Annotation of OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi, Revision 1.8
1.8 ! takayama 1: %% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.7 2022/01/13 02:38:00 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
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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 ---
1.4 takayama 22: @setfilename asir-contrib-mt_gkz_man
1.1 takayama 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
1.8 ! takayama 41: @subtitle February 13, 2022
1.1 takayama 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::
1.6 takayama 126: * mt_gkz.fvec_to_conn_mat::
127: * mt_gkz.contiguity::
1.8 ! takayama 128: * mt_gkz.contiguity_by_fvec::
1.1 takayama 129: @end menu
130:
131: @node Pfaff equation for given cocycles,,, Pfaff equation
132: @section Pfaff equation for given cocycles
133:
134: @comment **********************************************************
135: @comment --- 関数 pfaff_eq
136: @node mt_gkz.pfaff_eq,,, Pfaff equation for given cocycles
137: @subsection @code{mt_gkz.pfaff_eq}
138: @comment --- 索引用キーワード
139: @findex mt_gkz.pfaff_eq
1.5 takayama 140: @findex mt_gkz.use_hilbert_driven
1.1 takayama 141:
142: @table @t
143: @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
144: :: It returns the Pfaff equation for the GKZ system defined by @var{A} and @var{Beta} with respect to cocycles defined by @var{Rvec}.
145: @end table
146:
147: @comment --- 引数の簡単な説明 ---
148: @table @var
149: @item return
150: a list of coefficients of the Pfaff equation with respect to the direction @var{DirX}
151: @item A
152: the matrix A of the GKZ system.
153: @item Beta
154: the parameter vector of the GKZ system.
155: @item Ap
156: See [MT2020].
157: @item Rvec
1.6 takayama 158: It is used to specify a basis of cocycles as explained below. See also [MT2020].
1.1 takayama 159: @item DirX
160: a list of dxi's.
161: @end table
162:
163: @comment --- ここで関数の詳しい説明 ---
164: @comment --- @itemize〜@end itemize は箇条書き ---
165: @comment --- @bullet は黒点付き ---
166: @itemize @bullet
167: @item
168: The independent variables are x1, x2, x3, ...
169: @item
170: When @var{Rvec}=[v_1, v_2, ..., v_r] where r is the rank of the GKZ system,
171: the set of the cocycles standing for Av_1, Av_2, ..., Av_r
172: (see [MT2020])
173: is supposed
174: to be the basis to construct the Pfaffian system.
1.6 takayama 175: The exponents @math{(q_\ell, q)} of the integral representation
176: @math{\int \prod h_\ell^{-q_\ell} x^q {{dx} \over {x}}}
177: is shifted by Av_i@math{=:A_{v_i}} as @math{(q_\ell,q)+A_{v_i}}.
1.1 takayama 178: Let a_1, a_2, ..., a_n be the column vectors of the matrix A
179: and v be a column vector (x_1, x_2, ..., x_n)^T.
180: Av is defined as a_1 x_1 + a_2 x_2 + ... + a_n x_n.
181: @item
182: When the columns of @var{A} are expressed as
183: @math{e_i \otimes \alpha_{i_j}},
184: the columns of @var{Ap} is
185: @math{e_i \otimes 0} where $e_i$ is the i-th unit vector.
186: See [MT2020] on the definition of @var{Ap}.
187: Here are some examples.
188: When @var{A} is
189: @verbatim
190: [[1,1,0,0],
191: [0,0,1,1],
192: [0,1,0,1]]
193: @end verbatim
194: @var{Ap} is
195: @verbatim
196: [[1,1,0,0],
197: [0,0,1,1],
198: [0,0,0,0]] <-- zero row
199: @end verbatim
200: When @var{A} is
201: @verbatim
202: [[1,1,1,0,0,0],
203: [0,0,0,1,1,1],
204: [0,1,0,0,1,0],
205: [0,0,1,0,0,1]
206: ]
207: @end verbatim
208: @var{Ap} is
209: @verbatim
210: [[1,1,1,0,0,0],
211: [0,0,0,1,1,1],
212: [0,0,0,0,0,0], <-- zero row
213: [0,0,0,0,0,0] <-- zero row
214: ]
215: @end verbatim
216: See also page 223 of [SST2000].
217: @item
218: Option @var{xrule}. When the option @var{xrule} is given,
219: the x variables specified by this option are specialized to numbers.
220: @item
221: Option @var{shift}. When the matrix @var{A} is not normal
222: (the associated toric ideal is not normal), a proper shift vector
223: must be given to obtain an element of the b-ideal. Or, use the option
224: @var{b_ideal} below. See [SST1999] on the theory.
225: @item
226: Option @var{b_ideal}. When the matrix @var{A} is not normal,
227: the option @code{b_ideal=1} obtains b-ideals and the first element
228: of each b-ideal is used as the b-function. The option @var{shift}
229: is ignored.
230: @item
231: Option @var{cg}. A constant matrix given by this option is used
232: for the Gauge transformation of the Pfaffian system.
233: In other words, the basis of cocycles specified by @var{Rvec}
234: is transformed by the constant matrix given by this option.
1.5 takayama 235: @item
236: By mt_gkz.use_hilbert_driven(Rank), the rank of the GKZ system is assumed to be
237: Rank. It makes the computation of Groebner basis by yang.rr faster.
238: This option is disabled by mt_gkz.use_hilbert_driven(0);
1.1 takayama 239: @end itemize
240:
241: @comment --- @example〜@end example は実行例の表示 ---
242: Example: Gauss hypergeometric system, see [GM2020] example ??.
243: @example
244: [1883] import("mt_gkz.rr");
245: [2657] PP=mt_gkz.pfaff_eq(A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]],
246: Beta=[-g1,-g2,-c],
247: Ap = [[1,1,0,0],[0,0,1,1],[0,0,0,0]],
248: Rvec = [[1,0,0,0],[0,0,1,0]],
249: DirX=[dx4,dx3] | xrule=[[x1,1],[x2,1]],
250: cg=matrix_list_to_matrix([[1,0],[-1,1]]))$
251:
252: 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]
253: -- snip --
254: [2658] PP[0];
255: [ (g2*x3-g2)/(x4-x3) (g2*x3)/(x4-x3) ]
256: [ ((-g2*x3-c+g2)*x4+(c-g1)*x3+g1)/(x4^2-x3*x4)
257: ((-g2*x3-c)*x4+(c-g1)*x3)/(x4^2-x3*x4) ]
258: [2659] PP[1];
259: [ (-g2*x4+g2)/(x4-x3) (-g2*x4)/(x4-x3) ]
260: [ ((g2*x3+c-g2-1)*x4+(-c+g1+1)*x3-g1)/(x3*x4-x3^2)
261: ((g2*x3+c-g2-1)*x4+(-c+g1+g2+1)*x3)/(x3*x4-x3^2) ]
262: @end example
263:
264: @*
265:
266: Example: The role of shift.
267: When the toric ideal is not normal, a proper shift vector
268: must be given with the option @code{shift} to find an element of the b-ideal.
269: @example
270: [1882] load("mt_gkz.rr");
271: [1883] A=[[1,1,1,1],[0,1,3,4]];
272: [[1,1,1,1],[0,1,3,4]]
273: [1884] Ap=[[1,1,1,1],[0,0,0,0]];
274: [[1,1,1,1],[0,0,0,0]]
275: [1885] Rvec=[[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
276: [[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
277: [2674] P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
278: | xrule=[[x1,1],[x2,2],[x3,4]] )$
279: dx remains
280: stopped in step_up at line 342 in file "./mt_gkz/saito-b.rr"
281: 342 if (type(dn(Ans)) > 1) error("dx remains");
282: (debug) quit
283: // Since the toric ideal for A is not normal, it stops with the error.
284: [2675] P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
285: | shift=[1,0],xrule=[[x1,1],[x2,2],[x3,4]])$
286: // It works.
