Annotation of OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi, Revision 1.5
1.5 ! takayama 1: %% $OpenXM: OpenXM/src/asir-contrib/packages/doc/mt_gkz/mt_gkz-en.texi,v 1.4 2021/03/29 05:08:01 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
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
1.5 ! takayama 137: @findex mt_gkz.use_hilbert_driven
1.1 takayama 138:
139: @table @t
140: @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
141: :: It returns the Pfaff equation for the GKZ system defined by @var{A} and @var{Beta} with respect to cocycles defined by @var{Rvec}.
142: @end table
143:
144: @comment --- 引数の簡単な説明 ---
145: @table @var
146: @item return
147: a list of coefficients of the Pfaff equation with respect to the direction @var{DirX}
148: @item A
149: the matrix A of the GKZ system.
150: @item Beta
151: the parameter vector of the GKZ system.
152: @item Ap
153: See [MT2020].
154: @item Rvec
155: It is used to specify a basis of cocycles. See [MT2020]
156: @item DirX
157: a list of dxi's.
158: @end table
159:
160: @comment --- ここで関数の詳しい説明 ---
161: @comment --- @itemize〜@end itemize は箇条書き ---
162: @comment --- @bullet は黒点付き ---
163: @itemize @bullet
164: @item
165: The independent variables are x1, x2, x3, ...
166: @item
167: When @var{Rvec}=[v_1, v_2, ..., v_r] where r is the rank of the GKZ system,
168: the set of the cocycles standing for Av_1, Av_2, ..., Av_r
169: (see [MT2020])
170: is supposed
171: to be the basis to construct the Pfaffian system.
172: Let a_1, a_2, ..., a_n be the column vectors of the matrix A
173: and v be a column vector (x_1, x_2, ..., x_n)^T.
174: Av is defined as a_1 x_1 + a_2 x_2 + ... + a_n x_n.
175: @item
176: When the columns of @var{A} are expressed as
177: @math{e_i \otimes \alpha_{i_j}},
178: the columns of @var{Ap} is
179: @math{e_i \otimes 0} where $e_i$ is the i-th unit vector.
180: See [MT2020] on the definition of @var{Ap}.
181: Here are some examples.
182: When @var{A} is
183: @verbatim
184: [[1,1,0,0],
185: [0,0,1,1],
186: [0,1,0,1]]
187: @end verbatim
188: @var{Ap} is
189: @verbatim
190: [[1,1,0,0],
191: [0,0,1,1],
192: [0,0,0,0]] <-- zero row
193: @end verbatim
194: When @var{A} is
195: @verbatim
196: [[1,1,1,0,0,0],
197: [0,0,0,1,1,1],
198: [0,1,0,0,1,0],
199: [0,0,1,0,0,1]
200: ]
201: @end verbatim
202: @var{Ap} is
203: @verbatim
204: [[1,1,1,0,0,0],
205: [0,0,0,1,1,1],
206: [0,0,0,0,0,0], <-- zero row
207: [0,0,0,0,0,0] <-- zero row
208: ]
209: @end verbatim
210: See also page 223 of [SST2000].
211: @item
212: Option @var{xrule}. When the option @var{xrule} is given,
213: the x variables specified by this option are specialized to numbers.
214: @item
215: Option @var{shift}. When the matrix @var{A} is not normal
216: (the associated toric ideal is not normal), a proper shift vector
217: must be given to obtain an element of the b-ideal. Or, use the option
218: @var{b_ideal} below. See [SST1999] on the theory.
219: @item
220: Option @var{b_ideal}. When the matrix @var{A} is not normal,
221: the option @code{b_ideal=1} obtains b-ideals and the first element
222: of each b-ideal is used as the b-function. The option @var{shift}
223: is ignored.
224: @item
225: Option @var{cg}. A constant matrix given by this option is used
226: for the Gauge transformation of the Pfaffian system.
227: In other words, the basis of cocycles specified by @var{Rvec}
228: is transformed by the constant matrix given by this option.
1.5 ! takayama 229: @item
! 230: By mt_gkz.use_hilbert_driven(Rank), the rank of the GKZ system is assumed to be
! 231: Rank. It makes the computation of Groebner basis by yang.rr faster.
! 232: This option is disabled by mt_gkz.use_hilbert_driven(0);
1.1 takayama 233: @end itemize
234:
235: @comment --- @example〜@end example は実行例の表示 ---
236: Example: Gauss hypergeometric system, see [GM2020] example ??.
