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1.3     ! takayama    1: %% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.2 2021/01/20 08:17:54 takayama Exp $
1.1       takayama    2: %% xetex mt_gkz-en.texi   (.texi までつける. )
                      3: %% @math{tex形式の数式}
                      4: %% 参考: http://www.fan.gr.jp/~ring/doc/texinfo/texinfo-ja_14.html#SEC183
                      5: %% @tex{tex形式で書いたもの}
                      6: %%https://www.gnu.org/software/texinfo/manual/texinfo/html_node/_0040TeX-_0040LaTeX.html
                      7:
                      8: %% 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.
                      9: \input texinfo-ja
                     10: @iftex
                     11: @catcode`@#=6
                     12: @def@fref#1{@xrefX[#1,,@code{#1},,,]}
                     13: @def@b#1{{@bf #1}}
                     14: @catcode`@#=@other
                     15: @end iftex
                     16: @overfullrule=0pt
                     17: @documentlanguage en
                     18: @c -*-texinfo-*-
                     19: @comment --- おまじない終り ---
                     20:
                     21: @comment --- GNU info ファイルの名前 ---
                     22: @setfilename mt_gkz_man
                     23:
                     24: @comment --- タイトル ---
                     25: @settitle GKZ hypergeometric system
                     26:
                     27: @comment --- おまじない ---
                     28: @ifinfo
                     29: @macro fref{name}
                     30: @ref{\name\,,@code{\name\}}
                     31: @end macro
                     32: @end ifinfo
                     33:
                     34: @titlepage
                     35: @comment --- おまじない終り ---
                     36:
                     37: @comment --- タイトル, バージョン, 著者名, 著作権表示 ---
                     38: @title GKZ hypergeometric system
                     39: @subtitle Pfaffian system (Pfaff equation), contiguity relations, cohomology intersection
                     40: @subtitle Version 1.0
                     41: @subtitle January 20, 2021
                     42:
                     43: @author  by S-J. Matsubara-Heo, N.Takayama
                     44: @page
                     45: @vskip 0pt plus 1filll
                     46: Copyright @copyright{} Risa/Asir committers
                     47: 2004--2020. All rights reserved.
                     48: @end titlepage
                     49:
                     50: @comment --- おまじない ---
                     51: @synindex vr fn
                     52: @comment --- おまじない終り ---
                     53:
                     54: @comment --- @node は GNU info, HTML 用 ---
                     55: @comment --- @node  の引数は node-name,  next,  previous,  up ---
                     56: @node Top,, (dir), (dir)
                     57:
                     58: @comment --- @menu は GNU info, HTML 用 ---
                     59: @comment --- chapter 名を正確に並べる ---
                     60:
                     61: @menu
                     62: * About this document::
                     63: * Pfaff equation::
                     64: * b function::
                     65: * Utilities::
                     66: * Index::
                     67: @end menu
                     68:
                     69: @comment --- chapter の開始 ---
                     70: @comment --- 親 chapter 名を正確に. 親がない場合は Top ---
                     71: @node About this document,,, Top
                     72: @chapter About this document
                     73:
                     74: This document explains Risa/Asir functions for GKZ hypergeometric system
                     75: (A-hypergeometric system). @*    @comment 強制改行
                     76: Loading the package:
                     77: @example
                     78: import("mt_gkz.rr");
                     79: @end example
                     80: @noindent
                     81: References cited in this document.
                     82: @itemize @bullet
                     83: @item [MT2020]
                     84: Saiei-Jaeyeong Matsubara-Heo, Nobuki Takayama,
                     85: Algorithms for Pfaffian Systems and Cohomology Intersection Numbers of Hypergeometric Integrals,
                     86: Lecture Notes in Computer Science 12097 (2020), 73--84.
                     87: Errata is posted on @uref{http://arxiv.org/abs/???}.
                     88: E-attachments can be obtainable at
                     89: @uref{http://www.math.kobe-u.ac.jp/OpenXM/Math/intersection2}
                     90: @item [GM2020]
                     91: Yoshiaki Goto, Saiei-Jaeyeong Matsubara-Heo,
                     92: Homology and cohomology intersection numbers of GKZ systems, arXiv:2006.07848
                     93: @item [SST1999]
                     94: M.Saito, B.Sturmfels, N.Takayama, Hypergeometric polynomials
                     95: and integer programming, Compositio Mathematica, 155 (1999), 185--204
                     96: @item [SST2000]
                     97: M.Saito, B.Sturmfels, N.Takayama, Groebner Deformations of Hypergeometric
                     98: Differential Equations. Springer, 2000.
                     99: @end itemize
                    100:
                    101: References for maple packages IntegrableConnections and OreMorphisms.
                    102: @itemize @bullet
                    103: @item [BCEW]
                    104: M.Barkatou, T.Cluzeau, C.El Bacha, J.-A.Weil,
                    105: IntegrableConnections – a maple package for computing closed form solutions of integrable connections
                    106: (2012). @uref{https://www.unilim.fr/pages perso/thomas.cluzeau/Packages/IntegrableConnections/PDS.html}
                    107: @item [CQ]
                    108: T.Cluzeau and A.Quadrat,
                    109: 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}
                    110: @item [CQ08]
                    111: T.Cluzeau, A.Quadrat, Factoring and decomposing a class of linear functional
                    112: systems, Linear Algebra and its Applications (LAA), 428(1): 324-381, 2008.
                    113: @end itemize
                    114:
                    115:
                    116:
                    117: @node Pfaff equation,,, Top
                    118: @chapter Pfaff equation
                    119:
                    120: @menu
                    121: * mt_gkz.pfaff_eq::
                    122: * mt_gkz.ff::
                    123: * mt_gkz.ff1::
                    124: * mt_gkz.ff2::
                    125: * mt_gkz.rvec_to_fvec::
                    126: @end menu
                    127:
                    128: @node Pfaff equation for given cocycles,,, Pfaff equation
                    129: @section Pfaff equation for given cocycles
                    130:
                    131: @comment **********************************************************
                    132: @comment --- 関数 pfaff_eq
                    133: @node mt_gkz.pfaff_eq,,, Pfaff equation for given cocycles
                    134: @subsection @code{mt_gkz.pfaff_eq}
                    135: @comment --- 索引用キーワード
                    136: @findex mt_gkz.pfaff_eq
                    137:
                    138: @table @t
                    139: @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
                    140: :: It returns the Pfaff equation for the GKZ system defined by @var{A} and @var{Beta} with respect to cocycles defined by @var{Rvec}.
                    141: @end table
                    142:
                    143: @comment --- 引数の簡単な説明 ---
                    144: @table @var
                    145: @item return
                    146: a list of coefficients of the Pfaff equation with respect to the direction @var{DirX}
                    147: @item A
                    148: the matrix A of the GKZ system.
                    149: @item Beta
                    150: the parameter vector of the GKZ system.
                    151: @item Ap
                    152: See [MT2020].
                    153: @item Rvec
                    154: It is used to specify a basis of cocycles. See [MT2020]
                    155: @item DirX
                    156: a list of dxi's.
                    157: @end table
                    158:
                    159: @comment --- ここで関数の詳しい説明 ---
                    160: @comment --- @itemize〜@end itemize は箇条書き ---
                    161: @comment --- @bullet は黒点付き ---
                    162: @itemize @bullet
                    163: @item
                    164: The independent variables are x1, x2, x3, ...
                    165: @item
                    166: When @var{Rvec}=[v_1, v_2, ..., v_r] where r is the rank of the GKZ system,
                    167: the set of the cocycles standing for Av_1, Av_2, ..., Av_r
                    168: (see [MT2020])
                    169: is supposed
                    170: to be the basis to construct the Pfaffian system.
                    171: Let a_1, a_2, ..., a_n be the column vectors of the matrix A
                    172: and v be a column vector (x_1, x_2, ..., x_n)^T.
                    173: Av is defined as a_1 x_1 + a_2 x_2 + ... + a_n x_n.
                    174: @item
                    175: When the columns of @var{A} are expressed as
                    176: @math{e_i \otimes \alpha_{i_j}},
                    177: the columns of @var{Ap} is
                    178: @math{e_i \otimes 0} where $e_i$ is the i-th unit vector.
                    179: See [MT2020] on the definition of @var{Ap}.
                    180: Here are some examples.
                    181: When @var{A} is
                    182: @verbatim
                    183: [[1,1,0,0],
                    184:  [0,0,1,1],
                    185:  [0,1,0,1]]
                    186: @end verbatim
                    187: @var{Ap} is
                    188: @verbatim
                    189: [[1,1,0,0],
                    190:  [0,0,1,1],
                    191:  [0,0,0,0]] <-- zero row
                    192: @end verbatim
                    193: When @var{A} is
                    194: @verbatim
                    195: [[1,1,1,0,0,0],
                    196:  [0,0,0,1,1,1],
                    197:  [0,1,0,0,1,0],
                    198:  [0,0,1,0,0,1]
                    199: ]
                    200: @end verbatim
                    201: @var{Ap} is
                    202: @verbatim
                    203: [[1,1,1,0,0,0],
                    204:  [0,0,0,1,1,1],
                    205:  [0,0,0,0,0,0], <-- zero row
                    206:  [0,0,0,0,0,0]  <-- zero row
                    207: ]
                    208: @end verbatim
                    209: See also page 223 of [SST2000].