287: @end example
288:
289: @comment --- 参照(リンク)を書く ---
290: @table @t
291: @item Refer to
292: @ref{mt_gkz.ff1}
293: @ref{mt_gkz.ff2}
294: @ref{mt_gkz.ff}
295: @ref{mt_gkz.rvec_to_fvec}
296: @end table
297: @comment mt_gkz.pfaff_eq の説明おわり. あとはこれの繰り返し.
298:
299:
300: @comment --- 個々の関数の説明 ---
301: @comment --- section 名を正確に ---
302: @node mt_gkz.ff2,,, Pfaff equation for given cocycles
303: @node mt_gkz.ff1,,, Pfaff equation for given cocycles
304: @node mt_gkz.ff,,, Pfaff equation for given cocycles
305: @subsection @code{mt_gkz.ff2}, @code{mt_gkz.ff1}, @code{mt_gkz.ff}
306: @comment --- 索引用キーワード
307: @findex mt_gkz.ff2
308: @findex mt_gkz.ff1
309: @findex mt_gkz.ff
310:
311: @table @t
1.7 takayama 312: @item mt_gkz.ff(@var{Rvec0},@var{A},@var{Ap},@var{Beta})
1.1 takayama 313: @item mt_gkz.ff1(@var{Rvec0},@var{A},@var{Beta},@var{Ap})
314: @item mt_gkz.ff2(@var{Rvec0},@var{A},@var{Beta},@var{Ap},@var{BF},@var{C})
315: :: @code{ff} returns a differential operator whose action to 1 gives
316: the cocycle defined by @var{Rvec0}
317: @end table
318:
319: @comment --- 引数の簡単な説明 ---
320: @table @var
321: @item return
322: @code{ff} returns a differential operator whose action to 1 of @math{M_A(\beta)}
323: gives the cocycle defined by @var{Rvec0}.
324: @item return
325: @code{ff1} returns a composite of step-down operators for the positive part
326: of @var{Rvec0}
327: @item return
328: @code{ff2} returns a composite of step-up operators for the positive part
329: of @var{Rvec0}
330: @item Rvec0
331: An element of @var{Rvec} explained in @ref{mt_gkz.pfaff_eq}.
332: @item BF
333: the list of b-functions to all directions.
334: @item C
335: the list of the step up operators for all a_1, a_2, ..., a_n.
336: @end table
337: Other arguments are same with those of @code{pfaff_eq}.
338:
339: @comment --- ここで関数の詳しい説明 ---
340: @comment --- @itemize〜@end itemize は箇条書き ---
341: @comment --- @bullet は黒点付き ---
342: @itemize @bullet
343: @item
344: The function @code{ff} generates the list of b-functions and the list of
345: step up operators and store them in the cache variable.
346: They can be obtained by calling as @code{S=mt_gkz.get_bf_step_up()}
347: where S[0] is the list of b-functions and S[1] is the list of step up
348: operators.
349: Step up operators are obtained by the algorithm given in [SST1999].
350: @item
351: Option nf. When nf=1, the output operator is reduced to the normal form
352: with respect to the Groebner basis of the GKZ system of the graded reverse
353: lexicographic order.
354: @item
355: Option shift. See @ref{mt_gkz.pfaff_eq}.
356: @item
357: Internal info: The function @code{mt_gkz.bb} gives the constant so that
358: the step up and step down operators (contiguity operators) give
359: contiguity relations for the integral representation in [MT2020].
360: Note that @code{mt_gkz.ff1} and @code{mt_gkz.ff2} give contiguity
361: relations which are constant multiple of those for hypergeometric
362: polynomials.
363: @item
364: Internal info: @code{mt_gkz.step_up} generates step up operators
365: of [SST1999] from b-functions by utilizing @code{mt_gkz.bf2euler}
366: and @code{mt_gkz.toric}.
367: @end itemize
368:
369: @comment --- @example〜@end example は実行例の表示 ---
370: Example: Step up operators compatible with the integral representation in [MT2020].
371: The function hgpoly_res defined in @code{check-by-hgpoly.rr} returns
372: a multiple of the hypergeometric polynomial which agrees with
373: the residue times a power of @math{2\pi \sqrt{-1}}
374: of the integral representation.
375: See [SST1999].
376: @example
377: [1883] import("mt_gkz.rr")$
378: [3175] load("mt_gkz/check-by-hgpoly.rr")$
379: [3176] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
380: [3177] B=newvect(3,[5,4,7])$ Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
381: [3179] Beta=[b1,b2,b3]$ R=[0,0,-1,0]$
382: [3180] F2=hgpoly_res(A,B,2); // HG polynomial. 2 is the number of e_i's.
383: 10*x1^2*x2^3*x4^4+20*x1*x2^4*x3*x4^3+6*x2^5*x3^2*x4^2
384: [3182] mt_gkz.ff(R,A,Ap,Beta); // the operator standing for R
385: (x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
386: [3184] S=mt_gkz.get_bf_step_up(A); // b-function and non-reduced step up op's
387: [[ 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 ],
388: [ x2*x3*dx4+x1*x3*dx3+x1*x2*dx2+x1^2*dx1+x1
389: x2*x4*dx4+x1*x4*dx3+x2^2*dx2+x1*x2*dx1+x2
390: x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3
391: x4^2*dx4+x3*x4*dx3+x2*x4*dx2+x2*x3*dx1+x4 ]]
392: [3185] Fvec=mt_gkz.ff2(R,A,Beta,Ap,S[0],S[1]);
393: (x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
394: [3188] Fvec = base_replace(Fvec,assoc(Beta,vtol(B)));
395: 1/3*x3*x4*dx4+1/3*x3^2*dx3+1/3*x1*x4*dx2+1/3*x1*x3*dx1+1/3*x3
396: [3189] R32d = odiff_act(Fvec,F2,[x1,x2,x3,x4]); // Act Fvec to the hg-poly
397: 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
398: [3190] red(R32d/hgpoly_res(A,B+newvect(3,[0,1,0]),2));
399: // R32d agrees with the HG polynomial with Beta=[5,4,7]+[0,1,0].
400: 1
401: @end example
402:
403: @comment --- 参照(リンク)を書く ---
404: @table @t
405: @item Refer to
406: @ref{mt_gkz.pfaff_eq}
407: @end table
408: @comment おわり.
409:
410: @comment --- 個々の関数の説明 --- Ref:2020-11-09-tw-cohom-progs.goodnotes
411: @comment --- section 名を正確に ---
412: @node mt_gkz.rvec_to_fvec,,, Pfaff equation for given cocycles
413: @subsection @code{mt_gkz.rvec_to_fvec}
414: @comment --- 索引用キーワード
415: @findex mt_gkz.rvec_to_fvec
416:
417: @table @t
418: @item mt_gkz.rvec_to_fvec(@var{Rvec},@var{A},@var{Ap},@var{Beta})
419: :: It returns a set of differential operators standing for @var{Rvec}.
420: @end table
421:
422: @comment --- 引数の簡単な説明 ---
423: @table @var
424: @item return
425: It returns a set of differential operators of which action to
426: @math{1 \in M_A(\beta)} give cocycles specified by @var{Rvec}.
427: @item A, Ap, Beta
428: Same with @ref{mt_gkz.pfaff_eq}
429: @end table
430:
431: @comment --- ここで関数の詳しい説明 ---
432: @comment --- @itemize〜@end itemize は箇条書き ---
433: @comment --- @bullet は黒点付き ---
434: @itemize @bullet
435: @item
436: Internal info: this function builds the set of operators by calling
437: @ref{mt_gkz.ff}.
438: @end itemize
439:
440: @comment --- @example〜@end example は実行例の表示 ---
441: Example: The following two expressions are congruent because
442: @math{2a_1-a_2-a_3+a_4=a_1} for this @code{A}.