237: @example
238: [1883] import("mt_gkz.rr");
239: [2657] PP=mt_gkz.pfaff_eq(A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]],
240: Beta=[-g1,-g2,-c],
241: Ap = [[1,1,0,0],[0,0,1,1],[0,0,0,0]],
242: Rvec = [[1,0,0,0],[0,0,1,0]],
243: DirX=[dx4,dx3] | xrule=[[x1,1],[x2,1]],
244: cg=matrix_list_to_matrix([[1,0],[-1,1]]))$
245:
246: 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]
247: -- snip --
248: [2658] PP[0];
249: [ (g2*x3-g2)/(x4-x3) (g2*x3)/(x4-x3) ]
250: [ ((-g2*x3-c+g2)*x4+(c-g1)*x3+g1)/(x4^2-x3*x4)
251: ((-g2*x3-c)*x4+(c-g1)*x3)/(x4^2-x3*x4) ]
252: [2659] PP[1];
253: [ (-g2*x4+g2)/(x4-x3) (-g2*x4)/(x4-x3) ]
254: [ ((g2*x3+c-g2-1)*x4+(-c+g1+1)*x3-g1)/(x3*x4-x3^2)
255: ((g2*x3+c-g2-1)*x4+(-c+g1+g2+1)*x3)/(x3*x4-x3^2) ]
256: @end example
257:
258: @*
259:
260: Example: The role of shift.
261: When the toric ideal is not normal, a proper shift vector
262: must be given with the option @code{shift} to find an element of the b-ideal.
263: @example
264: [1882] load("mt_gkz.rr");
265: [1883] A=[[1,1,1,1],[0,1,3,4]];
266: [[1,1,1,1],[0,1,3,4]]
267: [1884] Ap=[[1,1,1,1],[0,0,0,0]];
268: [[1,1,1,1],[0,0,0,0]]
269: [1885] Rvec=[[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
270: [[0,0,0,0],[0,0,1,0],[0,0,0,1],[0,0,0,2]];
271: [2674] P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
272: | xrule=[[x1,1],[x2,2],[x3,4]] )$
273: dx remains
274: stopped in step_up at line 342 in file "./mt_gkz/saito-b.rr"
275: 342 if (type(dn(Ans)) > 1) error("dx remains");
276: (debug) quit
277: // Since the toric ideal for A is not normal, it stops with the error.
278: [2675] P=mt_gkz.pfaff_eq(A,[b1,b2],Ap,Rvec,DirX=[dx4]
279: | shift=[1,0],xrule=[[x1,1],[x2,2],[x3,4]])$
280: // It works.
281: @end example
282:
283: @comment --- 参照(リンク)を書く ---
284: @table @t
285: @item Refer to
286: @ref{mt_gkz.ff1}
287: @ref{mt_gkz.ff2}
288: @ref{mt_gkz.ff}
289: @ref{mt_gkz.rvec_to_fvec}
290: @end table
291: @comment mt_gkz.pfaff_eq の説明おわり. あとはこれの繰り返し.
292:
293:
294: @comment --- 個々の関数の説明 ---
295: @comment --- section 名を正確に ---
296: @node mt_gkz.ff2,,, Pfaff equation for given cocycles
297: @node mt_gkz.ff1,,, Pfaff equation for given cocycles
298: @node mt_gkz.ff,,, Pfaff equation for given cocycles
299: @subsection @code{mt_gkz.ff2}, @code{mt_gkz.ff1}, @code{mt_gkz.ff}
300: @comment --- 索引用キーワード
301: @findex mt_gkz.ff2
302: @findex mt_gkz.ff1
303: @findex mt_gkz.ff
304:
305: @table @t
306: @item mt_gkz.ff(@var{Rvec0},@var{A},@var{Beta},@var{Ap})
307: @item mt_gkz.ff1(@var{Rvec0},@var{A},@var{Beta},@var{Ap})
308: @item mt_gkz.ff2(@var{Rvec0},@var{A},@var{Beta},@var{Ap},@var{BF},@var{C})
309: :: @code{ff} returns a differential operator whose action to 1 gives
310: the cocycle defined by @var{Rvec0}
311: @end table
312:
313: @comment --- 引数の簡単な説明 ---
314: @table @var
315: @item return
316: @code{ff} returns a differential operator whose action to 1 of @math{M_A(\beta)}
317: gives the cocycle defined by @var{Rvec0}.
