3188] Fvec = base_replace(Fvec,assoc(Beta,vtol(B)));
1/3*x3*x4*dx4+1/3*x3^2*dx3+1/3*x1*x4*dx2+1/3*x1*x3*dx1+1/3*x3
[3189] R32d = odiff_act(Fvec,F2,[x1,x2,x3,x4]); // Act Fvec to the hg-poly
10*x1^3*x2^2*x4^5+50*x1^2*x2^3*x3*x4^4+50*x1*x2^4*x3^2*x4^3+10*x2^5*x3^3*x4^2
[3190] red(R32d/hgpoly_res(A,B+newvect(3,[0,1,0]),2));
// R32d agrees with the HG polynomial with Beta=[5,4,7]+[0,1,0].
1
@end example
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.pfaff_eq}
@end table
@comment おわり.
@comment --- 個々の関数の説明 --- Ref:2020-11-09-tw-cohom-progs.goodnotes
@comment --- section 名を正確に ---
@node mt_gkz.rvec_to_fvec,,, Pfaff equation for given cocycles
@subsection @code{mt_gkz.rvec_to_fvec}
@comment --- 索引用キーワード
@findex mt_gkz.rvec_to_fvec
@table @t
@item mt_gkz.rvec_to_fvec(@var{Rvec},@var{A},@var{Ap},@var{Beta})
:: It returns a set of differential operators standing for @var{Rvec}.
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
It returns a set of differential operators of which action to
@math{1 \in M_A(\beta)} give cocycles specified by @var{Rvec}.
@item A, Ap, Beta
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
Internal info: this function builds the set of operators by calling
@ref{mt_gkz.ff}.
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example: The following two expressions are congruent because
@math{2a_1-a_2-a_3+a_4=a_1} for this @code{A}.
@example
[1883] import("mt_gkz.rr");
[3191] mt_gkz.rvec_to_fvec([[2,-1,-1,1],[0,0,1,0]],
[[1,1,0,0],[0,0,1,1],[0,1,0,1]],
[[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
[(x2*x3*x4^2*dx1^2*dx4^3+((x1*x3*x4^2+x2*x3^2*x4)*dx1^2*dx3
+(x1*x2*x4^2+x2^2*x3*x4)*dx1^2*dx2+(x1^2*x4^2+2*x1*x2*x3*x4+x2^2*x3^2)*dx1^3
+(x1*x4^2+3*x2*x3*x4)*dx1^2)*dx4^2+(x1*x3^2*x4*dx1^2*dx3^2
+((x1^2*x3*x4+x1*x2*x3^2)*dx1^3+(3*x1*x3*x4+x2*x3^2)*dx1^2)*dx3
+x1*x2^2*x4*dx1^2*dx2^2+((x1^2*x2*x4+x1*x2^2*x3)*dx1^3
+(3*x1*x2*x4+x2^2*x3)*dx1^2)*dx2+x1^2*x2*x3*dx1^4
+(x1^2*x4+3*x1*x2*x3)*dx1^3+(x1*x4+x2*x3)*dx1^2)*dx4)
/(b3*b2*b1^3+(b3*b2^2+(-b3^2-2*b3)*b2)*b1^2+(-b3*b2^2+(b3^2+b3)*b2)*b1),
(dx3)/(b2)]
[3192] mt_gkz.rvec_to_fvec([[1,0,0,0],[0,0,1,0]],
[[1,1,0,0],[0,0,1,1],[0,1,0,1]],
[[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
[(dx1)/(b1),(dx3)/(b2)]
@end example
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.pfaff_eq}
@end table
@comment おわり.
@comment --- fvec_to_conn_mat
@comment --- section 名を正確に ---
@node mt_gkz.fvec_to_conn_mat,,, Pfaff equation for given cocycles
@subsection @code{mt_gkz.fvec_to_conn_mat}
@comment --- 索引用キーワード
@findex mt_gkz.fvec_to_conn_mat
@table @t
@item mt_gkz.fvec_to_conn_mat(@var{Fvec},@var{A},@var{Beta},@var{DirX})
:: It returns the coefficient matrices of the basis
@var{Fvec} or @var{DirX}[I]*@var{Fvec} in terms of the set of the standard basis.
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
It returns the coefficient matrices of the basis
@var{Fvec} or @var{DirX}[I]*@var{Fvec} in terms of the set of the standard basis of the Groebner basis explained below.
@item A Beta
Same with @ref{mt_gkz.pfaff_eq}.
@item DirX
When @var{DirX} is 1, this function returns the matrix which expresses
@var{Fvec} in terms of the set of the standard monomials of
the Groebner basis of the GKZ system in the ring of rational function
coefficients with respect to the graded reverse lexicographic order.