                    210: @item
                    211: Option @var{xrule}. When the option @var{xrule} is given,
                    212: the x variables specified by this option are specialized to numbers.
                    213: @item
                    214: Option @var{shift}. When the matrix @var{A} is not normal
                    215: (the associated toric ideal is not normal), a proper shift vector
                    216: must be given to obtain an element of the b-ideal. Or, use the option
                    217: @var{b_ideal} below. See [SST1999] on the theory.
                    218: @item
                    219: Option @var{b_ideal}. When the matrix @var{A} is not normal,
                    220: the option @code{b_ideal=1} obtains b-ideals and the first element
                    221: of each b-ideal is used as the b-function. The option @var{shift}
                    222: is ignored.
                    223: @item
                    224: Option @var{cg}. A constant matrix given by this option is used
                    225: for the Gauge transformation of the Pfaffian system.
                    226: In other words, the basis of cocycles specified by @var{Rvec}
                    227: is transformed by the constant matrix given by this option.
                    228: @end itemize
                    229:
                    230: @comment --- @example〜@end example は実行例の表示 ---
                    231: Example: Gauss hypergeometric system, see [GM2020] example ??.
                    232: @example
                    233: [1883] import("mt_gkz.rr");
                    234: [2657] PP=mt_gkz.pfaff_eq(A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]],
                    235:              Beta=[-g1,-g2,-c],
                    236:              Ap = [[1,1,0,0],[0,0,1,1],[0,0,0,0]],
                    237:              Rvec = [[1,0,0,0],[0,0,1,0]],
                    238:              DirX=[dx4,dx3] | xrule=[[x1,1],[x2,1]],
                    239:              cg=matrix_list_to_matrix([[1,0],[-1,1]]))$
                    240:
                    241: 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]
                    242: -- snip --
                    243: [2658] PP[0];
                    244: [ (g2*x3-g2)/(x4-x3) (g2*x3)/(x4-x3) ]
                    245: [ ((-g2*x3-c+g2)*x4+(c-g1)*x3+g1)/(x4^2-x3*x4)
                    246:            ((-g2*x3-c)*x4+(c-g1)*x3)/(x4^2-x3*x4) ]
                    247: [2659] PP[1];
                    248: [ (-g2*x4+g2)/(x4-x3) (-g2*x4)/(x4-x3) ]
                    249: [ ((g2*x3+c-g2-1)*x4+(-c+g1+1)*x3-g1)/(x3*x4-x3^2)
                    250:      ((g2*x3+c-g2-1)*x4+(-c+g1+g2+1)*x3)/(x3*x4-x3^2) ]
                    251: @end example
                    252:
                    253: @*
                    254:
                    255: Example: The role of shift.
                    256: When the toric ideal is not normal, a proper shift vector
                    257: must be given with the option @code{shift} to find an element of the b-ideal.
                    258: @example
                    259: [1882] load("mt_gkz.rr");
                    260: [1883] A=[[1,1,1,1],[0,1,3,4]];
                    261:   [[1,1,1,1],[0,1,3,4]]
                    262: [1884] Ap=[[1,1,1,1],[0,0,0,0]];
                    263:   [[1,1,1,1],[0,0,0,0]]
                    264: [1885] Rvec=[[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
                    265:   [[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
                    266: [2674] P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
                    267:     | xrule=[[x1,1],[x2,2],[x3,4]] )$
                    268: dx remains
                    269: stopped in step_up at line 342 in file "./mt_gkz/saito-b.rr"
                    270: 342    if (type(dn(Ans)) > 1) error("dx remains");
                    271: (debug) quit
                    272: // Since the toric ideal for A is not normal, it stops with the error.
                    273: [2675]  P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
                    274:    | shift=[1,0],xrule=[[x1,1],[x2,2],[x3,4]])$
                    275: // It works.
                    276: @end example
                    277:
                    278: @comment --- 参照(リンク)を書く ---
                    279: @table @t
                    280: @item Refer to
                    281: @ref{mt_gkz.ff1}
                    282: @ref{mt_gkz.ff2}
                    283: @ref{mt_gkz.ff}
                    284: @ref{mt_gkz.rvec_to_fvec}
                    285: @end table
                    286: @comment mt_gkz.pfaff_eq の説明おわり. あとはこれの繰り返し.
                    287:
                    288:
                    289: @comment --- 個々の関数の説明 ---
                    290: @comment --- section 名を正確に ---
                    291: @node mt_gkz.ff2,,, Pfaff equation for given cocycles
                    292: @node mt_gkz.ff1,,, Pfaff equation for given cocycles
                    293: @node mt_gkz.ff,,, Pfaff equation for given cocycles
                    294: @subsection @code{mt_gkz.ff2}, @code{mt_gkz.ff1}, @code{mt_gkz.ff}
                    295: @comment --- 索引用キーワード
                    296: @findex mt_gkz.ff2
                    297: @findex mt_gkz.ff1
                    298: @findex mt_gkz.ff
                    299:
                    300: @table @t
                    301: @item mt_gkz.ff(@var{Rvec0},@var{A},@var{Beta},@var{Ap})
                    302: @item mt_gkz.ff1(@var{Rvec0},@var{A},@var{Beta},@var{Ap})
                    303: @item mt_gkz.ff2(@var{Rvec0},@var{A},@var{Beta},@var{Ap},@var{BF},@var{C})
                    304: :: @code{ff} returns a differential operator whose action to 1 gives
                    305: the cocycle defined by @var{Rvec0}
                    306: @end table
                    307:
                    308: @comment --- 引数の簡単な説明 ---
                    309: @table @var
                    310: @item return
                    311: @code{ff} returns a differential operator whose action to 1 of @math{M_A(\beta)}
                    312: gives the cocycle defined by @var{Rvec0}.
                    313: @item return
                    314: @code{ff1} returns a composite of step-down operators for the positive part
                    315: of @var{Rvec0}
                    316: @item return
                    317: @code{ff2} returns a composite of step-up operators for the positive part
                    318: of @var{Rvec0}
                    319: @item Rvec0
                    320: An element of @var{Rvec} explained in @ref{mt_gkz.pfaff_eq}.
                    321: @item BF
                    322: the list of b-functions to all directions.
                    323: @item C
                    324: the list of the step up operators for all a_1, a_2, ..., a_n.
                    325: @end table
                    326: Other arguments are same with those of @code{pfaff_eq}.
                    327:
                    328: @comment --- ここで関数の詳しい説明 ---
                    329: @comment --- @itemize〜@end itemize は箇条書き ---
                    330: @comment --- @bullet は黒点付き ---
                    331: @itemize @bullet
                    332: @item
                    333: The function @code{ff} generates the list of b-functions and the list of
                    334: step up operators and store them in the cache variable.
                    335: They can be obtained by calling as @code{S=mt_gkz.get_bf_step_up()}
                    336: where S[0] is the list of b-functions and S[1] is the list of step up
                    337: operators.
                    338: Step up operators are obtained by the algorithm given in [SST1999].
                    339: @item
                    340: Option nf. When nf=1, the output operator is reduced to the normal form
                    341: with respect to the Groebner basis of the GKZ system of the graded reverse
                    342: lexicographic order.
                    343: @item
                    344: Option shift. See @ref{mt_gkz.pfaff_eq}.
                    345: @item
                    346: Internal info: The function @code{mt_gkz.bb} gives the constant so that
                    347: the step up and step down operators (contiguity operators) give
                    348: contiguity relations for the integral representation in [MT2020].
                    349: Note that @code{mt_gkz.ff1} and @code{mt_gkz.ff2} give contiguity
                    350: relations which are constant multiple of those for hypergeometric
                    351: polynomials.
                    352: @item
                    353: Internal info: @code{mt_gkz.step_up} generates step up operators
                    354: of [SST1999] from b-functions by utilizing @code{mt_gkz.bf2euler}
                    355: and @code{mt_gkz.toric}.
                    356: @end itemize
                    357:
                    358: @comment --- @example〜@end example は実行例の表示 ---
                    359: Example: Step up operators compatible with the integral representation in [MT2020].
                    360: The function hgpoly_res defined in @code{check-by-hgpoly.rr} returns
                    361: a multiple of the hypergeometric polynomial which agrees with
                    362: the residue times a power of @math{2\pi \sqrt{-1}}
                    363: of the integral representation.
                    364: See [SST1999].