443: @example
444: [1883] import("mt_gkz.rr");
445: [3191] mt_gkz.rvec_to_fvec([[2,-1,-1,1],[0,0,1,0]],
446: [[1,1,0,0],[0,0,1,1],[0,1,0,1]],
447: [[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
448: [(x2*x3*x4^2*dx1^2*dx4^3+((x1*x3*x4^2+x2*x3^2*x4)*dx1^2*dx3
449: +(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
450: +(x1*x4^2+3*x2*x3*x4)*dx1^2)*dx4^2+(x1*x3^2*x4*dx1^2*dx3^2
451: +((x1^2*x3*x4+x1*x2*x3^2)*dx1^3+(3*x1*x3*x4+x2*x3^2)*dx1^2)*dx3
452: +x1*x2^2*x4*dx1^2*dx2^2+((x1^2*x2*x4+x1*x2^2*x3)*dx1^3
453: +(3*x1*x2*x4+x2^2*x3)*dx1^2)*dx2+x1^2*x2*x3*dx1^4
454: +(x1^2*x4+3*x1*x2*x3)*dx1^3+(x1*x4+x2*x3)*dx1^2)*dx4)
455: /(b3*b2*b1^3+(b3*b2^2+(-b3^2-2*b3)*b2)*b1^2+(-b3*b2^2+(b3^2+b3)*b2)*b1),
456: (dx3)/(b2)]
457: [3192] mt_gkz.rvec_to_fvec([[1,0,0,0],[0,0,1,0]],
458: [[1,1,0,0],[0,0,1,1],[0,1,0,1]],
459: [[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
460: [(dx1)/(b1),(dx3)/(b2)]
461: @end example
462:
463: @comment --- 参照(リンク)を書く ---
464: @table @t
465: @item Refer to
466: @ref{mt_gkz.pfaff_eq}
467: @end table
468: @comment おわり.
469:
470: @comment --- fvec_to_conn_mat
471: @comment --- section 名を正確に ---
472: @node mt_gkz.fvec_to_conn_mat,,, Pfaff equation for given cocycles
473: @subsection @code{mt_gkz.fvec_to_conn_mat}
474: @comment --- 索引用キーワード
475: @findex mt_gkz.fvec_to_conn_mat
476:
477: @table @t
478: @item mt_gkz.fvec_to_conn_mat(@var{Fvec},@var{A},@var{Beta},@var{DirX})
479: :: It returns the coefficient matrices of the basis
480: @var{Fvec} or @var{DirX}[I]*@var{Fvec} in terms of the set of the standard basis.
481: @end table
482:
483: @comment --- 引数の簡単な説明 ---
484: @table @var
485: @item return
486: It returns the coefficient matrices of the basis
487: @var{Fvec} or @var{DirX}[I]*@var{Fvec} in terms of the set of the standard basis of the Groebner basis explained below.
488: @item A Beta
489: Same with @ref{mt_gkz.pfaff_eq}.
490: @item DirX
491: When @var{DirX} is 1, this function returns the matrix which expresses
492: @var{Fvec} in terms of the set of the standard monomials of
493: the Groebner basis of the GKZ system in the ring of rational function
494: coefficients with respect to the graded reverse lexicographic order.
495: In other cases, it returns the coefficient matrices of
496: @var{DirX}[I]'s*@var{Fvec} in terms of the set of the standard basis of the Groebner basis.
497: @end table
498:
499: @comment --- ここで関数の詳しい説明 ---
500: @comment --- @itemize〜@end itemize は箇条書き ---
501: @comment --- @bullet は黒点付き ---
502: @itemize @bullet
503: @item
504: It utilizes a Groebner basis computation by the package @code{yang.rr}
505: and @code{yang.reduction} to obtain connection matrices.
506: @item
507: This function calls some utility functions
508: @code{mt_gkz.dmul(Op1,Op2,XvarList)} (multiplication of @code{Op1} and @code{Op2}
509: and @code{mt_gkz.index_vars(x,Start,End | no_=1)}
510: which generates indexed variables without the underbar ``_''.
511: @item
512: We note here some other utility functions in this section:
513: @code{mt_gkz.check_compatibility(P,Q,X,Y)},
514: which checkes if the sytem d/dX-P, d/dY-Q is compatible.
515: @end itemize
516:
517: @comment --- @example〜@end example は実行例の表示 ---
518: Example: The following example illustrates how mt_gkz.pfaff_eq
519: obtains connection matrices.
520: @example
521: [1883] import("mt_gkz.rr");
522: [3201] V=mt_gkz.index_vars(x,1,4 | no_=1);
523: [x1,x2,x3,x4]
524: [3202] mt_gkz.dmul(dx1,x1^2,V);
525: x1^2*dx1+2*x1
526: [3204] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
527: Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
528: Beta= [b1,b2,b3]$
529: Rvec = [[1,0,0,0],[0,0,1,0]]$
530: Fvec = mt_gkz.rvec_to_fvec(Rvec,A,Ap,Beta)$
531: /* Express cocyles Rvec
532: by elements Fvec in the Weyl algebra by contiguity relations. */
533: Cg = matrix_list_to_matrix([[1,0],[1,-1]])$
534: [3208] NN=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,1);
535: // Express Fvec by the standard monomials Std=NN[1].
536: 1 ooo 2 .ooo
537: [[ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
538: [ (-x4)/(b1*x2) (1)/(x3) ],[dx4,1]]
539: [3209] Std=NN[1];
540: [dx4,1]
541: [3173] NN=NN[0];
542: [ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
543: [ (-x4)/(b2*x3) (1)/(x3) ]
544: [3174] NN1=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,dx1)[0];
545: // Express dx1*Fvec by the standard monomials Std.
546: 1 ooo 2 .ooo
547: [ ((2*b1+b2-b3-1)*x1*x4^2+(-b1+b3+1)*x2*x3*x4)/(b1*x1^3*x4-b1*x1^2*x2*x3)
548: ((b1^2+(-2*b3-1)*b1-b3*b2+b3^2+b3)*x1*x4
549: +(-b1^2+(2*b3+1)*b1-b3^2-b3)*x2*x3)/(b1*x1^3*x4-b1*x1^2*x2*x3) ]
550: [(b1 (-b1*x1*x4^2-b2*x2*x3*x4)/(b2*x1^2*x3*x4-b2*x1*x2*x3^2)
551: (b1*x1*x4+(-b1+b3)*x2*x3)/(x1^2*x3*x4-x1*x2*x3^2) ]
552: [3188] P1=map(red,Cg*NN1*matrix_inverse(NN)*matrix_inverse(Cg));
553: [ ((-b2*x3+(b1+b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3)/(x1^2*x4-x1*x2*x3)
554: (b2*x3*x4)/(x1^2*x4-x1*x2*x3) ]
555: [ ((-b2*x3+(b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3+b1*x1*x2)/(x1^2*x4-x1*x2*x3)
556: ((b2*x3+b1*x1)*x4)/(x1^2*x4-x1*x2*x3) ]
557:
558: [3191] mt_gkz.pfaff_eq(A,Beta,Ap,Rvec,[dx1]|cg=Cg)[0]-P1;
559: [ 0 0 ]
560: [ 0 0 ] // P1 agrees with the output of mt_gkz.pfaff_eq.
561: @end example
562:
563: @comment --- 参照(リンク)を書く ---
564: @table @t
565: @item Refer to
566: @ref{mt_gkz.pfaff_eq}
567: @end table
568: @comment おわり.
569:
1.6 takayama 570: @comment --- contiguity
571: @comment --- section 名を正確に ---
572: @node mt_gkz.contiguity,,, Pfaff equation for given cocycles
1.8 ! takayama 573: @node mt_gkz.contiguity_by_fvec,,, Pfaff equation for given cocycles
! 574: @subsection @code{mt_gkz.contiguity}, @code{mt_gkz.contiguity_by_fvec}
1.6 takayama 575: @comment --- 索引用キーワード
576: @findex mt_gkz.contiguity
1.8 ! takayama 577: @findex mt_gkz.contiguity_by_fvec
1.6 takayama 578:
579: @table @t
580: @item mt_gkz.contiguity(@var{A},@var{Beta},@var{Ap},@var{Rvec1},@var{Rvec2})
581: :: It returns the coefficient matrix P that satisfies
582: @var{Rvec1} = P @var{Rvec2}.