In other cases, it returns the coefficient matrices of
@var{DirX}[I]'s*@var{Fvec} in terms of the set of the standard basis of the Groebner basis.
@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 utilizes a Groebner basis computation by the package @code{yang.rr}
and @code{yang.reduction} to obtain connection matrices.
@item
This function calls some utility functions
@code{mt_gkz.dmul(Op1,Op2,XvarList)} (multiplication of @code{Op1} and @code{Op2}
and @code{mt_gkz.index_vars(x,Start,End | no_=1)}
which generates indexed variables without the underbar ``_''.
@item
We note here some other utility functions in this section:
@code{mt_gkz.check_compatibility(P,Q,X,Y)},
which checkes if the sytem d/dX-P, d/dY-Q is compatible.
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example: The following example illustrates how mt_gkz.pfaff_eq
obtains connection matrices.
@example
[1883] import("mt_gkz.rr");
[3201] V=mt_gkz.index_vars(x,1,4 | no_=1);
[x1,x2,x3,x4]
[3202] mt_gkz.dmul(dx1,x1^2,V);
x1^2*dx1+2*x1
[3204] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
Beta= [b1,b2,b3]$
Rvec = [[1,0,0,0],[0,0,1,0]]$
Fvec = mt_gkz.rvec_to_fvec(Rvec,A,Ap,Beta)$
/* Express cocyles Rvec
by elements Fvec in the Weyl algebra by contiguity relations. */
Cg = matrix_list_to_matrix([[1,0],[1,-1]])$
[3208] NN=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,1);
// Express Fvec by the standard monomials Std=NN[1].
1 ooo 2 .ooo
[[ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
[ (-x4)/(b1*x2) (1)/(x3) ],[dx4,1]]
[3209] Std=NN[1];
[dx4,1]
[3173] NN=NN[0];
[ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
[ (-x4)/(b2*x3) (1)/(x3) ]
[3174] NN1=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,dx1)[0];
// Express dx1*Fvec by the standard monomials Std.
1 ooo 2 .ooo
[ ((2*b1+b2-b3-1)*x1*x4^2+(-b1+b3+1)*x2*x3*x4)/(b1*x1^3*x4-b1*x1^2*x2*x3)
((b1^2+(-2*b3-1)*b1-b3*b2+b3^2+b3)*x1*x4
+(-b1^2+(2*b3+1)*b1-b3^2-b3)*x2*x3)/(b1*x1^3*x4-b1*x1^2*x2*x3) ]
[(b1 (-b1*x1*x4^2-b2*x2*x3*x4)/(b2*x1^2*x3*x4-b2*x1*x2*x3^2)
(b1*x1*x4+(-b1+b3)*x2*x3)/(x1^2*x3*x4-x1*x2*x3^2) ]
[3188] P1=map(red,Cg*NN1*matrix_inverse(NN)*matrix_inverse(Cg));
[ ((-b2*x3+(b1+b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3)/(x1^2*x4-x1*x2*x3)
(b2*x3*x4)/(x1^2*x4-x1*x2*x3) ]
[ ((-b2*x3+(b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3+b1*x1*x2)/(x1^2*x4-x1*x2*x3)
((b2*x3+b1*x1)*x4)/(x1^2*x4-x1*x2*x3) ]
[3191] mt_gkz.pfaff_eq(A,Beta,Ap,Rvec,[dx1]|cg=Cg)[0]-P1;
[ 0 0 ]
[ 0 0 ] // P1 agrees with the output of mt_gkz.pfaff_eq.
@end example
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.pfaff_eq}
@end table
@comment おわり.
@comment --- contiguity
@comment --- section 名を正確に ---
@node mt_gkz.contiguity,,, Pfaff equation for given cocycles
@node mt_gkz.contiguity_by_fvec,,, Pfaff equation for given cocycles
@subsection @code{mt_gkz.contiguity}, @code{mt_gkz.contiguity_by_fvec}
@comment --- 索引用キーワード
@findex mt_gkz.contiguity
@findex mt_gkz.contiguity_by_fvec
@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}.
@item mt_gkz.contiguity_by_fvec(@var{A},@var{Beta},@var{Ap},@var{Fvec1},@var{Fvec2})
:: It returns the coefficient matrix P that satisfies
@var{Fvec1} = P @var{Fvec2}.