                    365: @example
                    366: [1883] import("mt_gkz.rr")$
                    367: [3175] load("mt_gkz/check-by-hgpoly.rr")$
                    368: [3176]  A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
                    369: [3177]  B=newvect(3,[5,4,7])$ Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
                    370: [3179]  Beta=[b1,b2,b3]$  R=[0,0,-1,0]$
                    371: [3180]  F2=hgpoly_res(A,B,2);  // HG polynomial. 2 is the number of e_i's.
                    372:   10*x1^2*x2^3*x4^4+20*x1*x2^4*x3*x4^3+6*x2^5*x3^2*x4^2
                    373: [3182]  mt_gkz.ff(R,A,Ap,Beta); // the operator standing for R
                    374:   (x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
                    375: [3184] S=mt_gkz.get_bf_step_up(A); // b-function and non-reduced step up op's
                    376:   [[ 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 ],
                    377:   [ x2*x3*dx4+x1*x3*dx3+x1*x2*dx2+x1^2*dx1+x1
                    378:     x2*x4*dx4+x1*x4*dx3+x2^2*dx2+x1*x2*dx1+x2
                    379:     x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3
                    380:     x4^2*dx4+x3*x4*dx3+x2*x4*dx2+x2*x3*dx1+x4 ]]
                    381: [3185] Fvec=mt_gkz.ff2(R,A,Beta,Ap,S[0],S[1]);
                    382:   (x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
                    383: [3188] Fvec = base_replace(Fvec,assoc(Beta,vtol(B)));
                    384:   1/3*x3*x4*dx4+1/3*x3^2*dx3+1/3*x1*x4*dx2+1/3*x1*x3*dx1+1/3*x3
                    385: [3189] R32d = odiff_act(Fvec,F2,[x1,x2,x3,x4]); // Act Fvec to the hg-poly
                    386:   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
                    387: [3190] red(R32d/hgpoly_res(A,B+newvect(3,[0,1,0]),2));
                    388:    // R32d agrees with the HG polynomial with Beta=[5,4,7]+[0,1,0].
                    389: 1
                    390: @end example
                    391:
                    392: @comment --- 参照(リンク)を書く ---
                    393: @table @t
                    394: @item Refer to
                    395: @ref{mt_gkz.pfaff_eq}
                    396: @end table
                    397: @comment おわり.
                    398:
                    399: @comment --- 個々の関数の説明 ---  Ref:2020-11-09-tw-cohom-progs.goodnotes
                    400: @comment --- section 名を正確に ---
                    401: @node mt_gkz.rvec_to_fvec,,, Pfaff equation for given cocycles
                    402: @subsection @code{mt_gkz.rvec_to_fvec}
                    403: @comment --- 索引用キーワード
                    404: @findex mt_gkz.rvec_to_fvec
                    405:
                    406: @table @t
                    407: @item mt_gkz.rvec_to_fvec(@var{Rvec},@var{A},@var{Ap},@var{Beta})
                    408: :: It returns a set of differential operators standing for @var{Rvec}.
                    409: @end table
                    410:
                    411: @comment --- 引数の簡単な説明 ---
                    412: @table @var
                    413: @item return
                    414: It returns a set of differential operators of which action to
                    415: @math{1 \in M_A(\beta)} give cocycles specified by @var{Rvec}.
                    416: @item A, Ap, Beta
                    417: Same with @ref{mt_gkz.pfaff_eq}
                    418: @end table
                    419:
                    420: @comment --- ここで関数の詳しい説明 ---
                    421: @comment --- @itemize〜@end itemize は箇条書き ---
                    422: @comment --- @bullet は黒点付き ---
                    423: @itemize @bullet
                    424: @item
                    425: Internal info: this function builds the set of operators by calling
                    426: @ref{mt_gkz.ff}.
                    427: @end itemize
                    428:
                    429: @comment --- @example〜@end example は実行例の表示 ---
                    430: Example: The following two expressions are congruent because
                    431: @math{2a_1-a_2-a_3+a_4=a_1} for this @code{A}.
                    432: @example
                    433: [1883] import("mt_gkz.rr");
                    434: [3191] mt_gkz.rvec_to_fvec([[2,-1,-1,1],[0,0,1,0]],
                    435:  [[1,1,0,0],[0,0,1,1],[0,1,0,1]],
                    436:  [[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
                    437: [(x2*x3*x4^2*dx1^2*dx4^3+((x1*x3*x4^2+x2*x3^2*x4)*dx1^2*dx3
                    438:  +(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
                    439:  +(x1*x4^2+3*x2*x3*x4)*dx1^2)*dx4^2+(x1*x3^2*x4*dx1^2*dx3^2
                    440:  +((x1^2*x3*x4+x1*x2*x3^2)*dx1^3+(3*x1*x3*x4+x2*x3^2)*dx1^2)*dx3
                    441:  +x1*x2^2*x4*dx1^2*dx2^2+((x1^2*x2*x4+x1*x2^2*x3)*dx1^3
                    442:  +(3*x1*x2*x4+x2^2*x3)*dx1^2)*dx2+x1^2*x2*x3*dx1^4
                    443:  +(x1^2*x4+3*x1*x2*x3)*dx1^3+(x1*x4+x2*x3)*dx1^2)*dx4)
                    444:  /(b3*b2*b1^3+(b3*b2^2+(-b3^2-2*b3)*b2)*b1^2+(-b3*b2^2+(b3^2+b3)*b2)*b1),
                    445:  (dx3)/(b2)]
                    446: [3192] mt_gkz.rvec_to_fvec([[1,0,0,0],[0,0,1,0]],
                    447:    [[1,1,0,0],[0,0,1,1],[0,1,0,1]],
                    448:    [[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
                    449: [(dx1)/(b1),(dx3)/(b2)]
                    450: @end example
                    451:
                    452: @comment --- 参照(リンク)を書く ---
                    453: @table @t
                    454: @item Refer to
                    455: @ref{mt_gkz.pfaff_eq}
                    456: @end table
                    457: @comment おわり.
                    458:
                    459: @comment --- fvec_to_conn_mat
                    460: @comment --- section 名を正確に ---
                    461: @node mt_gkz.fvec_to_conn_mat,,, Pfaff equation for given cocycles
                    462: @subsection @code{mt_gkz.fvec_to_conn_mat}
                    463: @comment --- 索引用キーワード
                    464: @findex mt_gkz.fvec_to_conn_mat
                    465:
                    466: @table @t
                    467: @item mt_gkz.fvec_to_conn_mat(@var{Fvec},@var{A},@var{Beta},@var{DirX})
                    468: :: It returns the coefficient matrices of the basis
                    469: @var{Fvec} or @var{DirX}[I]*@var{Fvec} in terms of the set of the standard basis.
                    470: @end table
                    471:
                    472: @comment --- 引数の簡単な説明 ---
                    473: @table @var
                    474: @item return
                    475: It returns the coefficient matrices of the basis
                    476: @var{Fvec} or @var{DirX}[I]*@var{Fvec} in terms of the set of the standard basis of the Groebner basis explained below.
                    477: @item A Beta
                    478: Same with @ref{mt_gkz.pfaff_eq}.
                    479: @item DirX
                    480: When @var{DirX} is 1, this function returns the matrix which expresses
                    481: @var{Fvec} in terms of the set of the standard monomials of
                    482: the Groebner basis of the GKZ system in the ring of rational function
                    483: coefficients with respect to the graded reverse lexicographic order.
                    484: In other cases, it returns the coefficient matrices of
                    485: @var{DirX}[I]'s*@var{Fvec} in terms of the set of the standard basis of the Groebner basis.
                    486: @end table
                    487:
                    488: @comment --- ここで関数の詳しい説明 ---
                    489: @comment --- @itemize〜@end itemize は箇条書き ---
                    490: @comment --- @bullet は黒点付き ---
                    491: @itemize @bullet
                    492: @item
                    493: It utilizes a Groebner basis computation by the package @code{yang.rr}
                    494: and @code{yang.reduction} to obtain connection matrices.
                    495: @item
                    496: This function calls some utility functions
                    497: @code{mt_gkz.dmul(Op1,Op2,XvarList)} (multiplication of @code{Op1} and @code{Op2}
                    498: and @code{mt_gkz.index_vars(x,Start,End | no_=1)}
                    499: which generates indexed variables without the underbar ``_''.
                    500: @item
                    501: We note here some other utility functions in this section:
                    502: @code{mt_gkz.check_compatibility(P,Q,X,Y)},
                    503: which checkes if the sytem d/dX-P, d/dY-Q is compatible.
                    504: @end itemize
                    505:
                    506: @comment --- @example〜@end example は実行例の表示 ---
                    507: Example: The following example illustrates how mt_gkz.pfaff_eq
                    508: obtains connection matrices.