1.8 ! takayama 583: @item mt_gkz.contiguity_by_fvec(@var{A},@var{Beta},@var{Ap},@var{Fvec1},@var{Fvec2})
! 584: :: It returns the coefficient matrix P that satisfies
! 585: @var{Fvec1} = P @var{Fvec2}.
1.6 takayama 586: @end table
587:
588: @comment --- 引数の簡単な説明 ---
589: @table @var
590: @item return
1.8 ! takayama 591: The coefficient matrix P that satisfies @var{Rvec1} = P @var{Rvec2}
! 592: or @var{Fvec1}=P @var{Fvec2}
1.6 takayama 593: @item A Beta Ap Rvec1 Rvec2
594: Same with @ref{mt_gkz.pfaff_eq}.
595: @end table
596:
597: @comment --- ここで関数の詳しい説明 ---
598: @comment --- @itemize〜@end itemize は箇条書き ---
599: @comment --- @bullet は黒点付き ---
600: @itemize @bullet
601: @item
602: It returns the contiguity relation between
603: @var{Rvec1} and @var{Rvec2}
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: [3200] PP=mt_gkz.contiguity(A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]],
611: Beta=[-g1,-g2,-c],
612: Ap = [[1,1,0,0],[0,0,1,1],[0,0,0,0]],
613: Rvec1 = [[1,0,0,0],[0,0,1,0]],
614: Rvec2 = [[0,0,1,0],[1,0,0,0]]);
1.8 ! takayama 615:
! 616: [3366] Fvec411=mt_gkz.rvec_to_fvec(Rvec411=[[1,1,0]],
! 617: A=[[1,1,1],[1,0,1],[0,1,1]],
! 618: Ap=[[1,1,1],[0,0,0],[0,0,0]],
! 619: Beta=[eps,-eps*del,-eps*del])$
! 620: Fvec411d=[mt_gkz.dmul(dx1,Fvec411[0],[x1,x2,x3])];
! 621: [(dx1^2*dx2)/(eps^2-eps)]
! 622: [3367] mt_gkz.contiguity_by_fvec(A,Beta,Ap,Fvec411d,Fvec411);
! 623: 1 .ooo
! 624: [ ((del+1)*eps-1)/(x1) ]
1.6 takayama 625: @end example
626:
627: @comment --- 参照(リンク)を書く ---
628: @table @t
629: @item Refer to
630: @ref{mt_gkz.pfaff_eq}
631: @ref{mt_gkz.fvec_to_conn_mat}
632: @end table
633: @comment おわり.
634:
1.1 takayama 635: @comment ---------- New Chapter ---------------
636: @node b function,,, Top
637: @chapter b function
638:
639: @menu
640: * mt_gkz.bf::
641: * mt_gkz.bf::
642: @end menu
643:
644: @node b function and facet polynomial,,, b function
645: @section b function and facet polynomial
646: @comment ------- bf
647: @comment **********************************************************
648: @comment --- 個々の関数の説明 ---
649: @comment --- section 名を正確に ---
650: @node mt_gkz.bf,,, b function and facet polynomial
651: @subsection @code{mt_gkz.bf}
652: @comment --- 索引用キーワード
653: @findex mt_gkz.bf
654:
655: @table @t
656: @item mt_gkz.bf(@var{A},@var{Facet_poly},@var{II0})
657: :: It returns the b-function with respect to the direction @var{II0}.
658: @end table
659:
660: @comment --- 引数の簡単な説明 ---
661: @table @var
662: @item return
663: 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.
664: @item A
665: the matrix A of the GKZ system.
666: @item Facet_poly
667: The set of facet polynomials of the convex hull of @var{A}.
668: @item II0
669: Direction expressed as 0, 1, 2, ... (not 1, 2, 3, ...) to obtain the b function.
670: @end table
671:
672: @comment --- ここで関数の詳しい説明 ---
673: @comment --- @itemize〜@end itemize は箇条書き ---
674: @comment --- @bullet は黒点付き ---
675: @itemize @bullet
676: @item
677: See [SST1999] on the b-function introduced Saito and b-ideal.
678: @item
679: The facet polynomial must be primitive.
680: @end itemize
681:
682: @comment --- @example〜@end example は実行例の表示 ---
683: Example:
684: @example
685: [1883] import("mt_gkz.rr");
686:
687: [3193] A;
688: [[1,1,0,0],[0,0,1,1],[0,1,0,1]]
689: [3194] Fpoly=mt_gkz.facet_poly(A);
690: [[s_3,s_1,s_2-s_3+s_1,s_2],[[0,0,1],[1,0,0],[1,1,-1],[0,1,0]]]
691: [3196] mt_gkz.bf(A,Fpoly,0);
692: s_1*s_2-s_1*s_3+s_1^2
693: [3197] mt_gkz.bf(A,Fpoly,1);
694: s_1*s_3
695: @end example
696:
697: @comment --- 参照(リンク)を書く ---
698: @table @t
699: @item Refer to
700: @ref{mt_gkz.ff}
701: @ref{mt_gkz.facet_poly}
702: @end table
703: @comment おわり.
704:
705: @comment ------ facet_poly
706: @comment --- 個々の関数の説明 ---
707: @comment --- section 名を正確に ---
708: @node mt_gkz.facet_polyl,,, b function and facet polynomial
709: @subsection @code{mt_gkz.facet_poly}
710: @comment --- 索引用キーワード
711: @findex mt_gkz.facet_poly
712:
713: @table @t
714: @item mt_gkz.facet_poly(@var{A})
715: :: It returns the set of facet polynomials and their normal vectors of
716: the cone defined by @var{A}.
717: @end table
718:
719: @comment --- 引数の簡単な説明 ---
720: @table @var
721: @item return
722: It returns the set of facet polynomials and their normal vectors of
723: the cone generated by the column vectors of the matrix @var{A}.
724: @item A
725: the matrix A of the GKZ system.
726: @end table
727:
728: @comment --- ここで関数の詳しい説明 ---
729: @comment --- @itemize〜@end itemize は箇条書き ---
730: @comment --- @bullet は黒点付き ---
731: @itemize @bullet
732: @item
733: The facet polynomial f is primitive. In other words,
734: all f(a_i) is integer and min f(a_i)=1 for a_i's not being on f=0.
735: where a_i is the i-th column vector of the matrix @var{A}.
736: It can be checked by @code{mt_gkz.is_primitive(At,Facets)}
737: where @var{At} is the transpose of @var{A} and
738: @var{Facets} is the second return value of this function.
739: @item
740: This function utilizes the system polymake @uref{https://polymake.org}
741: on our server.
742: @end itemize
743:
744: @comment --- @example〜@end example は実行例の表示 ---
745: Example:
746: @example
747: [1883] import("mt_gkz.rr");
748: [1884] mt_gkz.facet_poly([[1,1,1,1],[0,1,2,3]]);
749: oohg_native=0, oohg_curl=1
750: [[s_2,-s_2+3*s_1],[[0,1],[3,-1]]]
751: @end example
752:
753: @comment --- 参照(リンク)を書く ---
754: @table @t
755: @item Refer to
756: @ref{mt_gkz.bf}
757: @end table
758: @comment おわり.