@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}
or @var{Fvec1}=P @var{Fvec2}
@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]]);
[3366] Fvec411=mt_gkz.rvec_to_fvec(Rvec411=[[1,1,0]],
A=[[1,1,1],[1,0,1],[0,1,1]],
Ap=[[1,1,1],[0,0,0],[0,0,0]],
Beta=[eps,-eps*del,-eps*del])$
Fvec411d=[mt_gkz.dmul(dx1,Fvec411[0],[x1,x2,x3])];
[(dx1^2*dx2)/(eps^2-eps)]
[3367] mt_gkz.contiguity_by_fvec(A,Beta,Ap,Fvec411d,Fvec411);
1 .ooo
[ ((del+1)*eps-1)/(x1) ]
@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
@menu
* mt_gkz.bf::
* mt_gkz.bf::
@end menu
@node b function and facet polynomial,,, b function
@section b function and facet polynomial
@comment ------- bf
@comment **********************************************************
@comment --- 個々の関数の説明 ---
@comment --- section 名を正確に ---
@node mt_gkz.bf,,, b function and facet polynomial
@subsection @code{mt_gkz.bf}
@comment --- 索引用キーワード
@findex mt_gkz.bf
@table @t
@item mt_gkz.bf(@var{A},@var{Facet_poly},@var{II0})
:: It returns the b-function with respect to the direction @var{II0}.
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
It returns the b-function introduced Saito with respect to the direction @var{II0} in case of @var{A} is normal or an element of b-ideal when a proper shift vector is given in case of @var{A} is not normal.
@item A
the matrix A of the GKZ system.
@item Facet_poly
The set of facet polynomials of the convex hull of @var{A}.
@item II0
Direction expressed as 0, 1, 2, ... (not 1, 2, 3, ...) to obtain the b function.
@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
See [SST1999] on the b-function introduced Saito and b-ideal.
@item
The facet polynomial must be primitive.
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example:
@example
[1883] import("mt_gkz.rr");
[3193] A;
[[1,1,0,0],[0,0,1,1],[0,1,0,1]]
[3194] Fpoly=mt_gkz.facet_poly(A);
[[s_3,s_1,s_2-s_3+s_1,s_2],[[0,0,1],[1,0,0],[1,1,-1],[0,1,0]]]
[3196] mt_gkz.bf(A,Fpoly,0);
s_1*s_2-s_1*s_3+s_1^2
[3197] mt_gkz.bf(A,Fpoly,1);
s_1*s_3
@end example
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.ff}
@ref{mt_gkz.facet_poly}
@end table
@comment おわり.
@comment ------ facet_poly
@comment --- 個々の関数の説明 ---
@comment --- section 名を正確に ---
@node mt_gkz.facet_polyl,,, b function and facet polynomial
@subsection @code{mt_gkz.facet_poly}
@comment --- 索引用キーワード
@findex mt_gkz.facet_poly
@table @t
@item mt_gkz.facet_poly(@var{A})
:: It returns the set of facet polynomials and their normal vectors of
the cone defined by @var{A}.
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
It returns the set of facet polynomials and their normal vectors of
the cone generated by the column vectors of the matrix @var{A}.
@item A
the matrix A of the GKZ system.
@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
The facet polynomial f is primitive. In other words,
all f(a_i) is integer and min f(a_i)=1 for a_i's not being on f=0.
where a_i is the i-th column vector of the matrix @var{A}.
It can be checked by @code{mt_gkz.is_primitive(At,Facets)}
where @var{At} is the transpose of @var{A} and
@var{Facets} is the second return value of this function.
@item
This function utilizes the system polymake @uref{https://polymake.org}
on our server.
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example:
@example
[1883] import("mt_gkz.rr");
[1884] mt_gkz.facet_poly([[1,1,1,1],[0,1,2,3]]);
oohg_native=0, oohg_curl=1
[[s_2,-s_2+3*s_1],[[0,1],[3,-1]]]
@end example
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.bf}
@end table
@comment おわり.
@comment ---------- New Chapter ---------------
@node utilities,,, Top
@chapter Utilities
@menu
* mt_gkz.reduce_by_toric::
* mt_gkz.tk_base_equal::
* mt_gkz.dp_op_to_coef_vec::
* mt_gkz.yang_gkz_buch::
* mt_gkz.p_true_nf_rat::
* mt_gkz.mdiff::
* mt_gkz.dvar::
* mt_gkz.ord_xi::
* 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
@section Some utility functions
@node mt_gkz.reduce_by_toric,,, some utility functions
@node mt_gkz.tk_base_equal,,, some utility functions
@node mt_gkz.dp_op_to_coef_vec,,, some utility functions
@node mt_gkz.yang_gkz_buch,,, some utility functions
@node mt_gkz.p_true_nf_rat,,, some utility functions
@node mt_gkz.mdiff,,, some utility functions
@node mt_gkz.dvar,,, some utility functions
@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
@node mt_gkz.cbase_by_euler,,, some utility functions
@findex mt_gkz.reduce_by_toric
@findex mt_gkz.tk_base_equal
@findex mt_gkz.dp_op_to_coef_vec
@findex mt_gkz.yang_gkz_buch
@findex mt_gkz.p_true_nf_rat
@findex mt_gkz.mdiff
@findex mt_gkz.dvar
@findex mt_gkz.ord_xi
@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
the source code.