                    509: @example
                    510: [1883] import("mt_gkz.rr");
                    511: [3201] V=mt_gkz.index_vars(x,1,4 | no_=1);
                    512:   [x1,x2,x3,x4]
                    513: [3202] mt_gkz.dmul(dx1,x1^2,V);
                    514:   x1^2*dx1+2*x1
                    515: [3204] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
                    516:    Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
                    517:    Beta= [b1,b2,b3]$
                    518:    Rvec = [[1,0,0,0],[0,0,1,0]]$
                    519:    Fvec = mt_gkz.rvec_to_fvec(Rvec,A,Ap,Beta)$
                    520:     /* Express cocyles Rvec
                    521:        by elements Fvec in the Weyl algebra by contiguity relations. */
                    522:    Cg = matrix_list_to_matrix([[1,0],[1,-1]])$
                    523: [3208] NN=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,1);
                    524:   // Express Fvec by the standard monomials Std=NN[1].
                    525:  1 ooo 2 .ooo
                    526:   [[ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
                    527:   [ (-x4)/(b1*x2) (1)/(x3) ],[dx4,1]]
                    528: [3209] Std=NN[1];
                    529:   [dx4,1]
                    530: [3173] NN=NN[0];
                    531:   [ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
                    532:   [ (-x4)/(b2*x3) (1)/(x3) ]
                    533: [3174] NN1=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,dx1)[0];
                    534:   // Express dx1*Fvec by the standard monomials Std.
                    535:  1 ooo 2 .ooo
                    536:   [ ((2*b1+b2-b3-1)*x1*x4^2+(-b1+b3+1)*x2*x3*x4)/(b1*x1^3*x4-b1*x1^2*x2*x3)
                    537:     ((b1^2+(-2*b3-1)*b1-b3*b2+b3^2+b3)*x1*x4
                    538:      +(-b1^2+(2*b3+1)*b1-b3^2-b3)*x2*x3)/(b1*x1^3*x4-b1*x1^2*x2*x3) ]
                    539:   [(b1 (-b1*x1*x4^2-b2*x2*x3*x4)/(b2*x1^2*x3*x4-b2*x1*x2*x3^2)
                    540:    (b1*x1*x4+(-b1+b3)*x2*x3)/(x1^2*x3*x4-x1*x2*x3^2) ]
                    541: [3188] P1=map(red,Cg*NN1*matrix_inverse(NN)*matrix_inverse(Cg));
                    542:   [ ((-b2*x3+(b1+b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3)/(x1^2*x4-x1*x2*x3)
                    543:      (b2*x3*x4)/(x1^2*x4-x1*x2*x3) ]
                    544:   [ ((-b2*x3+(b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3+b1*x1*x2)/(x1^2*x4-x1*x2*x3)
                    545:     ((b2*x3+b1*x1)*x4)/(x1^2*x4-x1*x2*x3) ]
                    546:
                    547: [3191] mt_gkz.pfaff_eq(A,Beta,Ap,Rvec,[dx1]|cg=Cg)[0]-P1;
                    548:  [ 0 0 ]
                    549:  [ 0 0 ]  // P1 agrees with the output of mt_gkz.pfaff_eq.
                    550: @end example
                    551:
                    552: @comment --- 参照(リンク)を書く ---
                    553: @table @t
                    554: @item Refer to
                    555: @ref{mt_gkz.pfaff_eq}
                    556: @end table
                    557: @comment おわり.
                    558:
                    559: @comment ---------- New Chapter  ---------------
                    560: @node b function,,, Top
                    561: @chapter b function
                    562:
                    563: @menu
                    564: * mt_gkz.bf::
                    565: * mt_gkz.bf::
                    566: @end menu
                    567:
                    568: @node b function and facet polynomial,,, b function
                    569: @section b function and facet polynomial
                    570: @comment ------- bf
                    571: @comment **********************************************************
                    572: @comment --- 個々の関数の説明 ---
                    573: @comment --- section 名を正確に ---
                    574: @node mt_gkz.bf,,, b function and facet polynomial
                    575: @subsection @code{mt_gkz.bf}
                    576: @comment --- 索引用キーワード
                    577: @findex mt_gkz.bf
                    578:
                    579: @table @t
                    580: @item mt_gkz.bf(@var{A},@var{Facet_poly},@var{II0})
                    581: :: It returns the b-function with respect to the direction @var{II0}.
                    582: @end table
                    583:
                    584: @comment --- 引数の簡単な説明 ---
                    585: @table @var
                    586: @item return
                    587: 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.
                    588: @item A
                    589: the matrix A of the GKZ system.
                    590: @item Facet_poly
                    591: The set of facet polynomials of the convex hull of @var{A}.
                    592: @item II0
                    593: Direction expressed as 0, 1, 2, ... (not 1, 2, 3, ...) to obtain the b function.
                    594: @end table
                    595:
                    596: @comment --- ここで関数の詳しい説明 ---
                    597: @comment --- @itemize〜@end itemize は箇条書き ---
                    598: @comment --- @bullet は黒点付き ---
                    599: @itemize @bullet
                    600: @item
                    601: See [SST1999] on the b-function introduced Saito and b-ideal.
                    602: @item
                    603: The facet polynomial must be primitive.
                    604: @end itemize
                    605:
                    606: @comment --- @example〜@end example は実行例の表示 ---
                    607: Example:
                    608: @example
                    609: [1883] import("mt_gkz.rr");
                    610:
                    611: [3193] A;
                    612:   [[1,1,0,0],[0,0,1,1],[0,1,0,1]]
                    613: [3194] Fpoly=mt_gkz.facet_poly(A);
                    614:   [[s_3,s_1,s_2-s_3+s_1,s_2],[[0,0,1],[1,0,0],[1,1,-1],[0,1,0]]]
                    615: [3196] mt_gkz.bf(A,Fpoly,0);
                    616:   s_1*s_2-s_1*s_3+s_1^2
                    617: [3197] mt_gkz.bf(A,Fpoly,1);
                    618:   s_1*s_3
                    619: @end example
                    620:
                    621: @comment --- 参照(リンク)を書く ---
                    622: @table @t
                    623: @item Refer to
                    624: @ref{mt_gkz.ff}
                    625: @ref{mt_gkz.facet_poly}
                    626: @end table
                    627: @comment おわり.
                    628:
                    629: @comment ------ facet_poly
                    630: @comment --- 個々の関数の説明 ---
                    631: @comment --- section 名を正確に ---
                    632: @node mt_gkz.facet_polyl,,, b function and facet polynomial
                    633: @subsection @code{mt_gkz.facet_poly}
                    634: @comment --- 索引用キーワード
                    635: @findex mt_gkz.facet_poly
                    636:
                    637: @table @t
                    638: @item mt_gkz.facet_poly(@var{A})
                    639: :: It returns the set of facet polynomials and their normal vectors of
                    640: the cone defined by  @var{A}.
                    641: @end table
                    642:
                    643: @comment --- 引数の簡単な説明 ---
                    644: @table @var
                    645: @item return
                    646: It returns the set of facet polynomials and their normal vectors of
                    647: the cone generated by the column vectors of the matrix @var{A}.
                    648: @item A
                    649: the matrix A of the GKZ system.
                    650: @end table
                    651:
                    652: @comment --- ここで関数の詳しい説明 ---
                    653: @comment --- @itemize〜@end itemize は箇条書き ---
                    654: @comment --- @bullet は黒点付き ---
                    655: @itemize @bullet
                    656: @item
                    657: The facet polynomial f is primitive. In other words,
                    658: all f(a_i) is integer and min f(a_i)=1 for a_i's not being on f=0.
                    659: where a_i is the i-th column vector of the matrix @var{A}.
                    660: It can be checked by @code{mt_gkz.is_primitive(At,Facets)}
                    661: where @var{At} is the transpose of @var{A} and
                    662: @var{Facets} is the second return value of this function.
                    663: @item
                    664: This function utilizes the system polymake @uref{https://polymake.org}
                    665: on our server.
                    666: @end itemize
                    667:
                    668: @comment --- @example〜@end example は実行例の表示 ---
                    669: Example:
                    670: @example
                    671: [1883] import("mt_gkz.rr");
                    672: [1884] mt_gkz.facet_poly([[1,1,1,1],[0,1,2,3]]);
                    673:   oohg_native=0, oohg_curl=1
                    674:   [[s_2,-s_2+3*s_1],[[0,1],[3,-1]]]
                    675: @end example
                    676:
                    677: @comment --- 参照(リンク)を書く ---
                    678: @table @t
                    679: @item Refer to
                    680: @ref{mt_gkz.bf}
                    681: @end table
                    682: @comment おわり.