759:
760: @comment ---------- New Chapter ---------------
761: @node utilities,,, Top
762: @chapter Utilities
763:
764: @menu
765: * mt_gkz.reduce_by_toric::
766: * mt_gkz.tk_base_equal::
767: * mt_gkz.dp_op_to_coef_vec::
768: * mt_gkz.yang_gkz_buch::
769: * mt_gkz.p_true_nf_rat::
770: * mt_gkz.mdiff::
771: * mt_gkz.dvar::
772: * mt_gkz.ord_xi::
773: * mt_gkz.get_check_fvec::
774: * mt_gkz.get_bf_step_up::
1.2 takayama 775: * mt_gkz.mytoric_ideal::
1.5 takayama 776: * mt_gkz.cbase_by_euler::
1.1 takayama 777: @end menu
778:
779: @node some utility functions,,, utilities
780: @section Some utility functions
781:
782: @node mt_gkz.reduce_by_toric,,, some utility functions
783: @node mt_gkz.tk_base_equal,,, some utility functions
784: @node mt_gkz.dp_op_to_coef_vec,,, some utility functions
785: @node mt_gkz.yang_gkz_buch,,, some utility functions
786: @node mt_gkz.p_true_nf_rat,,, some utility functions
787: @node mt_gkz.mdiff,,, some utility functions
788: @node mt_gkz.dvar,,, some utility functions
789: @node mt_gkz.ord_xi,,, some utility functions
790: @node mt_gkz.get_check_fvec,,, some utility functions
791: @node mt_gkz.get_bf_step_up,,, some utility functions
1.2 takayama 792: @node mt_gkz.mytoric_ideal,,, some utility functions
1.5 takayama 793: @node mt_gkz.cbase_by_euler,,, some utility functions
1.1 takayama 794:
795: @findex mt_gkz.reduce_by_toric
796: @findex mt_gkz.tk_base_equal
797: @findex mt_gkz.dp_op_to_coef_vec
798: @findex mt_gkz.yang_gkz_buch
799: @findex mt_gkz.p_true_nf_rat
800: @findex mt_gkz.mdiff
801: @findex mt_gkz.dvar
802: @findex mt_gkz.ord_xi
803: @findex mt_gkz.get_check_fvec
804: @findex mt_gkz.get_bf_step_up
1.2 takayama 805: @findex mt_gkz.mytoric_ideal
1.5 takayama 806: @findex mt_gkz.cbase_by_euler
1.1 takayama 807:
808: @comment --- @example〜@end example は実行例の表示 ---
809: We only show examples on these functions. As for details, please see
810: the source code.
811: @example
812: [1883] import("mt_gkz.rr");
813: [2667] mt_gkz.dvar([x1,x2]); // it generates variables starting with d
814: [dx1,dx2]
815: [2669] mt_gkz.p_true_nf_rat((1/3)*x^3-1,[x^2-1],[x],0);
816: [x-3,3] // p_true_nf does not accept rational number coefficients
817: [2670] A=[[1,1,1,1],[0,1,3,4]];
818: [[1,1,1,1],[0,1,3,4]]
819: [2671] mt_gkz.reduce_by_toric(dx3^4,A);
820: dx1*dx4^3 // reduction by toric ideal defined by A
821: [2672] nk_toric.toric_ideal(A);
822: [-x1*x4+x2*x3,-x2*x4^2+x3^3,x2^2*x4-x1*x3^2,-x1^2*x3+x2^3]
823: [2673] mt_gkz.yang_gkz_buch(A,[b1,b2]); // Groebner basis of GKZ system by yang.rr
824: 1 o 2 ..o 3 ..oooooooo 4 o 6 ooo 9 o
825: [[[(x2)*<<0,1,0,0>>+(3*x3)*<<0,0,1,0>>+ ---snip ---*<<0,0,0,0>>,1]],
826: [dx1,dx2,dx3,dx4],
827: [(1)*<<0,0,0,2>>,(1)*<<0,0,1,0>>,(1)*<<0,0,0,1>>,(1)*<<0,0,0,0>>]]
828:
829: [2674] mt_gkz.dp_op_to_coef_vec([x1*<<1,0>>+x1*x2*<<0,1>>,x1+1],[<<1,0>>,<<0,1>>]);
830: // x1+1 is the denominator
831: [ (x1)/(x1+1) (x1*x2)/(x1+1) ]
832: [2675] mt_gkz.tk_base_is_equal([1,2],[1,2]);
833: 1
834: [2676] mt_gkz.tk_base_is_equal([1,2],[1,x,y]);
835: 0
836: [2677] mt_gkz.mdiff(sin(x),x,1);
837: cos(x)
838: [2678] mt_gkz.mdiff(sin(x),x,2); //2nd derivative
839: -sin(x)
840: [3164] mt_gkz.ord_xi(V=[x1,x2,x3],II=1);
841: // matrix to define graded lexicographic order so that V[II] is the smallest.
842: [ 1 1 1 ]
843: [ 0 -1 0 ]
844: [ -1 0 0 ]
845: [3166] load("mt_gkz/check-by-hgpoly.rr");
846: [3187] check_0123(); // check the pfaffian for the A below by hg-polynomial.
847: A=[[1,1,1,1],[0,1,2,3]]
848: Ap=[[1,1,1,1],[0,0,0,0]]
849: --- snip ---
850: Bfunctions= --- snip ---
851: 0 (vector) is expected:
852: [[ 0 0 0 ],[ 0 0 0 ]]
853: [3188] mt_gkz.get_check_fvec();
854: // get the basis of cocycles used in terms of differential operators.
855: [1,(dx4)/(b1),(dx4^2)/(b1^2-b1)]
856: [3189] mt_gkz.clear_bf();
857: 0
858: [3190] mt_gkz.get_bf_step_up(A=[[1,1,1,1],[0,1,2,3]]);
859: // b-functions and step-up operators.
860: // Option b_ideal=1 or shift=... may be used for non-normal case.
861: [[ -s_2^3+(9*s_1-3)*s_2^2+ ---snip---
862: -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 ],
863: [ x3^3*dx4^2+ ---snip---
864: 3*x3^2*x4*dx4^2+ --- snip---]]
1.2 takayama 865: [3191] mt_gkz.mytoric_ideal(0 | use_4ti2=1);
866: // 4ti2 is used to obtain a generator set of the toric ideal
867: // defined by the matrix A
868: [3192] mt_gkz.mytoric_ideal(0 | use_4ti2=0);
869: // A slower method is used to obtain a generator set of the toric ideal
870: // defined by the matrix A. 4ti2 is not needed. Default.
1.5 takayama 871: [3193] mt_gkz.cbase_by_euler(A=[[1,1,1,1],[0,1,3,4]]);
872: // Cohomology basis of the GKZ system defined by A for generic beta.
873: // Basis is given by a set of Euler operators.
874: // It is an implementation of the algorithm in http://dx.doi.org/10.1016/j.aim.2016.10.021
875: // beta is set by random numbers. Option: no_prob=1
876:
1.1 takayama 877: @end example
878:
879:
880:
881:
882:
883:
884:
885:
886:
887:
888:
889:
890:
891:
892:
893:
894:
895:
896:
897:
898:
899:
900:
901:
902:
903:
904:
905:
906:
907:
908:
909:
910:
911:
912:
913:
914:
915:
916:
917:
918:
919:
920:
921:
922:
923:
924:
925:
926:
927:
928:
929:
930:
931:
932:
933:
934: @comment ここから追加版
935:
936: @node Cohomology intersection numbers,,, Top
937: @chapter Cohomology intersection numbers
938:
939: @menu
940: * mt_gkz.kronecker_prd::
941: * mt_gkz.secondary_eq::
942: * mt_gkz.generate_maple_file_IC::
943: * mt_gkz.generate_maple_file_MR::
1.2 takayama 944: * mt_gkz.principal_normalizing_constant::
1.1 takayama 945: @end menu
946:
947:
948:
949:
950:
951:
952: @node Secondary equation,,, Cohomology intersection numbers
953: @section Secondary equation
954:
955: @comment **********************************************************
956: @comment --- 関数 pfaff_eq
957: @node mt_gkz.kronecker_prd,,, Secondary equation
958: @subsection @code{mt_gkz.kronecker_prd}
959: @comment --- 索引用キーワード
960: @findex mt_gkz.kronecker_prd
961:
962: @table @t
963: @item mt_gkz.kronecker_prd(@var{A},@var{B})
964: :: It returns the Kronecker product of @var{A} and @var{B}.