@example
[1883] import("mt_gkz.rr");
[2667] mt_gkz.dvar([x1,x2]); // it generates variables starting with d
[dx1,dx2]
[2669] mt_gkz.p_true_nf_rat((1/3)*x^3-1,[x^2-1],[x],0);
[x-3,3] // p_true_nf does not accept rational number coefficients
[2670] A=[[1,1,1,1],[0,1,3,4]];
[[1,1,1,1],[0,1,3,4]]
[2671] mt_gkz.reduce_by_toric(dx3^4,A);
dx1*dx4^3 // reduction by toric ideal defined by A
[2672] nk_toric.toric_ideal(A);
[-x1*x4+x2*x3,-x2*x4^2+x3^3,x2^2*x4-x1*x3^2,-x1^2*x3+x2^3]
[2673] mt_gkz.yang_gkz_buch(A,[b1,b2]); // Groebner basis of GKZ system by yang.rr
1 o 2 ..o 3 ..oooooooo 4 o 6 ooo 9 o
[[[(x2)*<<0,1,0,0>>+(3*x3)*<<0,0,1,0>>+ ---snip ---*<<0,0,0,0>>,1]],
[dx1,dx2,dx3,dx4],
[(1)*<<0,0,0,2>>,(1)*<<0,0,1,0>>,(1)*<<0,0,0,1>>,(1)*<<0,0,0,0>>]]
[2674] mt_gkz.dp_op_to_coef_vec([x1*<<1,0>>+x1*x2*<<0,1>>,x1+1],[<<1,0>>,<<0,1>>]);
// x1+1 is the denominator
[ (x1)/(x1+1) (x1*x2)/(x1+1) ]
[2675] mt_gkz.tk_base_is_equal([1,2],[1,2]);
1
[2676] mt_gkz.tk_base_is_equal([1,2],[1,x,y]);
0
[2677] mt_gkz.mdiff(sin(x),x,1);
cos(x)
[2678] mt_gkz.mdiff(sin(x),x,2); //2nd derivative
-sin(x)
[3164] mt_gkz.ord_xi(V=[x1,x2,x3],II=1);
// matrix to define graded lexicographic order so that V[II] is the smallest.
[ 1 1 1 ]
[ 0 -1 0 ]
[ -1 0 0 ]
[3166] load("mt_gkz/check-by-hgpoly.rr");
[3187] check_0123(); // check the pfaffian for the A below by hg-polynomial.
A=[[1,1,1,1],[0,1,2,3]]
Ap=[[1,1,1,1],[0,0,0,0]]
--- snip ---
Bfunctions= --- snip ---
0 (vector) is expected:
[[ 0 0 0 ],[ 0 0 0 ]]
[3188] mt_gkz.get_check_fvec();
// get the basis of cocycles used in terms of differential operators.
[1,(dx4)/(b1),(dx4^2)/(b1^2-b1)]
[3189] mt_gkz.clear_bf();
0
[3190] mt_gkz.get_bf_step_up(A=[[1,1,1,1],[0,1,2,3]]);
// b-functions and step-up operators.
// Option b_ideal=1 or shift=... may be used for non-normal case.
[[ -s_2^3+(9*s_1-3)*s_2^2+ ---snip---
-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.
[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
@comment ここから追加版
@node Cohomology intersection numbers,,, Top
@chapter Cohomology intersection numbers
@menu
* mt_gkz.kronecker_prd::
* mt_gkz.secondary_eq::
* mt_gkz.generate_maple_file_IC::
* mt_gkz.generate_maple_file_MR::
* mt_gkz.principal_normalizing_constant::
@end menu
@node Secondary equation,,, Cohomology intersection numbers
@section Secondary equation
@comment **********************************************************
@comment --- 関数 pfaff_eq
@node mt_gkz.kronecker_prd,,, Secondary equation
@subsection @code{mt_gkz.kronecker_prd}
@comment --- 索引用キーワード
@findex mt_gkz.kronecker_prd
@table @t
@item mt_gkz.kronecker_prd(@var{A},@var{B})
:: It returns the Kronecker product of @var{A} and @var{B}.
@end table
@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} (@uref{https://en.wikipedia.org/wiki/Kronecker_product}).