                    683:
                    684: @comment ---------- New Chapter  ---------------
                    685: @node utilities,,, Top
                    686: @chapter Utilities
                    687:
                    688: @menu
                    689: * mt_gkz.reduce_by_toric::
                    690: * mt_gkz.tk_base_equal::
                    691: * mt_gkz.dp_op_to_coef_vec::
                    692: * mt_gkz.yang_gkz_buch::
                    693: * mt_gkz.p_true_nf_rat::
                    694: * mt_gkz.mdiff::
                    695: * mt_gkz.dvar::
                    696: * mt_gkz.ord_xi::
                    697: * mt_gkz.get_check_fvec::
                    698: * mt_gkz.get_bf_step_up::
1.2       takayama  699: * mt_gkz.mytoric_ideal::
1.1       takayama  700: @end menu
                    701:
                    702: @node some utility functions,,, utilities
                    703: @section Some utility functions
                    704:
                    705: @node mt_gkz.reduce_by_toric,,, some utility functions
                    706: @node mt_gkz.tk_base_equal,,, some utility functions
                    707: @node mt_gkz.dp_op_to_coef_vec,,, some utility functions
                    708: @node mt_gkz.yang_gkz_buch,,, some utility functions
                    709: @node mt_gkz.p_true_nf_rat,,, some utility functions
                    710: @node mt_gkz.mdiff,,, some utility functions
                    711: @node mt_gkz.dvar,,, some utility functions
                    712: @node mt_gkz.ord_xi,,, some utility functions
                    713: @node mt_gkz.get_check_fvec,,, some utility functions
                    714: @node mt_gkz.get_bf_step_up,,, some utility functions
1.2       takayama  715: @node mt_gkz.mytoric_ideal,,, some utility functions
1.1       takayama  716:
                    717: @findex mt_gkz.reduce_by_toric
                    718: @findex mt_gkz.tk_base_equal
                    719: @findex mt_gkz.dp_op_to_coef_vec
                    720: @findex mt_gkz.yang_gkz_buch
                    721: @findex mt_gkz.p_true_nf_rat
                    722: @findex mt_gkz.mdiff
                    723: @findex mt_gkz.dvar
                    724: @findex mt_gkz.ord_xi
                    725: @findex mt_gkz.get_check_fvec
                    726: @findex mt_gkz.get_bf_step_up
1.2       takayama  727: @findex mt_gkz.mytoric_ideal
1.1       takayama  728:
                    729: @comment --- @example〜@end example は実行例の表示 ---
                    730: We only show examples on these functions. As for details, please see
                    731: the source code.
                    732: @example
                    733: [1883] import("mt_gkz.rr");
                    734: [2667] mt_gkz.dvar([x1,x2]);  // it generates variables starting with d
                    735:   [dx1,dx2]
                    736: [2669] mt_gkz.p_true_nf_rat((1/3)*x^3-1,[x^2-1],[x],0);
                    737:   [x-3,3]     // p_true_nf does not accept rational number coefficients
                    738: [2670] A=[[1,1,1,1],[0,1,3,4]];
                    739:   [[1,1,1,1],[0,1,3,4]]
                    740: [2671] mt_gkz.reduce_by_toric(dx3^4,A);
                    741:   dx1*dx4^3   // reduction by toric ideal defined by A
                    742: [2672] nk_toric.toric_ideal(A);
                    743:   [-x1*x4+x2*x3,-x2*x4^2+x3^3,x2^2*x4-x1*x3^2,-x1^2*x3+x2^3]
                    744: [2673] mt_gkz.yang_gkz_buch(A,[b1,b2]);  // Groebner basis of GKZ system by yang.rr
                    745:  1 o 2 ..o 3 ..oooooooo 4 o 6 ooo 9 o
                    746: [[[(x2)*<<0,1,0,0>>+(3*x3)*<<0,0,1,0>>+ ---snip ---*<<0,0,0,0>>,1]],
                    747: [dx1,dx2,dx3,dx4],
                    748: [(1)*<<0,0,0,2>>,(1)*<<0,0,1,0>>,(1)*<<0,0,0,1>>,(1)*<<0,0,0,0>>]]
                    749:
                    750: [2674] mt_gkz.dp_op_to_coef_vec([x1*<<1,0>>+x1*x2*<<0,1>>,x1+1],[<<1,0>>,<<0,1>>]);
                    751:   // x1+1 is the denominator
                    752:   [ (x1)/(x1+1) (x1*x2)/(x1+1) ]
                    753: [2675] mt_gkz.tk_base_is_equal([1,2],[1,2]);
                    754:  1
                    755: [2676] mt_gkz.tk_base_is_equal([1,2],[1,x,y]);
                    756:  0
                    757: [2677] mt_gkz.mdiff(sin(x),x,1);
                    758:  cos(x)
                    759: [2678] mt_gkz.mdiff(sin(x),x,2); //2nd derivative
                    760:  -sin(x)
                    761: [3164] mt_gkz.ord_xi(V=[x1,x2,x3],II=1);
                    762: // matrix to define graded lexicographic order so that V[II] is the smallest.
                    763: [ 1 1 1 ]
                    764: [ 0 -1 0 ]
                    765: [ -1 0 0 ]
                    766: [3166] load("mt_gkz/check-by-hgpoly.rr");
                    767: [3187] check_0123(); // check the pfaffian for the A below by hg-polynomial.
                    768:   A=[[1,1,1,1],[0,1,2,3]]
                    769:   Ap=[[1,1,1,1],[0,0,0,0]]
                    770:   --- snip ---
                    771:   Bfunctions= --- snip ---
                    772:   0 (vector) is expected:
                    773:   [[ 0 0 0 ],[ 0 0 0 ]]
                    774: [3188] mt_gkz.get_check_fvec();
                    775:  // get the basis of cocycles used in terms of differential operators.
                    776:   [1,(dx4)/(b1),(dx4^2)/(b1^2-b1)]
                    777: [3189] mt_gkz.clear_bf();
                    778: 0
                    779: [3190] mt_gkz.get_bf_step_up(A=[[1,1,1,1],[0,1,2,3]]);
                    780: // b-functions and step-up operators.
                    781: // Option b_ideal=1 or shift=... may be used for non-normal case.
                    782:   [[ -s_2^3+(9*s_1-3)*s_2^2+ ---snip---
                    783:      -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 ],
                    784:   [ x3^3*dx4^2+ ---snip---
                    785:     3*x3^2*x4*dx4^2+ --- snip---]]
1.2       takayama  786: [3191] mt_gkz.mytoric_ideal(0 | use_4ti2=1);
                    787: // 4ti2 is used to obtain a generator set of the toric ideal
                    788: // defined by the matrix A
                    789: [3192] mt_gkz.mytoric_ideal(0 | use_4ti2=0);
                    790: // A slower method is used to obtain a generator set of the toric ideal
                    791: // defined by the matrix A.  4ti2 is not needed. Default.
1.1       takayama  792: @end example
                    793:
                    794:
                    795:
                    796:
                    797:
                    798:
                    799:
                    800:
                    801:
                    802:
                    803:
                    804:
                    805:
                    806:
                    807:
                    808:
                    809:
                    810:
                    811:
                    812:
                    813:
                    814:
                    815:
                    816:
                    817:
                    818:
                    819:
                    820:
                    821:
                    822:
                    823:
                    824:
                    825:
                    826:
                    827:
                    828:
                    829:
                    830:
                    831:
                    832:
                    833:
                    834:
                    835:
                    836:
                    837:
                    838:
                    839:
                    840:
                    841:
                    842:
                    843:
                    844:
                    845:
                    846:
                    847:
                    848:
                    849: @comment ここから追加版
                    850:
                    851: @node Cohomology intersection numbers,,, Top
                    852: @chapter Cohomology intersection numbers
                    853:
                    854: @menu
                    855: * mt_gkz.kronecker_prd::
                    856: * mt_gkz.secondary_eq::
                    857: * mt_gkz.generate_maple_file_IC::
                    858: * mt_gkz.generate_maple_file_MR::
1.2       takayama  859: * mt_gkz.principal_normalizing_constant::
1.1       takayama  860: @end menu
                    861:
                    862:
                    863:
                    864:
                    865:
                    866:
                    867: @node Secondary equation,,, Cohomology intersection numbers
                    868: @section Secondary equation
                    869:
                    870: @comment **********************************************************
                    871: @comment --- 関数 pfaff_eq
                    872: @node mt_gkz.kronecker_prd,,, Secondary equation
                    873: @subsection @code{mt_gkz.kronecker_prd}
                    874: @comment --- 索引用キーワード
                    875: @findex mt_gkz.kronecker_prd
                    876:
                    877: @table @t
                    878: @item mt_gkz.kronecker_prd(@var{A},@var{B})
                    879: :: It returns the Kronecker product of @var{A} and @var{B}.
                    880: @end table
                    881:
                    882: @comment --- 引数の簡単な説明 ---
                    883: @table @var
                    884: @item return
1.2       takayama  885: a matrix which is equal to the Kronecker product of @var{A} and @var{B} (@uref{https://en.wikipedia.org/wiki/Kronecker_product}).