318: @item return
319: @code{ff1} returns a composite of step-down operators for the positive part
320: of @var{Rvec0}
321: @item return
322: @code{ff2} returns a composite of step-up operators for the positive part
323: of @var{Rvec0}
324: @item Rvec0
325: An element of @var{Rvec} explained in @ref{mt_gkz.pfaff_eq}.
326: @item BF
327: the list of b-functions to all directions.
328: @item C
329: the list of the step up operators for all a_1, a_2, ..., a_n.
330: @end table
331: Other arguments are same with those of @code{pfaff_eq}.
332:
333: @comment --- ここで関数の詳しい説明 ---
334: @comment --- @itemize〜@end itemize は箇条書き ---
335: @comment --- @bullet は黒点付き ---
336: @itemize @bullet
337: @item
338: The function @code{ff} generates the list of b-functions and the list of
339: step up operators and store them in the cache variable.
340: They can be obtained by calling as @code{S=mt_gkz.get_bf_step_up()}
341: where S[0] is the list of b-functions and S[1] is the list of step up
342: operators.
343: Step up operators are obtained by the algorithm given in [SST1999].
344: @item
345: Option nf. When nf=1, the output operator is reduced to the normal form
346: with respect to the Groebner basis of the GKZ system of the graded reverse
347: lexicographic order.
348: @item
349: Option shift. See @ref{mt_gkz.pfaff_eq}.
350: @item
351: Internal info: The function @code{mt_gkz.bb} gives the constant so that
352: the step up and step down operators (contiguity operators) give
353: contiguity relations for the integral representation in [MT2020].
354: Note that @code{mt_gkz.ff1} and @code{mt_gkz.ff2} give contiguity
355: relations which are constant multiple of those for hypergeometric
356: polynomials.
357: @item
358: Internal info: @code{mt_gkz.step_up} generates step up operators
359: of [SST1999] from b-functions by utilizing @code{mt_gkz.bf2euler}
360: and @code{mt_gkz.toric}.
361: @end itemize
362:
363: @comment --- @example〜@end example は実行例の表示 ---
364: Example: Step up operators compatible with the integral representation in [MT2020].
365: The function hgpoly_res defined in @code{check-by-hgpoly.rr} returns
366: a multiple of the hypergeometric polynomial which agrees with
367: the residue times a power of @math{2\pi \sqrt{-1}}
368: of the integral representation.
369: See [SST1999].
370: @example
371: [1883] import("mt_gkz.rr")$
372: [3175] load("mt_gkz/check-by-hgpoly.rr")$
373: [3176] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
374: [3177] B=newvect(3,[5,4,7])$ Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
375: [3179] Beta=[b1,b2,b3]$ R=[0,0,-1,0]$
376: [3180] F2=hgpoly_res(A,B,2); // HG polynomial. 2 is the number of e_i's.
377: 10*x1^2*x2^3*x4^4+20*x1*x2^4*x3*x4^3+6*x2^5*x3^2*x4^2
378: [3182] mt_gkz.ff(R,A,Ap,Beta); // the operator standing for R
379: (x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
380: [3184] S=mt_gkz.get_bf_step_up(A); // b-function and non-reduced step up op's
381: [[ 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 ],
382: [ x2*x3*dx4+x1*x3*dx3+x1*x2*dx2+x1^2*dx1+x1
383: x2*x4*dx4+x1*x4*dx3+x2^2*dx2+x1*x2*dx1+x2
384: x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3
385: x4^2*dx4+x3*x4*dx3+x2*x4*dx2+x2*x3*dx1+x4 ]]
386: [3185] Fvec=mt_gkz.ff2(R,A,Beta,Ap,S[0],S[1]);
387: (x3*x4*dx4+x3^2*dx3+x1*x4*dx2+x1*x3*dx1+x3)/(b1+b2-b3+1)
388: [3188] Fvec = base_replace(Fvec,assoc(Beta,vtol(B)));
389: 1/3*x3*x4*dx4+1/3*x3^2*dx3+1/3*x1*x4*dx2+1/3*x1*x3*dx1+1/3*x3
390: [3189] R32d = odiff_act(Fvec,F2,[x1,x2,x3,x4]); // Act Fvec to the hg-poly
391: 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
392: [3190] red(R32d/hgpoly_res(A,B+newvect(3,[0,1,0]),2));
393: // R32d agrees with the HG polynomial with Beta=[5,4,7]+[0,1,0].
394: 1
395: @end example
396:
397: @comment --- 参照(リンク)を書く ---
398: @table @t
399: @item Refer to
400: @ref{mt_gkz.pfaff_eq}
401: @end table
402: @comment おわり.