@item A,B
list
@end table
@comment --- @example〜@end example は実行例の表示 ---
@example
[2644] A=[[a,b],[c,d]];
[[a,b],[c,d]]
[2645] B=[[e,f],[g,h]];
[[e,f],[g,h]]
[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 ]
[ g*c h*c g*d h*d ]
@end example
@node mt_gkz.secondary_eq,,, Secondary equation
@subsection @code{mt_gkz.secondary_eq}
@comment --- 索引用キーワード
@findex mt_gkz.secondary_eq
@table @t
@item mt_gkz.secondary_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
:: It returns the secondary equation with respect to cocycles defined by Rvec.
@end table
@table @var
@item return
a list of coefficients of the Pfaffian system corresponding to the secondary equation (cf. equation (8) of [MT2020]).
@item A,Beta,Ap,Rvec,DirX
see @code{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
The secondary equation is originally a Pfaffian system for an unkwon @math{r} by @math{r} matrix @math{I} with @math{r=}length(Rvec). We set @math{Y=(I_{11},I_{12},...,I_{1r},I_{21},I_{22},...)^T}. Then, the secondary equation can be seen as a Pfaffian system @math{{dY\over dx_i}=A_iY} with DirX=@math{\{dx_i\}_i}. The function mt_gkz.secondary_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX}) outputs a list obtained by aligning the matrices @math{A_i}.
@item
Let @math{F:=(\omega_i)_i} be a column vector whose entries are given by the cohomology classes specified by entries of Rvec. Then, @code{pfaff_eq} computes the Pfaffian matrices @math{P_i} so that @math{{dF\over dx_i}=P_iF}. If @math{Q_i} denotes the matrix obtained by replacing Beta by -Beta, we have @math{A_i=}@code{mt_gkz.kronecker_prd}(E,@math{P_i})+@code{mt_gkz.kronecker_prd}(@math{Q_i},E) where E is the identity matrix of size length(Rvec).
@item Options xrule, shift, b_ideal,cg.
Same as @code{pfaff_eq}.
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example:
@example
[2647] Beta=[b1,b2,b3]$
[2648] DirX=[dx1,dx4]$
[2649] Rvec=[[1,0,0,0],[0,0,1,0]]$
[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=mt_gkz.secondary_eq(A,Beta,Ap,Rvec,DirX|xrule=Xrule)$
--snip--
[2654] length(P);
2
[2655] P[0];
[[(-2*x1^3*x4^2+4*x1^2*x4-2*x1)/(x1^4*x4^2-2*x1^3*x4+x1^2),(b2*x4)/(x1^2*x4-x1),
(-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
((b2-4/3)*x1^2*x4^2+(-b1-b2+8/3)*x1*x4+b1-4/3)/(x1^3*x4^2-2*x1^2*x4+x1),0,
(-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
((-b2-2/3)*x1^2*x4^2+(b1+b2+4/3)*x1*x4-b1-2/3)/(x1^3*x4^2-2*x1^2*x4+x1),
(b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]
<--- Paffian matrix in x1 direction.
[2656] P[1];
[[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
((b2-1/3)*x1^2*x4^2+(-b1-b2+2/3)*x1*x4+b1-1/3)/(x1^2*x4^3-2*x1*x4^2+x4),0,
(-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
((-b2+1/3)*x1^2*x4^2+(b1+b2-2/3)*x1*x4-b1+1/3)/(x1^2*x4^3-2*x1*x4^2+x4),
(b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]]
<--- Paffian matrix in x4 direction.
@end example
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.pfaff_eq}
@end table
@comment おわり.
@node mt_gkz.generate_maple_file_IC,,, Secondary equation
@subsection @code{mt_gkz.generate_maple_file_IC}
@comment --- 索引用キーワード
@findex mt_gkz.generate_maple_file_IC
@table @t
@item mt_gkz.generate_maple_file_IC(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
:: It returns the maple input for a solver of a Pfaffian system IntegrableConnections[RationalSolutions].
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
a maple input file for the function IntegrableConnections[RationalSolutions] (cf. [BCEW]) for the Pfaffian system mt_gkz.secondary_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX}).
@item A,Beta,Ap,Rvec,DirX
see @code{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
A maple package IntegrableConnections is available in [BCEW]. In order to use IntegrableConnections, you need to add the global path to the file IntegrableConnections.m to libname on maple. See [BCEW].
@item
If Beta contains unkwon variables, they are regarded as generic parameters. For example, if Beta=[b1,b2,1/5,1/7,b5,...], parameters are [b1,b2,b5,...].
@item Options xrule, shift, b_ideal,cg.
Same as @code{pfaff_eq}.
@item Option filename.