1.1       takayama  886: @item A,B
                    887: list
                    888: @end table
                    889:
                    890:
                    891: @comment --- @example〜@end example は実行例の表示 ---
                    892:
                    893: @example
                    894: [2644]  A=[[a,b],[c,d]];
                    895: [[a,b],[c,d]]
                    896: [2645] B=[[e,f],[g,h]];
                    897: [[e,f],[g,h]]
1.2       takayama  898: [2646] mt_gkz.kronecker_prd(A,B);
1.1       takayama  899: [ e*a f*a e*b f*b ]
                    900: [ g*a h*a g*b h*b ]
                    901: [ e*c f*c e*d f*d ]
                    902: [ g*c h*c g*d h*d ]
                    903: @end example
                    904:
                    905:
                    906:
                    907:
                    908:
                    909:
                    910:
                    911:
                    912: @node mt_gkz.secondary_eq,,, Secondary equation
                    913: @subsection @code{mt_gkz.secondary_eq}
                    914: @comment --- 索引用キーワード
                    915: @findex mt_gkz.secondary_eq
                    916:
                    917: @table @t
                    918: @item mt_gkz.secondary_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
                    919: :: It returns the secondary equation with respect to cocycles defined by Rvec.
                    920: @end table
                    921:
                    922:
                    923: @table @var
                    924: @item return
                    925: a list of coefficients of the Pfaffian system corresponding to the secondary equation (cf. equation (8) of [MT2020]).
                    926: @item A,Beta,Ap,Rvec,DirX
                    927: see @code{pfaff_eq}
                    928: @end table
                    929:
                    930:
                    931:
                    932: @comment --- ここで関数の詳しい説明 ---
                    933: @comment --- @itemize〜@end itemize は箇条書き ---
                    934: @comment --- @bullet は黒点付き ---
                    935: @itemize @bullet
                    936: @item
                    937: 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}.
                    938: @item
                    939: 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).
                    940: @item Options xrule, shift, b_ideal,cg.
                    941: Same as @code{pfaff_eq}.
                    942: @end itemize
                    943:
                    944: @comment --- @example〜@end example は実行例の表示 ---
                    945: Example:
                    946: @example
                    947: [2647] Beta=[b1,b2,b3]$
                    948: [2648] DirX=[dx1,dx4]$
                    949: [2649] Rvec=[[1,0,0,0],[0,0,1,0]]$
                    950: [2650] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
                    951: [2651] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
                    952: [2652] Xrule=[[x2,1],[x3,1]]$
1.2       takayama  953: [2653] P=mt_gkz.secondary_eq(A,Beta,Ap,Rvec,DirX|xrule=Xrule)$
1.1       takayama  954: --snip--
                    955: [2654] length(P);
                    956: 2
                    957: [2655] P[0];
                    958: [[(-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),
                    959: (-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
                    960: ((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,
                    961: (-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
                    962: ((-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),
                    963: (b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]
                    964: <--- Paffian matrix in x1 direction.
                    965: [2656] P[1];
                    966: [[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
                    967: ((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,
                    968: (-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
                    969: ((-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),
                    970: (b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]]
                    971: <--- Paffian matrix in x4 direction.
                    972: @end example
                    973:
                    974: @comment --- 参照(リンク)を書く ---
                    975: @table @t
                    976: @item Refer to
                    977: @ref{mt_gkz.pfaff_eq}
                    978: @end table
                    979: @comment おわり.
                    980:
                    981:
                    982:
                    983:
                    984: @node mt_gkz.generate_maple_file_IC,,, Secondary equation
                    985: @subsection @code{mt_gkz.generate_maple_file_IC}
                    986: @comment --- 索引用キーワード
                    987: @findex mt_gkz.generate_maple_file_IC
                    988:
                    989: @table @t
                    990: @item mt_gkz.generate_maple_file_IC(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
                    991: :: It returns the maple input for a solver of a Pfaffian system IntegrableConnections[RationalSolutions].
                    992: @end table
                    993:
                    994: @comment --- 引数の簡単な説明 ---
                    995: @table @var
                    996: @item return
                    997: 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}).
                    998: @item A,Beta,Ap,Rvec,DirX
                    999: see @code{pfaff_eq}.
                   1000: @end table
                   1001:
                   1002: @comment --- ここで関数の詳しい説明 ---
                   1003: @comment --- @itemize〜@end itemize は箇条書き ---
                   1004: @comment --- @bullet は黒点付き ---
                   1005: @itemize @bullet
                   1006: @item
                   1007: 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].
                   1008: @item
                   1009: 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,...].
                   1010: @item Options xrule, shift, b_ideal,cg.
                   1011: Same as @code{pfaff_eq}.
                   1012: @item Option filename.
                   1013: 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".
                   1014: @end itemize
                   1015:
                   1016: @comment --- @example〜@end example は実行例の表示 ---
                   1017: Example:
                   1018: @example
                   1019: [2681] Beta=[b1,b2,1/3]$
                   1020: [2682] DirX=[dx1,dx4]$
                   1021: [2683] Rvec=[[1,0,0,0],[0,0,1,0]]$
                   1022: [2684] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
                   1023: [2685] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
                   1024: [2687] Xrule=[[x2,1],[x3,1]]$
1.2       takayama 1025: [2688] mt_gkz.generate_maple_file_IC(A,Beta,Ap,Rvec,DirX|xrule=Xrule,filename="Test.ml")$
1.1       takayama 1026:
                   1027:
                   1028: //A file named Test.ml is automatically generated as follows:
                   1029:
                   1030:
                   1031:
                   1032: with(OreModules);
                   1033: with(IntegrableConnections);
                   1034: with(linalg);
                   1035: C:=[Matrix([[(-2*x1^3*x4^2+4*x1^2*x4-2*x1)/(x1^4*x4^2-2*x1^3*x4+x1^2),
                   1036: (b2*x4)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
                   1037: ((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,
                   1038: (-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
                   1039: ((-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),
                   1040: (b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]),
                   1041: Matrix([[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
                   1042: ((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,
                   1043: (-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
                   1044: ((-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),
                   1045: (b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]])];
                   1046: RatSols:=RationalSolutions(C,[x1,x4],['param',[b1,b2]]);
                   1047:
                   1048:
                   1049: /*
                   1050: If you run the output file on maple, you obtain a rational solution of
                   1051: the secondary equation.
                   1052: */
                   1053:
                   1054:          [b2*(3*b1-1)/(b1*x1^2)]
                   1055: RatSols:=[3*b2/x1              ]
                   1056:          [3*b2/x1              ]
                   1057:          [3*b2-1               ]
                   1058:
                   1059: /*
                   1060: Note that the 4 entries of this vector correspond to entries of a 2 by 2 matrix.
                   1061: They are aligned as (1,1), (1,2), (2,1) (2,2) from the top.
                   1062: */
                   1063: @end example
                   1064:
                   1065: @*
                   1066:
                   1067: @comment --- 参照(リンク)を書く ---
                   1068: @table @t
                   1069: @item Refer to
                   1070: @ref{mt_gkz.pfaff_eq}
                   1071: @end table
                   1072: @comment おわり.
                   1073:
                   1074:
                   1075:
                   1076:
                   1077:
                   1078:
                   1079:
                   1080:
                   1081: @node mt_gkz.generate_maple_file_MR,,, Secondary equation
                   1082: @subsection @code{mt_gkz.generate_maple_file_MR}
                   1083: @comment --- 索引用キーワード
                   1084: @findex mt_gkz.generate_maple_file_MR
                   1085:
                   1086: @table @t
                   1087: @item mt_gkz.generate_maple_file_MR(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX},@var{D1},@var{D2})
                   1088: :: It returns the maple input for a solver of a Pfaffian system MorphismsRat[OreMorphisms].
                   1089: @end table
                   1090:
                   1091: @comment --- 引数の簡単な説明 ---
                   1092: @table @var
                   1093: @item return
                   1094: 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.
                   1095: @item A,Beta,Ap,Rvec,DirX
                   1096: see @code{pfaff_eq}.
                   1097: @item D1,D2
                   1098: Positive integers. D1 (resp. D2) is the upper bound of the degree of the numerator (resp. denominator) of the solution.
                   1099: @end table
                   1100:
                   1101: @comment --- ここで関数の詳しい説明 ---
                   1102: @comment --- @itemize〜@end itemize は箇条書き ---
                   1103: @comment --- @bullet は黒点付き ---
                   1104: @itemize @bullet
                   1105: @item
                   1106: 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]).
                   1107: @item
                   1108: 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.
                   1109: @item Options xrule, shift, b_ideal,cg.
                   1110: Same as @code{pfaff_eq}.
                   1111: @item Option filename.
                   1112: You can specify the file name as in @code{generate_maple_file_IC}.
                   1113: @item
                   1114: 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].