403:
404: @comment --- 個々の関数の説明 --- Ref:2020-11-09-tw-cohom-progs.goodnotes
405: @comment --- section 名を正確に ---
406: @node mt_gkz.rvec_to_fvec,,, Pfaff equation for given cocycles
407: @subsection @code{mt_gkz.rvec_to_fvec}
408: @comment --- 索引用キーワード
409: @findex mt_gkz.rvec_to_fvec
410:
411: @table @t
412: @item mt_gkz.rvec_to_fvec(@var{Rvec},@var{A},@var{Ap},@var{Beta})
413: :: It returns a set of differential operators standing for @var{Rvec}.
414: @end table
415:
416: @comment --- 引数の簡単な説明 ---
417: @table @var
418: @item return
419: It returns a set of differential operators of which action to
420: @math{1 \in M_A(\beta)} give cocycles specified by @var{Rvec}.
421: @item A, Ap, Beta
422: Same with @ref{mt_gkz.pfaff_eq}
423: @end table
424:
425: @comment --- ここで関数の詳しい説明 ---
426: @comment --- @itemize〜@end itemize は箇条書き ---
427: @comment --- @bullet は黒点付き ---
428: @itemize @bullet
429: @item
430: Internal info: this function builds the set of operators by calling
431: @ref{mt_gkz.ff}.
432: @end itemize
433:
434: @comment --- @example〜@end example は実行例の表示 ---
435: Example: The following two expressions are congruent because
436: @math{2a_1-a_2-a_3+a_4=a_1} for this @code{A}.
437: @example
438: [1883] import("mt_gkz.rr");
439: [3191] mt_gkz.rvec_to_fvec([[2,-1,-1,1],[0,0,1,0]],
440: [[1,1,0,0],[0,0,1,1],[0,1,0,1]],
441: [[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
442: [(x2*x3*x4^2*dx1^2*dx4^3+((x1*x3*x4^2+x2*x3^2*x4)*dx1^2*dx3
443: +(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
444: +(x1*x4^2+3*x2*x3*x4)*dx1^2)*dx4^2+(x1*x3^2*x4*dx1^2*dx3^2
445: +((x1^2*x3*x4+x1*x2*x3^2)*dx1^3+(3*x1*x3*x4+x2*x3^2)*dx1^2)*dx3
446: +x1*x2^2*x4*dx1^2*dx2^2+((x1^2*x2*x4+x1*x2^2*x3)*dx1^3
447: +(3*x1*x2*x4+x2^2*x3)*dx1^2)*dx2+x1^2*x2*x3*dx1^4
448: +(x1^2*x4+3*x1*x2*x3)*dx1^3+(x1*x4+x2*x3)*dx1^2)*dx4)
449: /(b3*b2*b1^3+(b3*b2^2+(-b3^2-2*b3)*b2)*b1^2+(-b3*b2^2+(b3^2+b3)*b2)*b1),
450: (dx3)/(b2)]
451: [3192] mt_gkz.rvec_to_fvec([[1,0,0,0],[0,0,1,0]],
452: [[1,1,0,0],[0,0,1,1],[0,1,0,1]],
453: [[1,1,0,0],[0,0,1,1],[0,0,0,0]],[b1,b2,b3]);
454: [(dx1)/(b1),(dx3)/(b2)]
455: @end example
456:
457: @comment --- 参照(リンク)を書く ---
458: @table @t
459: @item Refer to
460: @ref{mt_gkz.pfaff_eq}
461: @end table
462: @comment おわり.
463:
464: @comment --- fvec_to_conn_mat
465: @comment --- section 名を正確に ---
466: @node mt_gkz.fvec_to_conn_mat,,, Pfaff equation for given cocycles
467: @subsection @code{mt_gkz.fvec_to_conn_mat}
468: @comment --- 索引用キーワード
469: @findex mt_gkz.fvec_to_conn_mat
470:
471: @table @t
472: @item mt_gkz.fvec_to_conn_mat(@var{Fvec},@var{A},@var{Beta},@var{DirX})
473: :: It returns the coefficient matrices of the basis
474: @var{Fvec} or @var{DirX}[I]*@var{Fvec} in terms of the set of the standard basis.
475: @end table
476:
477: @comment --- 引数の簡単な説明 ---
478: @table @var
479: @item return
480: It returns the coefficient matrices of the basis
481: @var{Fvec} or @var{DirX}[I]*@var{Fvec} in terms of the set of the standard basis of the Groebner basis explained below.