You can specify the file name by specifying the option variable filename. If you do not specify it, @code{generate_maple_file_IC} generates a file "auto-generated-IC.ml".
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example:
@example
[2681] Beta=[b1,b2,1/3]$
[2682] DirX=[dx1,dx4]$
[2683] Rvec=[[1,0,0,0],[0,0,1,0]]$
[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] 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:
with(OreModules);
with(IntegrableConnections);
with(linalg);
C:=[Matrix([[(-2*x1^3*x4^2+4*x1^2*x4-2*x1)/(x1^4*x4^2-2*x1^3*x4+x1^2),
(b2*x4)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
((b2-4/3)*x1^2*x4^2+(-b1-b2+8/3)*x1*x4+b1-4/3)/(x1^3*x4^2-2*x1^2*x4+x1),0,
(-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
((-b2-2/3)*x1^2*x4^2+(b1+b2+4/3)*x1*x4-b1-2/3)/(x1^3*x4^2-2*x1^2*x4+x1),
(b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]),
Matrix([[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
((b2-1/3)*x1^2*x4^2+(-b1-b2+2/3)*x1*x4+b1-1/3)/(x1^2*x4^3-2*x1*x4^2+x4),0,
(-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
((-b2+1/3)*x1^2*x4^2+(b1+b2-2/3)*x1*x4-b1+1/3)/(x1^2*x4^3-2*x1*x4^2+x4),
(b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]])];
RatSols:=RationalSolutions(C,[x1,x4],['param',[b1,b2]]);
/*
If you run the output file on maple, you obtain a rational solution of
the secondary equation.
*/
[b2*(3*b1-1)/(b1*x1^2)]
RatSols:=[3*b2/x1 ]
[3*b2/x1 ]
[3*b2-1 ]
/*
Note that the 4 entries of this vector correspond to entries of a 2 by 2 matrix.
They are aligned as (1,1), (1,2), (2,1) (2,2) from the top.
*/
@end example
@*
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.pfaff_eq}
@end table
@comment おわり.
@node mt_gkz.generate_maple_file_MR,,, Secondary equation
@subsection @code{mt_gkz.generate_maple_file_MR}
@comment --- 索引用キーワード
@findex mt_gkz.generate_maple_file_MR
@table @t
@item mt_gkz.generate_maple_file_MR(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX},@var{D1},@var{D2})
:: It returns the maple input for a solver of a Pfaffian system MorphismsRat[OreMorphisms].
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
a maple input file for the function MorphismsRat[OreMorphisms] (cf. [CQ]) for the Pfaffian system obtained by @code{secondary_eq}. If you run the output file on maple, you obtain a rational solution of the secondary equation.
@item A,Beta,Ap,Rvec,DirX
see @code{pfaff_eq}.
@item D1,D2
Positive integers. D1 (resp. D2) is the upper bound of the degree of the numerator (resp. denominator) of the solution.
@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
We use the same notation as the explanation of @code{generate_maple_file_IC}. Let @math{D} denote the ring of linear differential operators with coeffiecients in the field of rational functions. We consider @math{D}-modules @math{R:=D^{1\times l}/\sum_{dx_i\in DirX}D^{1\times l}(\partial_i E-P_i)} and @math{S:=D^{1\times l}/\sum_{dx_i\in DirX}D^{1\times l}(\partial_i E+Q_i^T)} where @math{l=}length(Rvec). Then, computing a rational solution of the secondary equation is equivalent to computing a @math{D}-morphism from @math{R} to @math{S} represented by rational function matrix (cf. pp12-13 of [CQ08]).
@item
A maple package OreMorphisms is available in [CQ]. In order to use OreMorphisms, you need to add the global path to the file OreMorphisms.m to libname on maple.
@item Options xrule, shift, b_ideal,cg.
Same as @code{pfaff_eq}.
@item Option filename.
You can specify the file name as in @code{generate_maple_file_IC}.
@item
The difference between @code{generate_maple_file_IC} and @code{generate_maple_file_MR} is the appearence of auxilliary variables D1 and D2. If you can guess the degree of the numerator and the denominator of the solution of the secondary equation, MorphismsRat[OreMorphisms] can be faster than RationalSolutions[IntegrableConnections].