                   1115: @end itemize
                   1116:
                   1117: @comment --- @example〜@end example は実行例の表示 ---
                   1118: Example:
                   1119: @example
                   1120: [2668] Beta=[b1,b2,1/3]$
                   1121: [2669] DirX=[dx1,dx4]$
                   1122: [2670] Rvec=[[1,0,0,0],[0,0,1,0]]$
                   1123: [2671] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
                   1124: [2672] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
                   1125: [2673] Xvar=[x1,x4]$
                   1126: [2674] Xrule=[[x2,1],[x3,1]]$
1.2       takayama 1127: [2675] mt_gkz.generate_maple_file_MR(A,Beta,Ap,Rvec,DirX,2,2|xrule=Xrule)$
1.1       takayama 1128:
                   1129:
                   1130: //A file "auto-generated-MR.ml" is automatically generated as follows:
                   1131:
                   1132:
                   1133: with(OreModules);
                   1134: with(OreMorphisms);
                   1135: with(linalg);
                   1136: Alg:=DefineOreAlgebra(diff=[dx1,x1],diff=[dx4,x4],polynom=[x1,x4],comm=[b1,b2]);
                   1137: P:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
                   1138: -Matrix([[((b1+b2-4/3)*x1*x4-b1+4/3)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1)],
                   1139: [(-b1)/(x1*x4-1),(b1*x4)/(x1*x4-1)],[(b2*x1)/(x1*x4-1),(-b2)/(x1*x4-1)],
                   1140: [(-b1*x1)/(x1*x4^2-x4),(1/3*x1*x4+b1-1/3)/(x1*x4^2-x4)]]);
                   1141: Q:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
                   1142: +Matrix([[((-b1-b2-2/3)*x1*x4+b1+2/3)/(x1^2*x4-x1),(b1)/(x1*x4-1)],
                   1143: [(b2*x4)/(x1^2*x4-x1),(-b1*x4)/(x1*x4-1)],[(-b2*x1)/(x1*x4-1),(b1*x1)/(x1*x4^2-x4)],
                   1144: [(b2)/(x1*x4-1),(-1/3*x1*x4-b1+1/3)/(x1*x4^2-x4)]]);
                   1145: RatSols:=MorphismsRat(P,Q,Alg,0,2,2);
                   1146:
                   1147: /*
                   1148: If you run the output file on maple, you obtain a vector RatSols.
                   1149: RatSols[1] is the rational solution of the secondary equation:
                   1150: */
                   1151:
                   1152: 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}})]
                   1153:             [@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}})]
                   1154:
                   1155: /*
                   1156: 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}.
                   1157: */
                   1158: @end example
                   1159:
                   1160: @*
                   1161:
                   1162: @comment --- 参照(リンク)を書く ---
                   1163: @table @t
                   1164: @item Refer to
                   1165: @ref{mt_gkz.pfaff_eq}, @ref{mt_gkz.generate_maple_file_IC}.
                   1166: @end table
                   1167: @comment おわり.
                   1168:
                   1169:
                   1170:
                   1171:
                   1172:
                   1173:
                   1174:
                   1175:
                   1176:
                   1177:
                   1178:
                   1179:
                   1180: @node Normalizing constant,,, Cohomology intersection numbers
                   1181: @section Normalizing the cohomology intersection matrix
                   1182:
                   1183:
                   1184: @node mt_gkz.principal_normalizing_constant,,, Normalizing constant
                   1185: @subsection @code{mt_gkz.principal_normalizing_constant}
                   1186: @comment --- 索引用キーワード
                   1187: @findex mt_gkz.principal_normalizing_constant
                   1188:
                   1189: @table @t
                   1190: @item mt_gkz.principal_normalizing_constant(@var{A},@var{T},@var{Beta},@var{K})
                   1191: :: It returns the normalizing constant of the cohomology intersection matrix in terms of a regular triangulation T.
                   1192: @end table
                   1193:
                   1194: @comment --- 引数の簡単な説明 ---
                   1195: @table @var
                   1196: @item return
1.3     ! takayama 1197: 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.
1.1       takayama 1198: @item A,Beta
                   1199: see @code{pfaff_eq}.
                   1200: @item T
                   1201: a regular triangulation of A.
                   1202: @item K
                   1203: The number of polynomial factors in the integrand. see [MT2020].
                   1204: @end table
                   1205:
                   1206: @comment --- ここで関数の詳しい説明 ---
                   1207: @comment --- @itemize〜@end itemize は箇条書き ---
                   1208: @comment --- @bullet は黒点付き ---
                   1209: @itemize @bullet
                   1210: @item
1.3     ! takayama 1211: 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}]}.
1.1       takayama 1212: @item
1.3     ! takayama 1213: One can find a regular triangulation by using a function @code{mt_gkz.regular_triangulation}.
        !          1214: @item
        !          1215: @code{mt_gkz.leading_terms} can be used more generally.
1.1       takayama 1216: @end itemize
                   1217:
                   1218: @comment --- @example〜@end example は実行例の表示 ---
                   1219: Example:
                   1220: @example
                   1221: [2676] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
                   1222: [2677] Beta=[b1,b2,b3]$
                   1223: [2678] K=2$
                   1224: [2679] T=[[1,2,3],[2,3,4]]$
1.2       takayama 1225: [2680] mt_gkz.principal_normalizing_constant(A,T,Beta,K);
1.1       takayama 1226: (-b1-b2)/(b3*b1+b3*b2-b3^2)
                   1227: @end example
                   1228:
                   1229: @comment --- 参照(リンク)を書く ---
                   1230: @table @t
                   1231: @item Refer to
1.3     ! takayama 1232: @ref{mt_gkz.leading_terms}.
1.1       takayama 1233: @end table
                   1234: @comment おわり.
                   1235:
                   1236:
                   1237:
                   1238:
                   1239:
                   1240:
                   1241:
                   1242:
1.3     ! takayama 1243: @node mt_gkz.leading_terms,,, Normalizing constant
        !          1244: @subsection @code{mt_gkz.leading_terms}
1.1       takayama 1245: @comment --- 索引用キーワード
                   1246: @findex mt_gkz.leading_terms
                   1247:
                   1248: @table @t
1.3     ! takayama 1249: @item mt_gkz.leading_terms(@var{A},@var{Beta},@var{W},@var{Q1},@var{Q2},@var{K},@var{N})
1.1       takayama 1250: :: It returns the W-leading terms of a cohomology intersection number specified by Q1 and Q2 up to W-degree=(minimum W-degree)+N.
                   1251: @end table
                   1252:
                   1253: @comment --- 引数の簡単な説明 ---
                   1254: @table @var
                   1255: @item return
1.3     ! takayama 1256: 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+....
1.1       takayama 1257: @item A,Beta
                   1258: see @code{pfaff_eq}.
                   1259: @item W
                   1260: a positive and integral weight vector.
                   1261: @item Q1,Q2
                   1262: @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}.
                   1263: @item K
                   1264: The number of polynomial factors in the integrand. see [MT2020].
                   1265: @item N
                   1266: A positive integer.
                   1267: @end table
                   1268:
                   1269: @comment --- ここで関数の詳しい説明 ---
                   1270: @comment --- @itemize〜@end itemize は箇条書き ---
                   1271: @comment --- @bullet は黒点付き ---
                   1272: @itemize @bullet
                   1273: @item
                   1274: 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}.
                   1275: @item
1.3     ! takayama 1276: 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].
1.1       takayama 1277: @item
                   1278: 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.
                   1279: @item Option xrule.
                   1280: Same as @code{pfaff_eq}.
                   1281: @end itemize
                   1282:
                   1283: @comment --- @example〜@end example は実行例の表示 ---
                   1284: Example:
                   1285: @example
                   1286: [2922] Beta=[b1,b2,1/3];
                   1287: [b1,b2,1/3]
                   1288: [2923] Q=[[1,0,0],[0,1,0]];
                   1289: [[1,0,0],[0,1,0]]
                   1290: [2924] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]];
                   1291: [[1,1,0,0],[0,0,1,1],[0,1,0,1]]
                   1292: [2925] W=[1,0,0,0];
                   1293: [1,0,0,0]
                   1294: [2926] K=2;
                   1295: 2
                   1296: [2927] N=2;
                   1297: 2
1.3     ! takayama 1298: [2928] NC=mt_gkz.leading_terms(A,Beta,W,Q[0],Q[1],K,N|xrule=[[x2,1],[x3,1],[x4,1]])$
1.1       takayama 1299: --snip--
                   1300: [2929] NC;
                   1301: [[(-3)/(x1),-5],[0,-4],[0,-3]]
                   1302:
                   1303:
                   1304: /*
                   1305: This output means that the W-leading term of the (1,2) entry of the cohomology
                   1306: 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
                   1307: intersection matrix is given by
                   1308: */
                   1309:
                   1310: [-(3*b1-1)/(b1*x1^2)  -3/x1        ]
                   1311: [-3/x1                -(3*b2-1)/b2]]
                   1312:
                   1313:
                   1314: //divided by  2@math{\pi\sqrt{-1}}.