482: @item A Beta
483: Same with @ref{mt_gkz.pfaff_eq}.
484: @item DirX
485: When @var{DirX} is 1, this function returns the matrix which expresses
486: @var{Fvec} in terms of the set of the standard monomials of
487: the Groebner basis of the GKZ system in the ring of rational function
488: coefficients with respect to the graded reverse lexicographic order.
489: In other cases, it returns the coefficient matrices of
490: @var{DirX}[I]'s*@var{Fvec} in terms of the set of the standard basis of the Groebner basis.
491: @end table
492:
493: @comment --- ここで関数の詳しい説明 ---
494: @comment --- @itemize〜@end itemize は箇条書き ---
495: @comment --- @bullet は黒点付き ---
496: @itemize @bullet
497: @item
498: It utilizes a Groebner basis computation by the package @code{yang.rr}
499: and @code{yang.reduction} to obtain connection matrices.
500: @item
501: This function calls some utility functions
502: @code{mt_gkz.dmul(Op1,Op2,XvarList)} (multiplication of @code{Op1} and @code{Op2}
503: and @code{mt_gkz.index_vars(x,Start,End | no_=1)}
504: which generates indexed variables without the underbar ``_''.
505: @item
506: We note here some other utility functions in this section:
507: @code{mt_gkz.check_compatibility(P,Q,X,Y)},
508: which checkes if the sytem d/dX-P, d/dY-Q is compatible.
509: @end itemize
510:
511: @comment --- @example〜@end example は実行例の表示 ---
512: Example: The following example illustrates how mt_gkz.pfaff_eq
513: obtains connection matrices.
514: @example
515: [1883] import("mt_gkz.rr");
516: [3201] V=mt_gkz.index_vars(x,1,4 | no_=1);
517: [x1,x2,x3,x4]
518: [3202] mt_gkz.dmul(dx1,x1^2,V);
519: x1^2*dx1+2*x1
520: [3204] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
521: Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
522: Beta= [b1,b2,b3]$
523: Rvec = [[1,0,0,0],[0,0,1,0]]$
524: Fvec = mt_gkz.rvec_to_fvec(Rvec,A,Ap,Beta)$
525: /* Express cocyles Rvec
526: by elements Fvec in the Weyl algebra by contiguity relations. */
527: Cg = matrix_list_to_matrix([[1,0],[1,-1]])$
528: [3208] NN=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,1);
529: // Express Fvec by the standard monomials Std=NN[1].
530: 1 ooo 2 .ooo
531: [[ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
532: [ (-x4)/(b1*x2) (1)/(x3) ],[dx4,1]]
533: [3209] Std=NN[1];
534: [dx4,1]
535: [3173] NN=NN[0];
536: [ (x4)/(b1*x1) (b1-b3)/(b1*x1) ]
537: [ (-x4)/(b2*x3) (1)/(x3) ]
538: [3174] NN1=mt_gkz.fvec_to_conn_mat(Fvec,A,Beta,dx1)[0];
539: // Express dx1*Fvec by the standard monomials Std.
540: 1 ooo 2 .ooo
541: [ ((2*b1+b2-b3-1)*x1*x4^2+(-b1+b3+1)*x2*x3*x4)/(b1*x1^3*x4-b1*x1^2*x2*x3)
542: ((b1^2+(-2*b3-1)*b1-b3*b2+b3^2+b3)*x1*x4
543: +(-b1^2+(2*b3+1)*b1-b3^2-b3)*x2*x3)/(b1*x1^3*x4-b1*x1^2*x2*x3) ]
544: [(b1 (-b1*x1*x4^2-b2*x2*x3*x4)/(b2*x1^2*x3*x4-b2*x1*x2*x3^2)
545: (b1*x1*x4+(-b1+b3)*x2*x3)/(x1^2*x3*x4-x1*x2*x3^2) ]
546: [3188] P1=map(red,Cg*NN1*matrix_inverse(NN)*matrix_inverse(Cg));
547: [ ((-b2*x3+(b1+b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3)/(x1^2*x4-x1*x2*x3)
548: (b2*x3*x4)/(x1^2*x4-x1*x2*x3) ]
549: [ ((-b2*x3+(b2-b3-1)*x1)*x4+(-b1+b3+1)*x2*x3+b1*x1*x2)/(x1^2*x4-x1*x2*x3)
550: ((b2*x3+b1*x1)*x4)/(x1^2*x4-x1*x2*x3) ]
551:
552: [3191] mt_gkz.pfaff_eq(A,Beta,Ap,Rvec,[dx1]|cg=Cg)[0]-P1;
553: [ 0 0 ]
554: [ 0 0 ] // P1 agrees with the output of mt_gkz.pfaff_eq.