965: @end table
966:
967: @comment --- 引数の簡単な説明 ---
968: @table @var
969: @item return
1.2 takayama 970: 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 971: @item A,B
972: list
973: @end table
974:
975:
976: @comment --- @example〜@end example は実行例の表示 ---
977:
978: @example
979: [2644] A=[[a,b],[c,d]];
980: [[a,b],[c,d]]
981: [2645] B=[[e,f],[g,h]];
982: [[e,f],[g,h]]
1.2 takayama 983: [2646] mt_gkz.kronecker_prd(A,B);
1.1 takayama 984: [ e*a f*a e*b f*b ]
985: [ g*a h*a g*b h*b ]
986: [ e*c f*c e*d f*d ]
987: [ g*c h*c g*d h*d ]
988: @end example
989:
990:
991:
992:
993:
994:
995:
996:
997: @node mt_gkz.secondary_eq,,, Secondary equation
998: @subsection @code{mt_gkz.secondary_eq}
999: @comment --- 索引用キーワード
1000: @findex mt_gkz.secondary_eq
1001:
1002: @table @t
1003: @item mt_gkz.secondary_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
1004: :: It returns the secondary equation with respect to cocycles defined by Rvec.
1005: @end table
1006:
1007:
1008: @table @var
1009: @item return
1010: a list of coefficients of the Pfaffian system corresponding to the secondary equation (cf. equation (8) of [MT2020]).
1011: @item A,Beta,Ap,Rvec,DirX
1012: see @code{pfaff_eq}
1013: @end table
1014:
1015:
1016:
1017: @comment --- ここで関数の詳しい説明 ---
1018: @comment --- @itemize〜@end itemize は箇条書き ---
1019: @comment --- @bullet は黒点付き ---
1020: @itemize @bullet
1021: @item
1022: 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}.
1023: @item
1024: 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).
1025: @item Options xrule, shift, b_ideal,cg.
1026: Same as @code{pfaff_eq}.
1027: @end itemize
1028:
1029: @comment --- @example〜@end example は実行例の表示 ---
1030: Example:
1031: @example
1032: [2647] Beta=[b1,b2,b3]$
1033: [2648] DirX=[dx1,dx4]$
1034: [2649] Rvec=[[1,0,0,0],[0,0,1,0]]$
1035: [2650] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1036: [2651] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
1037: [2652] Xrule=[[x2,1],[x3,1]]$
1.2 takayama 1038: [2653] P=mt_gkz.secondary_eq(A,Beta,Ap,Rvec,DirX|xrule=Xrule)$
1.1 takayama 1039: --snip--
1040: [2654] length(P);
1041: 2
1042: [2655] P[0];
1043: [[(-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),
1044: (-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
1045: ((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,
1046: (-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
1047: ((-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),
1048: (b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]
1049: <--- Paffian matrix in x1 direction.
1050: [2656] P[1];
1051: [[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
1052: ((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,
1053: (-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
1054: ((-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),
1055: (b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]]
1056: <--- Paffian matrix in x4 direction.
1057: @end example
1058:
1059: @comment --- 参照(リンク)を書く ---
1060: @table @t
1061: @item Refer to
1062: @ref{mt_gkz.pfaff_eq}
1063: @end table
1064: @comment おわり.
1065:
1066:
1067:
1068:
1069: @node mt_gkz.generate_maple_file_IC,,, Secondary equation
1070: @subsection @code{mt_gkz.generate_maple_file_IC}
1071: @comment --- 索引用キーワード
1072: @findex mt_gkz.generate_maple_file_IC
1073:
1074: @table @t
1075: @item mt_gkz.generate_maple_file_IC(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
1076: :: It returns the maple input for a solver of a Pfaffian system IntegrableConnections[RationalSolutions].
1077: @end table
1078:
1079: @comment --- 引数の簡単な説明 ---
1080: @table @var
1081: @item return
1082: 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}).
1083: @item A,Beta,Ap,Rvec,DirX
1084: see @code{pfaff_eq}.
1085: @end table
1086:
1087: @comment --- ここで関数の詳しい説明 ---
1088: @comment --- @itemize〜@end itemize は箇条書き ---
1089: @comment --- @bullet は黒点付き ---
1090: @itemize @bullet
1091: @item
1092: 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].
1093: @item
1094: 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,...].
1095: @item Options xrule, shift, b_ideal,cg.
1096: Same as @code{pfaff_eq}.
1097: @item Option filename.
1098: 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".
1099: @end itemize
1100:
1101: @comment --- @example〜@end example は実行例の表示 ---
1102: Example:
1103: @example
1104: [2681] Beta=[b1,b2,1/3]$
1105: [2682] DirX=[dx1,dx4]$
1106: [2683] Rvec=[[1,0,0,0],[0,0,1,0]]$
1107: [2684] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1108: [2685] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
1109: [2687] Xrule=[[x2,1],[x3,1]]$
1.2 takayama 1110: [2688] mt_gkz.generate_maple_file_IC(A,Beta,Ap,Rvec,DirX|xrule=Xrule,filename="Test.ml")$
1.1 takayama 1111:
1112:
1113: //A file named Test.ml is automatically generated as follows:
1114:
1115:
1116:
1117: with(OreModules);
1118: with(IntegrableConnections);
1119: with(linalg);
1120: C:=[Matrix([[(-2*x1^3*x4^2+4*x1^2*x4-2*x1)/(x1^4*x4^2-2*x1^3*x4+x1^2),
1121: (b2*x4)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
1122: ((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,
1123: (-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
1124: ((-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),
1125: (b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]),
1126: Matrix([[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
1127: ((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,
1128: (-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
1129: ((-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),
1130: (b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]])];
1131: RatSols:=RationalSolutions(C,[x1,x4],['param',[b1,b2]]);
1132:
1133:
1134: /*
1135: If you run the output file on maple, you obtain a rational solution of
1136: the secondary equation.
1137: */
1138:
1139: [b2*(3*b1-1)/(b1*x1^2)]
1140: RatSols:=[3*b2/x1 ]
1141: [3*b2/x1 ]
1142: [3*b2-1 ]
1143:
1144: /*
1145: Note that the 4 entries of this vector correspond to entries of a 2 by 2 matrix.
1146: They are aligned as (1,1), (1,2), (2,1) (2,2) from the top.
1147: */
1148: @end example
1149:
1150: @*
1151:
1152: @comment --- 参照(リンク)を書く ---
1153: @table @t
1154: @item Refer to
1155: @ref{mt_gkz.pfaff_eq}
1156: @end table
1157: @comment おわり.
1158:
1159:
1160:
1161:
1162:
1163:
1164:
1165:
1166: @node mt_gkz.generate_maple_file_MR,,, Secondary equation
1167: @subsection @code{mt_gkz.generate_maple_file_MR}
1168: @comment --- 索引用キーワード
1169: @findex mt_gkz.generate_maple_file_MR
1170:
1171: @table @t
1172: @item mt_gkz.generate_maple_file_MR(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX},@var{D1},@var{D2})
1173: :: It returns the maple input for a solver of a Pfaffian system MorphismsRat[OreMorphisms].
1174: @end table
1175:
1176: @comment --- 引数の簡単な説明 ---
1177: @table @var
1178: @item return
1179: 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.
1180: @item A,Beta,Ap,Rvec,DirX
1181: see @code{pfaff_eq}.
1182: @item D1,D2
1183: Positive integers. D1 (resp. D2) is the upper bound of the degree of the numerator (resp. denominator) of the solution.
1184: @end table
1185:
1186: @comment --- ここで関数の詳しい説明 ---
1187: @comment --- @itemize〜@end itemize は箇条書き ---
1188: @comment --- @bullet は黒点付き ---
1189: @itemize @bullet
1190: @item
1191: 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]).
1192: @item
1193: 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.
1194: @item Options xrule, shift, b_ideal,cg.
1195: Same as @code{pfaff_eq}.
1196: @item Option filename.