                   1315: @end example
                   1316:
                   1317:
                   1318: @comment --- 参照(リンク)を書く ---
                   1319: @table @t
                   1320: @item Refer to
1.3     ! takayama 1321: @ref{mt_gkz.leading_terms}, @ref{mt_gkz.generate_maple_file_IC}, @ref{mt_gkz.generate_maple_file_MR}.
1.1       takayama 1322: @end table
                   1323: @comment おわり.
                   1324:
                   1325:
                   1326:
                   1327:
                   1328:
                   1329: @node mt_gkz.leading_term_rat,,, Normalizing constant
                   1330: @subsection @code{mt_gkz.leading_term_rat}
                   1331: @comment --- 索引用キーワード
                   1332: @findex mt_gkz.leading_term_rat
                   1333:
                   1334: @table @t
                   1335: @item mt_gkz.leading_term_rat(@var{P},@var{W},@var{V})
                   1336: :: It returns the W-leading term of a rational function P depending on variables V.
                   1337: @end table
                   1338:
                   1339: @comment --- 引数の簡単な説明 ---
                   1340: @table @var
                   1341: @item return
                   1342: It returns the W-leading term of a rational function P.
                   1343: @item P
                   1344: a rational function.
                   1345: @item W
                   1346: a weight vector.
                   1347: @item V
                   1348: a list of variables of P.
                   1349: @end table
                   1350:
                   1351: @comment --- ここで関数の詳しい説明 ---
                   1352: @comment --- @itemize〜@end itemize は箇条書き ---
                   1353: @comment --- @bullet は黒点付き ---
                   1354: @itemize @bullet
                   1355: @item
1.3     ! takayama 1356: This function is supposed to be combined with @code{leading_terms} to compute the leading term of a cohomology intersection number.
1.1       takayama 1357: @item
                   1358: 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."
                   1359: @end itemize
                   1360:
                   1361:
                   1362: @comment --- 参照(リンク)を書く ---
                   1363: @table @t
                   1364: @item Refer to
1.3     ! takayama 1365: @ref{mt_gkz.leading_terms}.
1.1       takayama 1366: @end table
                   1367: @comment おわり.
                   1368:
                   1369:
                   1370:
                   1371:
                   1372:
                   1373:
                   1374: @node Regular triangulations,,, Cohomology intersection numbers
                   1375: @section Regular triangulations
                   1376:
                   1377: @comment --- 個々の関数の説明 ---
                   1378: @comment --- section 名を正確に ---
                   1379: @node mt_gkz.toric_gen_initial,,, Regular triangulations
                   1380: @node mt_gkz.regular_triangulation,,, Regular triangulations
                   1381: @node mt_gkz.top_standard_pairs,,, Regular triangulations
                   1382: @subsection @code{mt_gkz.toric_gen_initial}, @code{mt_gkz.regular_triangulation}, @code{mt_gkz.top_standard_pairs}
                   1383: @comment --- 索引用キーワード
                   1384: @findex mt_gkz.toric_gen_initial
                   1385: @findex mt_gkz.regular_triangulation
                   1386: @findex mt_gkz.top_standard_pairs
                   1387:
                   1388: @table @t
                   1389: @item mt_gkz.toric_gen_initial(@var{A},@var{W})
                   1390: @item mt_gkz.regular_triangulation(@var{A},@var{W})
                   1391: @item mt_gkz.top_standard_pairs(@var{A},@var{W})
                   1392: :: 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.
                   1393: @end table
                   1394:
                   1395: @comment --- 引数の簡単な説明 ---
                   1396: @table @var
                   1397: @item return
                   1398: @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}.
                   1399: @item return
                   1400: @code{regular_triangulation} returns a list of simplices of a regular triangulation @math{T_W} specified by the weight W.
                   1401: @item return
                   1402: @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.
                   1403: @item A
                   1404: a configuration matrix.
                   1405: @item W
                   1406: a positive weight vector.
                   1407: @end table
                   1408:
                   1409: @comment --- ここで関数の詳しい説明 ---
                   1410: @comment --- @itemize〜@end itemize は箇条書き ---
                   1411: @comment --- @bullet は黒点付き ---
                   1412: @itemize @bullet
                   1413: @item
                   1414: As for the definition of the standard pair, see Chapter 3 of [SST00].
                   1415: @item
                   1416: 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}}.
                   1417: @item
1.3     ! takayama 1418: 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}.
1.1       takayama 1419: @item
1.3     ! takayama 1420: These functions are utilized in @code{leading_terms}.
1.1       takayama 1421: @end itemize
                   1422:
                   1423: @comment --- @example〜@end example は実行例の表示 ---
                   1424: Example: An example of a non-unimodular triangulation and non-trivial standard pairs.
                   1425: @example
                   1426: [3256] A=[[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]];
                   1427: [[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]]
                   1428: [3257] W=[2,0,1,2,2];
                   1429: [2,0,1,2,2]
1.3     ! takayama 1430: [3258] mt_gkz.toric_gen_initial(A,W);
1.1       takayama 1431: --snip--
                   1432: [[x1*x5,x1*x4,x3^2*x4],[x1,x2,x3,x4,x5]]
1.3     ! takayama 1433: [3259] mt_gkz.regular_triangulation(A,W);
1.1       takayama 1434: --snip--
                   1435: [[2,4,5],[2,3,5],[1,2,3]]
1.3     ! takayama 1436: [3260] mt_gkz.top_standard_pairs(A,W);
1.1       takayama 1437: --snip--
                   1438: [[[[0,0],[0,1]],[2,4,5]],[[[0,0]],[2,3,5]],[[[0,0]],[1,2,3]]]
                   1439:
                   1440: /*
                   1441: This means that the regular triangulation of the configuration matrix A is
                   1442: given by @math{T=\{\{2,4,5\},\{2,3,5\},\{1,2,3\}\}}. The normalized volumes of these simplices
                   1443: are 2,1 and 1. Moreover, the top-dimensional standard pairs are
                   1444: @math{(1,\{2,4,5\}),(\partial_3,\{2,4,5\})}, @math{(1,\{2,3,5\})},@math{(1,\{1,2,3\})}.
                   1445: */
                   1446: @end example
                   1447:
                   1448:
                   1449:
                   1450: @comment --- 参照(リンク)を書く ---
                   1451: @table @t
                   1452: @item Refer to
1.3     ! takayama 1453: @ref{mt_gkz.leading_terms}.
1.1       takayama 1454: @end table
                   1455: @comment おわり.
                   1456:
                   1457:
                   1458:
                   1459:
                   1460:
                   1461:
                   1462:
                   1463: @comment --- おまじない ---
                   1464: @node Index,,, Top
                   1465: @unnumbered Index
                   1466: @printindex fn
                   1467: @printindex cp
                   1468: @iftex
                   1469: @vfill @eject
                   1470: @end iftex
                   1471: @summarycontents
                   1472: @contents
                   1473: @bye
                   1474: @comment --- おまじない終り ---
                   1475:
                   1476: @comment *********************************************************
                   1477: @comment ********* template
                   1478: @comment **********************************************************
                   1479: @comment --- 個々の関数の説明 ---
                   1480: @comment --- section 名を正確に ---
                   1481: @node mt_gkz.pfaff_eq,,, Pfaff equation for given cocycles
                   1482: @subsection @code{mt_gkz.pfaff_eq}
                   1483: @comment --- 索引用キーワード
                   1484: @findex mt_gkz.pfaff_eq
                   1485:
                   1486: @table @t
                   1487: @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
                   1488: :: It returns the Pfaff equation for the GKZ system defined by @var{A} and @var{Beta} with respect to cocycles defined by @var{Rvec}.
                   1489: @end table
                   1490:
                   1491: @comment --- 引数の簡単な説明 ---
                   1492: @table @var
                   1493: @item return
                   1494: a list of coefficients of the Pfaff equation with respect to the direction @var{DirX}
                   1495: @item A
                   1496: the matrix A of the GKZ system.
                   1497: @item Beta
                   1498: ...
                   1499: @end table
                   1500:
                   1501: @comment --- ここで関数の詳しい説明 ---
                   1502: @comment --- @itemize〜@end itemize は箇条書き ---
                   1503: @comment --- @bullet は黒点付き ---
                   1504: @itemize @bullet
                   1505: @item
                   1506: The independent variables are @code{x1,x2,x3,...} (@math{x_1, x_2, x_3, \ldots}).
                   1507: @end itemize
                   1508:
                   1509: @comment --- @example〜@end example は実行例の表示 ---
                   1510: Example: Gauss hypergeometric system, see [GM2020] example ??.
                   1511: @example
                   1512: [1883] import("mt_gkz.rr");
                   1513: @end example
                   1514:
                   1515: @comment --- 参照(リンク)を書く ---
                   1516: @table @t
                   1517: @item Refer to
                   1518: @ref{mt_gkz.pfaff_eq}
                   1519: @end table
                   1520: @comment おわり.

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