555: @end example
556:
557: @comment --- 参照(リンク)を書く ---
558: @table @t
559: @item Refer to
560: @ref{mt_gkz.pfaff_eq}
561: @end table
562: @comment おわり.
563:
564: @comment ---------- New Chapter ---------------
565: @node b function,,, Top
566: @chapter b function
567:
568: @menu
569: * mt_gkz.bf::
570: * mt_gkz.bf::
571: @end menu
572:
573: @node b function and facet polynomial,,, b function
574: @section b function and facet polynomial
575: @comment ------- bf
576: @comment **********************************************************
577: @comment --- 個々の関数の説明 ---
578: @comment --- section 名を正確に ---
579: @node mt_gkz.bf,,, b function and facet polynomial
580: @subsection @code{mt_gkz.bf}
581: @comment --- 索引用キーワード
582: @findex mt_gkz.bf
583:
584: @table @t
585: @item mt_gkz.bf(@var{A},@var{Facet_poly},@var{II0})
586: :: It returns the b-function with respect to the direction @var{II0}.
587: @end table
588:
589: @comment --- 引数の簡単な説明 ---
590: @table @var
591: @item return
592: 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.
593: @item A
594: the matrix A of the GKZ system.
595: @item Facet_poly
596: The set of facet polynomials of the convex hull of @var{A}.
597: @item II0
598: Direction expressed as 0, 1, 2, ... (not 1, 2, 3, ...) to obtain the b function.
599: @end table
600:
601: @comment --- ここで関数の詳しい説明 ---
602: @comment --- @itemize〜@end itemize は箇条書き ---
603: @comment --- @bullet は黒点付き ---
604: @itemize @bullet
605: @item
606: See [SST1999] on the b-function introduced Saito and b-ideal.
607: @item
608: The facet polynomial must be primitive.
609: @end itemize
610:
611: @comment --- @example〜@end example は実行例の表示 ---
612: Example:
613: @example
614: [1883] import("mt_gkz.rr");
615:
616: [3193] A;
617: [[1,1,0,0],[0,0,1,1],[0,1,0,1]]
618: [3194] Fpoly=mt_gkz.facet_poly(A);
619: [[s_3,s_1,s_2-s_3+s_1,s_2],[[0,0,1],[1,0,0],[1,1,-1],[0,1,0]]]
620: [3196] mt_gkz.bf(A,Fpoly,0);
621: s_1*s_2-s_1*s_3+s_1^2
622: [3197] mt_gkz.bf(A,Fpoly,1);
623: s_1*s_3
624: @end example
625:
626: @comment --- 参照(リンク)を書く ---
627: @table @t
628: @item Refer to
629: @ref{mt_gkz.ff}
630: @ref{mt_gkz.facet_poly}
631: @end table
632: @comment おわり.
633:
634: @comment ------ facet_poly
635: @comment --- 個々の関数の説明 ---
636: @comment --- section 名を正確に ---
637: @node mt_gkz.facet_polyl,,, b function and facet polynomial
638: @subsection @code{mt_gkz.facet_poly}
639: @comment --- 索引用キーワード
640: @findex mt_gkz.facet_poly
641:
642: @table @t
643: @item mt_gkz.facet_poly(@var{A})
644: :: It returns the set of facet polynomials and their normal vectors of
645: the cone defined by @var{A}.
646: @end table
647:
648: @comment --- 引数の簡単な説明 ---
649: @table @var
650: @item return
651: It returns the set of facet polynomials and their normal vectors of
652: the cone generated by the column vectors of the matrix @var{A}.
653: @item A
654: the matrix A of the GKZ system.
655: @end table
656:
657: @comment --- ここで関数の詳しい説明 ---
658: @comment --- @itemize〜@end itemize は箇条書き ---
659: @comment --- @bullet は黒点付き ---
660: @itemize @bullet
661: @item
662: The facet polynomial f is primitive. In other words,
663: all f(a_i) is integer and min f(a_i)=1 for a_i's not being on f=0.
664: where a_i is the i-th column vector of the matrix @var{A}.
665: It can be checked by @code{mt_gkz.is_primitive(At,Facets)}
666: where @var{At} is the transpose of @var{A} and
667: @var{Facets} is the second return value of this function.
668: @item
669: This function utilizes the system polymake @uref{https://polymake.org}
670: on our server.