1197: You can specify the file name as in @code{generate_maple_file_IC}.
1198: @item
1199: 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].
1200: @end itemize
1201:
1202: @comment --- @example〜@end example は実行例の表示 ---
1203: Example:
1204: @example
1205: [2668] Beta=[b1,b2,1/3]$
1206: [2669] DirX=[dx1,dx4]$
1207: [2670] Rvec=[[1,0,0,0],[0,0,1,0]]$
1208: [2671] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1209: [2672] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
1210: [2673] Xvar=[x1,x4]$
1211: [2674] Xrule=[[x2,1],[x3,1]]$
1.2 takayama 1212: [2675] mt_gkz.generate_maple_file_MR(A,Beta,Ap,Rvec,DirX,2,2|xrule=Xrule)$
1.1 takayama 1213:
1214:
1215: //A file "auto-generated-MR.ml" is automatically generated as follows:
1216:
1217:
1218: with(OreModules);
1219: with(OreMorphisms);
1220: with(linalg);
1221: Alg:=DefineOreAlgebra(diff=[dx1,x1],diff=[dx4,x4],polynom=[x1,x4],comm=[b1,b2]);
1222: P:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
1223: -Matrix([[((b1+b2-4/3)*x1*x4-b1+4/3)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1)],
1224: [(-b1)/(x1*x4-1),(b1*x4)/(x1*x4-1)],[(b2*x1)/(x1*x4-1),(-b2)/(x1*x4-1)],
1225: [(-b1*x1)/(x1*x4^2-x4),(1/3*x1*x4+b1-1/3)/(x1*x4^2-x4)]]);
1226: Q:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
1227: +Matrix([[((-b1-b2-2/3)*x1*x4+b1+2/3)/(x1^2*x4-x1),(b1)/(x1*x4-1)],
1228: [(b2*x4)/(x1^2*x4-x1),(-b1*x4)/(x1*x4-1)],[(-b2*x1)/(x1*x4-1),(b1*x1)/(x1*x4^2-x4)],
1229: [(b2)/(x1*x4-1),(-1/3*x1*x4-b1+1/3)/(x1*x4^2-x4)]]);
1230: RatSols:=MorphismsRat(P,Q,Alg,0,2,2);
1231:
1232: /*
1233: If you run the output file on maple, you obtain a vector RatSols.
1234: RatSols[1] is the rational solution of the secondary equation:
1235: */
1236:
1237: 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}})]
1238: [@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}})]
1239:
1240: /*
1241: 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}.
1242: */
1243: @end example
1244:
1245: @*
1246:
1247: @comment --- 参照(リンク)を書く ---
1248: @table @t
1249: @item Refer to
1250: @ref{mt_gkz.pfaff_eq}, @ref{mt_gkz.generate_maple_file_IC}.
1251: @end table
1252: @comment おわり.
1253:
1254:
1255:
1256:
1257:
1258:
1259:
1260:
1261:
1262:
1263:
1264:
1265: @node Normalizing constant,,, Cohomology intersection numbers
1266: @section Normalizing the cohomology intersection matrix
1267:
1268:
1269: @node mt_gkz.principal_normalizing_constant,,, Normalizing constant
1270: @subsection @code{mt_gkz.principal_normalizing_constant}
1271: @comment --- 索引用キーワード
1272: @findex mt_gkz.principal_normalizing_constant
1273:
1274: @table @t
1275: @item mt_gkz.principal_normalizing_constant(@var{A},@var{T},@var{Beta},@var{K})
1276: :: It returns the normalizing constant of the cohomology intersection matrix in terms of a regular triangulation T.
1277: @end table
1278:
1279: @comment --- 引数の簡単な説明 ---
1280: @table @var
1281: @item return
1.3 takayama 1282: 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 1283: @item A,Beta
1284: see @code{pfaff_eq}.
1285: @item T
1286: a regular triangulation of A.
1287: @item K
1288: The number of polynomial factors in the integrand. see [MT2020].
1289: @end table
1290:
1291: @comment --- ここで関数の詳しい説明 ---
1292: @comment --- @itemize〜@end itemize は箇条書き ---
1293: @comment --- @bullet は黒点付き ---
1294: @itemize @bullet
1295: @item
1.3 takayama 1296: 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 1297: @item
1.3 takayama 1298: One can find a regular triangulation by using a function @code{mt_gkz.regular_triangulation}.
1299: @item
1300: @code{mt_gkz.leading_terms} can be used more generally.
1.1 takayama 1301: @end itemize
1302:
1303: @comment --- @example〜@end example は実行例の表示 ---
1304: Example:
1305: @example
1306: [2676] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1307: [2677] Beta=[b1,b2,b3]$
1308: [2678] K=2$
1309: [2679] T=[[1,2,3],[2,3,4]]$
1.2 takayama 1310: [2680] mt_gkz.principal_normalizing_constant(A,T,Beta,K);
1.1 takayama 1311: (-b1-b2)/(b3*b1+b3*b2-b3^2)
1312: @end example
1313:
1314: @comment --- 参照(リンク)を書く ---
1315: @table @t
1316: @item Refer to
1.3 takayama 1317: @ref{mt_gkz.leading_terms}.
1.1 takayama 1318: @end table
1319: @comment おわり.
1320:
1321:
1322:
1323:
1324:
1325:
1326:
1327:
1.3 takayama 1328: @node mt_gkz.leading_terms,,, Normalizing constant
1329: @subsection @code{mt_gkz.leading_terms}
1.1 takayama 1330: @comment --- 索引用キーワード
1331: @findex mt_gkz.leading_terms
1332:
1333: @table @t
1.3 takayama 1334: @item mt_gkz.leading_terms(@var{A},@var{Beta},@var{W},@var{Q1},@var{Q2},@var{K},@var{N})
1.1 takayama 1335: :: It returns the W-leading terms of a cohomology intersection number specified by Q1 and Q2 up to W-degree=(minimum W-degree)+N.
1336: @end table
1337:
1338: @comment --- 引数の簡単な説明 ---
1339: @table @var
1340: @item return
1.3 takayama 1341: 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 1342: @item A,Beta
1343: see @code{pfaff_eq}.
1344: @item W
1345: a positive and integral weight vector.
1346: @item Q1,Q2
1347: @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}.
1348: @item K
1349: The number of polynomial factors in the integrand. see [MT2020].
1350: @item N
1351: A positive integer.
1352: @end table
1353:
1354: @comment --- ここで関数の詳しい説明 ---
1355: @comment --- @itemize〜@end itemize は箇条書き ---
1356: @comment --- @bullet は黒点付き ---
1357: @itemize @bullet
1358: @item
1359: 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}.
1360: @item
1.3 takayama 1361: 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 1362: @item
1363: 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.
1364: @item Option xrule.
1365: Same as @code{pfaff_eq}.
1366: @end itemize
1367:
1368: @comment --- @example〜@end example は実行例の表示 ---
1369: Example:
1370: @example
1371: [2922] Beta=[b1,b2,1/3];
1372: [b1,b2,1/3]
1373: [2923] Q=[[1,0,0],[0,1,0]];
1374: [[1,0,0],[0,1,0]]
1375: [2924] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]];
1376: [[1,1,0,0],[0,0,1,1],[0,1,0,1]]
1377: [2925] W=[1,0,0,0];
1378: [1,0,0,0]
1379: [2926] K=2;
1380: 2
1381: [2927] N=2;
1382: 2
1.3 takayama 1383: [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 1384: --snip--
1385: [2929] NC;
1386: [[(-3)/(x1),-5],[0,-4],[0,-3]]
1387:
1388:
1389: /*
1390: This output means that the W-leading term of the (1,2) entry of the cohomology
1391: 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
1392: intersection matrix is given by
1393: */
1394:
1395: [-(3*b1-1)/(b1*x1^2) -3/x1 ]
1396: [-3/x1 -(3*b2-1)/b2]]
1397:
1398:
1399: //divided by 2@math{\pi\sqrt{-1}}.