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example:
@example
[2668] Beta=[b1,b2,1/3]$
[2669] DirX=[dx1,dx4]$
[2670] Rvec=[[1,0,0,0],[0,0,1,0]]$
[2671] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
[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] 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:
with(OreModules);
with(OreMorphisms);
with(linalg);
Alg:=DefineOreAlgebra(diff=[dx1,x1],diff=[dx4,x4],polynom=[x1,x4],comm=[b1,b2]);
P:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
-Matrix([[((b1+b2-4/3)*x1*x4-b1+4/3)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1)],
[(-b1)/(x1*x4-1),(b1*x4)/(x1*x4-1)],[(b2*x1)/(x1*x4-1),(-b2)/(x1*x4-1)],
[(-b1*x1)/(x1*x4^2-x4),(1/3*x1*x4+b1-1/3)/(x1*x4^2-x4)]]);
Q:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
+Matrix([[((-b1-b2-2/3)*x1*x4+b1+2/3)/(x1^2*x4-x1),(b1)/(x1*x4-1)],
[(b2*x4)/(x1^2*x4-x1),(-b1*x4)/(x1*x4-1)],[(-b2*x1)/(x1*x4-1),(b1*x1)/(x1*x4^2-x4)],
[(b2)/(x1*x4-1),(-1/3*x1*x4-b1+1/3)/(x1*x4^2-x4)]]);
RatSols:=MorphismsRat(P,Q,Alg,0,2,2);
/*
If you run the output file on maple, you obtain a vector RatSols.
RatSols[1] is the rational solution of the secondary equation:
*/
RatSols[1]:=[(1/3)*@math{n_{2_{1_{3_1}}}}*(3*b1-1)/(b1*x1^2*@math{d_{6_1}}) @math{n_{2_{1_{3_1}}}}/(x1*@math{d_{6_1}})]
[@math{n_{2_{1_{3_1}}}}/(x1*@math{d_{6_1}}) (1/3)*@math{n_{2_{1_{3_1}}}}*(3*b2-1)/(b2*@math{d_{6_1}})]
/*
Here, @math{n_{2_{1_{3_1}}}} and @math{d_{6_1}} are arbitrary constants. We can take @math{n_{2_{1_{3_1}}}=3*b2} and @math{d_{6_1}=1} to obtain the rational solution of the secondary equation which is identical to the one obtained from @code{generate_maple_file_IC}.
*/
@end example
@*
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.pfaff_eq}, @ref{mt_gkz.generate_maple_file_IC}.
@end table
@comment おわり.
@node Normalizing constant,,, Cohomology intersection numbers
@section Normalizing the cohomology intersection matrix
@node mt_gkz.principal_normalizing_constant,,, Normalizing constant
@subsection @code{mt_gkz.principal_normalizing_constant}
@comment --- 索引用キーワード
@findex mt_gkz.principal_normalizing_constant
@table @t
@item mt_gkz.principal_normalizing_constant(@var{A},@var{T},@var{Beta},@var{K})
:: It returns the normalizing constant of the cohomology intersection matrix in terms of a regular triangulation T.
@end table
@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[{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
a regular triangulation of A.
@item K
The number of polynomial factors in the integrand. see [MT2020].
@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
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
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 ---
Example:
@example
[2676] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
[2677] Beta=[b1,b2,b3]$
[2678] K=2$
[2679] T=[[1,2,3],[2,3,4]]$
[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_terms}.
@end table
@comment おわり.
@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_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}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
a positive and integral weight vector.
@item Q1,Q2
@math{Q1=(q_1^\prime,q_1^{\prime\prime})^T}, @math{Q2=(q_2^\prime,q_2^{\prime\prime})^T} are integer vectors. The lengths of @math{q_1^\prime} and @math{q_2^\prime} are both equal to @math{K}.
@item K
The number of polynomial factors in the integrand. see [MT2020].
@item N
A positive integer.
@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
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}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.
Same as @code{pfaff_eq}.
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example:
@example
[2922] Beta=[b1,b2,1/3];
[b1,b2,1/3]
[2923] Q=[[1,0,0],[0,1,0]];
[[1,0,0],[0,1,0]]
[2924] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]];
[[1,1,0,0],[0,0,1,1],[0,1,0,1]]
[2925] W=[1,0,0,0];
[1,0,0,0]
[2926] K=2;
2
[2927] N=2;
2
[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]]
/*
This output means that the W-leading term of the (1,2) entry of the cohomology
intersection matrix is @math{(-3)/(x1)\times (2\pi\sqrt{-1})}. In view of examples of @code{generate_maple_file_IC} or @code{generate_maple_file_MR}, we can conclude that the cohomology
intersection matrix is given by
*/
[-(3*b1-1)/(b1*x1^2) -3/x1 ]
[-3/x1 -(3*b2-1)/b2]]
//divided by 2@math{\pi\sqrt{-1}}.
@end example
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@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.
@node mt_gkz.leading_term_rat,,, Normalizing constant
@subsection @code{mt_gkz.leading_term_rat}
@comment --- 索引用キーワード
@findex mt_gkz.leading_term_rat
@table @t
@item mt_gkz.leading_term_rat(@var{P},@var{W},@var{V})
:: It returns the W-leading term of a rational function P depending on variables V.