671: @end itemize
672:
673: @comment --- @example〜@end example は実行例の表示 ---
674: Example:
675: @example
676: [1883] import("mt_gkz.rr");
677: [1884] mt_gkz.facet_poly([[1,1,1,1],[0,1,2,3]]);
678: oohg_native=0, oohg_curl=1
679: [[s_2,-s_2+3*s_1],[[0,1],[3,-1]]]
680: @end example
681:
682: @comment --- 参照(リンク)を書く ---
683: @table @t
684: @item Refer to
685: @ref{mt_gkz.bf}
686: @end table
687: @comment おわり.
688:
689: @comment ---------- New Chapter ---------------
690: @node utilities,,, Top
691: @chapter Utilities
692:
693: @menu
694: * mt_gkz.reduce_by_toric::
695: * mt_gkz.tk_base_equal::
696: * mt_gkz.dp_op_to_coef_vec::
697: * mt_gkz.yang_gkz_buch::
698: * mt_gkz.p_true_nf_rat::
699: * mt_gkz.mdiff::
700: * mt_gkz.dvar::
701: * mt_gkz.ord_xi::
702: * mt_gkz.get_check_fvec::
703: * mt_gkz.get_bf_step_up::
1.2 takayama 704: * mt_gkz.mytoric_ideal::
1.5 ! takayama 705: * mt_gkz.cbase_by_euler::
1.1 takayama 706: @end menu
707:
708: @node some utility functions,,, utilities
709: @section Some utility functions
710:
711: @node mt_gkz.reduce_by_toric,,, some utility functions
712: @node mt_gkz.tk_base_equal,,, some utility functions
713: @node mt_gkz.dp_op_to_coef_vec,,, some utility functions
714: @node mt_gkz.yang_gkz_buch,,, some utility functions
715: @node mt_gkz.p_true_nf_rat,,, some utility functions
716: @node mt_gkz.mdiff,,, some utility functions
717: @node mt_gkz.dvar,,, some utility functions
718: @node mt_gkz.ord_xi,,, some utility functions
719: @node mt_gkz.get_check_fvec,,, some utility functions
720: @node mt_gkz.get_bf_step_up,,, some utility functions
1.2 takayama 721: @node mt_gkz.mytoric_ideal,,, some utility functions
1.5 ! takayama 722: @node mt_gkz.cbase_by_euler,,, some utility functions
1.1 takayama 723:
724: @findex mt_gkz.reduce_by_toric
725: @findex mt_gkz.tk_base_equal
726: @findex mt_gkz.dp_op_to_coef_vec
727: @findex mt_gkz.yang_gkz_buch
728: @findex mt_gkz.p_true_nf_rat
729: @findex mt_gkz.mdiff
730: @findex mt_gkz.dvar
731: @findex mt_gkz.ord_xi
732: @findex mt_gkz.get_check_fvec
733: @findex mt_gkz.get_bf_step_up
1.2 takayama 734: @findex mt_gkz.mytoric_ideal
1.5 ! takayama 735: @findex mt_gkz.cbase_by_euler
1.1 takayama 736:
737: @comment --- @example〜@end example は実行例の表示 ---
738: We only show examples on these functions. As for details, please see
739: the source code.
740: @example
741: [1883] import("mt_gkz.rr");
742: [2667] mt_gkz.dvar([x1,x2]); // it generates variables starting with d
743: [dx1,dx2]
744: [2669] mt_gkz.p_true_nf_rat((1/3)*x^3-1,[x^2-1],[x],0);
745: [x-3,3] // p_true_nf does not accept rational number coefficients
746: [2670] A=[[1,1,1,1],[0,1,3,4]];
747: [[1,1,1,1],[0,1,3,4]]
748: [2671] mt_gkz.reduce_by_toric(dx3^4,A);
749: dx1*dx4^3 // reduction by toric ideal defined by A
750: [2672] nk_toric.toric_ideal(A);
751: [-x1*x4+x2*x3,-x2*x4^2+x3^3,x2^2*x4-x1*x3^2,-x1^2*x3+x2^3]
752: [2673] mt_gkz.yang_gkz_buch(A,[b1,b2]); // Groebner basis of GKZ system by yang.rr
753: 1 o 2 ..o 3 ..oooooooo 4 o 6 ooo 9 o
754: [[[(x2)*<<0,1,0,0>>+(3*x3)*<<0,0,1,0>>+ ---snip ---*<<0,0,0,0>>,1]],
755: [dx1,dx2,dx3,dx4],
756: [(1)*<<0,0,0,2>>,(1)*<<0,0,1,0>>,(1)*<<0,0,0,1>>,(1)*<<0,0,0,0>>]]
757:
758: [2674] mt_gkz.dp_op_to_coef_vec([x1*<<1,0>>+x1*x2*<<0,1>>,x1+1],[<<1,0>>,<<0,1>>]);
759: // x1+1 is the denominator
760: [ (x1)/(x1+1) (x1*x2)/(x1+1) ]
761: [2675] mt_gkz.tk_base_is_equal([1,2],[1,2]);
762: 1
763: [2676] mt_gkz.tk_base_is_equal([1,2],[1,x,y]);
764: 0
765: [2677] mt_gkz.mdiff(sin(x),x,1);
766: cos(x)
767: [2678] mt_gkz.mdiff(sin(x),x,2); //2nd derivative
768: -sin(x)
769: [3164] mt_gkz.ord_xi(V=[x1,x2,x3],II=1);
770: // matrix to define graded lexicographic order so that V[II] is the smallest.