1400: @end example
1401:
1402:
1403: @comment --- 参照(リンク)を書く ---
1404: @table @t
1405: @item Refer to
1.3 takayama 1406: @ref{mt_gkz.leading_terms}, @ref{mt_gkz.generate_maple_file_IC}, @ref{mt_gkz.generate_maple_file_MR}.
1.1 takayama 1407: @end table
1408: @comment おわり.
1409:
1410:
1411:
1412:
1413:
1414: @node mt_gkz.leading_term_rat,,, Normalizing constant
1415: @subsection @code{mt_gkz.leading_term_rat}
1416: @comment --- 索引用キーワード
1417: @findex mt_gkz.leading_term_rat
1418:
1419: @table @t
1420: @item mt_gkz.leading_term_rat(@var{P},@var{W},@var{V})
1421: :: It returns the W-leading term of a rational function P depending on variables V.
1422: @end table
1423:
1424: @comment --- 引数の簡単な説明 ---
1425: @table @var
1426: @item return
1427: It returns the W-leading term of a rational function P.
1428: @item P
1429: a rational function.
1430: @item W
1431: a weight vector.
1432: @item V
1433: a list of variables of P.
1434: @end table
1435:
1436: @comment --- ここで関数の詳しい説明 ---
1437: @comment --- @itemize〜@end itemize は箇条書き ---
1438: @comment --- @bullet は黒点付き ---
1439: @itemize @bullet
1440: @item
1.3 takayama 1441: This function is supposed to be combined with @code{leading_terms} to compute the leading term of a cohomology intersection number.
1.1 takayama 1442: @item
1443: 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."
1444: @end itemize
1445:
1446:
1447: @comment --- 参照(リンク)を書く ---
1448: @table @t
1449: @item Refer to
1.3 takayama 1450: @ref{mt_gkz.leading_terms}.
1.1 takayama 1451: @end table
1452: @comment おわり.
1453:
1454:
1455:
1456:
1457:
1458:
1459: @node Regular triangulations,,, Cohomology intersection numbers
1460: @section Regular triangulations
1461:
1462: @comment --- 個々の関数の説明 ---
1463: @comment --- section 名を正確に ---
1464: @node mt_gkz.toric_gen_initial,,, Regular triangulations
1465: @node mt_gkz.regular_triangulation,,, Regular triangulations
1466: @node mt_gkz.top_standard_pairs,,, Regular triangulations
1467: @subsection @code{mt_gkz.toric_gen_initial}, @code{mt_gkz.regular_triangulation}, @code{mt_gkz.top_standard_pairs}
1468: @comment --- 索引用キーワード
1469: @findex mt_gkz.toric_gen_initial
1470: @findex mt_gkz.regular_triangulation
1471: @findex mt_gkz.top_standard_pairs
1472:
1473: @table @t
1474: @item mt_gkz.toric_gen_initial(@var{A},@var{W})
1475: @item mt_gkz.regular_triangulation(@var{A},@var{W})
1476: @item mt_gkz.top_standard_pairs(@var{A},@var{W})
1477: :: 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.
1478: @end table
1479:
1480: @comment --- 引数の簡単な説明 ---
1481: @table @var
1482: @item return
1483: @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}.
1484: @item return
1485: @code{regular_triangulation} returns a list of simplices of a regular triangulation @math{T_W} specified by the weight W.
1486: @item return
1487: @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.
1488: @item A
1489: a configuration matrix.
1490: @item W
1491: a positive weight vector.
1492: @end table
1493:
1494: @comment --- ここで関数の詳しい説明 ---
1495: @comment --- @itemize〜@end itemize は箇条書き ---
1496: @comment --- @bullet は黒点付き ---
1497: @itemize @bullet
1498: @item
1499: As for the definition of the standard pair, see Chapter 3 of [SST00].
1500: @item
1501: 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}}.
1502: @item
1.3 takayama 1503: 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 1504: @item
1.3 takayama 1505: These functions are utilized in @code{leading_terms}.
1.1 takayama 1506: @end itemize
1507:
1508: @comment --- @example〜@end example は実行例の表示 ---
1509: Example: An example of a non-unimodular triangulation and non-trivial standard pairs.
1510: @example
1511: [3256] A=[[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]];
1512: [[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]]
1513: [3257] W=[2,0,1,2,2];
1514: [2,0,1,2,2]
1.3 takayama 1515: [3258] mt_gkz.toric_gen_initial(A,W);
1.1 takayama 1516: --snip--
1517: [[x1*x5,x1*x4,x3^2*x4],[x1,x2,x3,x4,x5]]
1.3 takayama 1518: [3259] mt_gkz.regular_triangulation(A,W);
1.1 takayama 1519: --snip--
1520: [[2,4,5],[2,3,5],[1,2,3]]
1.3 takayama 1521: [3260] mt_gkz.top_standard_pairs(A,W);
1.1 takayama 1522: --snip--
1523: [[[[0,0],[0,1]],[2,4,5]],[[[0,0]],[2,3,5]],[[[0,0]],[1,2,3]]]
1524:
1525: /*
1526: This means that the regular triangulation of the configuration matrix A is
1527: given by @math{T=\{\{2,4,5\},\{2,3,5\},\{1,2,3\}\}}. The normalized volumes of these simplices
1528: are 2,1 and 1. Moreover, the top-dimensional standard pairs are
1529: @math{(1,\{2,4,5\}),(\partial_3,\{2,4,5\})}, @math{(1,\{2,3,5\})},@math{(1,\{1,2,3\})}.
1530: */
1531: @end example
1532:
1533:
1534:
1535: @comment --- 参照(リンク)を書く ---
1536: @table @t
1537: @item Refer to
1.3 takayama 1538: @ref{mt_gkz.leading_terms}.
1.1 takayama 1539: @end table
1540: @comment おわり.
1541:
1542:
1543:
1544:
1545:
1546:
1547:
1548: @comment --- おまじない ---
1549: @node Index,,, Top
1550: @unnumbered Index
1551: @printindex fn
1552: @printindex cp
1553: @iftex
1554: @vfill @eject
1555: @end iftex
1556: @summarycontents
1557: @contents
1558: @bye
1559: @comment --- おまじない終り ---
1560:
1561: @comment *********************************************************
1562: @comment ********* template
1563: @comment **********************************************************
1564: @comment --- 個々の関数の説明 ---
1565: @comment --- section 名を正確に ---
1566: @node mt_gkz.pfaff_eq,,, Pfaff equation for given cocycles
1567: @subsection @code{mt_gkz.pfaff_eq}
1568: @comment --- 索引用キーワード
1569: @findex mt_gkz.pfaff_eq
1570:
1571: @table @t
1572: @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
1573: :: It returns the Pfaff equation for the GKZ system defined by @var{A} and @var{Beta} with respect to cocycles defined by @var{Rvec}.
1574: @end table
1575:
1576: @comment --- 引数の簡単な説明 ---
1577: @table @var
1578: @item return
1579: a list of coefficients of the Pfaff equation with respect to the direction @var{DirX}
1580: @item A
1581: the matrix A of the GKZ system.
1582: @item Beta
1583: ...
1584: @end table
1585:
1586: @comment --- ここで関数の詳しい説明 ---
1587: @comment --- @itemize〜@end itemize は箇条書き ---
1588: @comment --- @bullet は黒点付き ---
1589: @itemize @bullet
1590: @item
1591: The independent variables are @code{x1,x2,x3,...} (@math{x_1, x_2, x_3, \ldots}).
1592: @end itemize
1593:
1594: @comment --- @example〜@end example は実行例の表示 ---
1595: Example: Gauss hypergeometric system, see [GM2020] example ??.
1596: @example
1597: [1883] import("mt_gkz.rr");
1598: @end example
1599:
1600: @comment --- 参照(リンク)を書く ---
1601: @table @t
1602: @item Refer to
1603: @ref{mt_gkz.pfaff_eq}
1604: @end table
1605: @comment おわり.
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