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
It returns the W-leading term of a rational function P.
@item P
a rational function.
@item W
a weight vector.
@item V
a list of variables of P.
@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
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
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.leading_terms}.
@end table
@comment おわり.
@node Regular triangulations,,, Cohomology intersection numbers
@section Regular triangulations
@comment --- 個々の関数の説明 ---
@comment --- section 名を正確に ---
@node mt_gkz.toric_gen_initial,,, Regular triangulations
@node mt_gkz.regular_triangulation,,, Regular triangulations
@node mt_gkz.top_standard_pairs,,, Regular triangulations
@subsection @code{mt_gkz.toric_gen_initial}, @code{mt_gkz.regular_triangulation}, @code{mt_gkz.top_standard_pairs}
@comment --- 索引用キーワード
@findex mt_gkz.toric_gen_initial
@findex mt_gkz.regular_triangulation
@findex mt_gkz.top_standard_pairs
@table @t
@item mt_gkz.toric_gen_initial(@var{A},@var{W})
@item mt_gkz.regular_triangulation(@var{A},@var{W})
@item mt_gkz.top_standard_pairs(@var{A},@var{W})
:: utility functions for computing ring theoretic invariants: generic initial ideal for the toric ideal specified by the matrix A and a weight W, its associated regular triangulation, and its associated top-dimensional standard pairs.
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
@code{toric_gen_initial} returns a list [L1,L2] of length 2. L1 is a list of generators of the W-initial ideal of the toric ideal @math{I_A} specified by A. L2 is a list of variables of @math{I_A}.
@item return
@code{regular_triangulation} returns a list of simplices of a regular triangulation @math{T_W} specified by the weight W.
@item return
@code{top_standard_pairs} returns a list of the form [[L1,S1],[L2,S2],...]. Each SI is a simplex of @math{T_W}. Each LI is a list of exponents.
@item A
a configuration matrix.
@item W
a positive weight vector.
@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
As for the definition of the standard pair, see Chapter 3 of [SST00].
@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_terms}.
@item
These functions are utilized in @code{leading_terms}.
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example: An example of a non-unimodular triangulation and non-trivial standard pairs.
@example
[3256] A=[[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];
[2,0,1,2,2]
[3258] mt_gkz.toric_gen_initial(A,W);
--snip--
[[x1*x5,x1*x4,x3^2*x4],[x1,x2,x3,x4,x5]]
[3259] mt_gkz.regular_triangulation(A,W);
--snip--
[[2,4,5],[2,3,5],[1,2,3]]
[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]]]
/*
This means that the regular triangulation of the configuration matrix A is
given by @math{T=\{\{2,4,5\},\{2,3,5\},\{1,2,3\}\}}. The normalized volumes of these simplices
are 2,1 and 1. Moreover, the top-dimensional standard pairs are
@math{(1,\{2,4,5\}),(\partial_3,\{2,4,5\})}, @math{(1,\{2,3,5\})},@math{(1,\{1,2,3\})}.
*/
@end example
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.leading_terms}.
@end table
@comment おわり.
@comment --- おまじない ---
@node Index,,, Top
@unnumbered Index
@printindex fn
@printindex cp
@iftex
@vfill @eject
@end iftex
@summarycontents
@contents
@bye
@comment --- おまじない終り ---
@comment *********************************************************
@comment ********* template
@comment **********************************************************
@comment --- 個々の関数の説明 ---
@comment --- section 名を正確に ---
@node mt_gkz.pfaff_eq,,, Pfaff equation for given cocycles
@subsection @code{mt_gkz.pfaff_eq}
@comment --- 索引用キーワード
@findex mt_gkz.pfaff_eq
@table @t
@item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
:: It returns the Pfaff equation for the GKZ system defined by @var{A} and @var{Beta} with respect to cocycles defined by @var{Rvec}.
@end table
@comment --- 引数の簡単な説明 ---
@table @var
@item return
a list of coefficients of the Pfaff equation with respect to the direction @var{DirX}
@item A
the matrix A of the GKZ system.
@item Beta
...
@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
The independent variables are @code{x1,x2,x3,...} (@math{x_1, x_2, x_3, \ldots}).
@end itemize
@comment --- @example〜@end example は実行例の表示 ---
Example: Gauss hypergeometric system, see [GM2020] example ??.
@example
[1883] import("mt_gkz.rr");
@end example
@comment --- 参照(リンク)を書く ---
@table @t
@item Refer to
@ref{mt_gkz.pfaff_eq}
@end table
@comment おわり.
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