771: [ 1 1 1 ]
772: [ 0 -1 0 ]
773: [ -1 0 0 ]
774: [3166] load("mt_gkz/check-by-hgpoly.rr");
775: [3187] check_0123(); // check the pfaffian for the A below by hg-polynomial.
776: A=[[1,1,1,1],[0,1,2,3]]
777: Ap=[[1,1,1,1],[0,0,0,0]]
778: --- snip ---
779: Bfunctions= --- snip ---
780: 0 (vector) is expected:
781: [[ 0 0 0 ],[ 0 0 0 ]]
782: [3188] mt_gkz.get_check_fvec();
783: // get the basis of cocycles used in terms of differential operators.
784: [1,(dx4)/(b1),(dx4^2)/(b1^2-b1)]
785: [3189] mt_gkz.clear_bf();
786: 0
787: [3190] mt_gkz.get_bf_step_up(A=[[1,1,1,1],[0,1,2,3]]);
788: // b-functions and step-up operators.
789: // Option b_ideal=1 or shift=... may be used for non-normal case.
790: [[ -s_2^3+(9*s_1-3)*s_2^2+ ---snip---
791: -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 ],
792: [ x3^3*dx4^2+ ---snip---
793: 3*x3^2*x4*dx4^2+ --- snip---]]
1.2 takayama 794: [3191] mt_gkz.mytoric_ideal(0 | use_4ti2=1);
795: // 4ti2 is used to obtain a generator set of the toric ideal
796: // defined by the matrix A
797: [3192] mt_gkz.mytoric_ideal(0 | use_4ti2=0);
798: // A slower method is used to obtain a generator set of the toric ideal
799: // defined by the matrix A. 4ti2 is not needed. Default.
1.5 ! takayama 800: [3193] mt_gkz.cbase_by_euler(A=[[1,1,1,1],[0,1,3,4]]);
! 801: // Cohomology basis of the GKZ system defined by A for generic beta.
! 802: // Basis is given by a set of Euler operators.
! 803: // It is an implementation of the algorithm in http://dx.doi.org/10.1016/j.aim.2016.10.021
! 804: // beta is set by random numbers. Option: no_prob=1
! 805:
1.1 takayama 806: @end example
807:
808:
809:
810:
811:
812:
813:
814:
815:
816:
817:
818:
819:
820:
821:
822:
823:
824:
825:
826:
827:
828:
829:
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831:
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833:
834:
835:
836:
837:
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839:
840:
841:
842:
843:
844:
845:
846:
847:
848:
849:
850:
851:
852:
853:
854:
855:
856:
857:
858:
859:
860:
861:
862:
863: @comment ここから追加版
864:
865: @node Cohomology intersection numbers,,, Top
866: @chapter Cohomology intersection numbers
867:
868: @menu
869: * mt_gkz.kronecker_prd::
870: * mt_gkz.secondary_eq::
871: * mt_gkz.generate_maple_file_IC::
872: * mt_gkz.generate_maple_file_MR::
1.2 takayama 873: * mt_gkz.principal_normalizing_constant::
1.1 takayama 874: @end menu
875:
876:
877:
878:
879:
880:
881: @node Secondary equation,,, Cohomology intersection numbers
882: @section Secondary equation
883:
884: @comment **********************************************************
885: @comment --- 関数 pfaff_eq
886: @node mt_gkz.kronecker_prd,,, Secondary equation
887: @subsection @code{mt_gkz.kronecker_prd}
888: @comment --- 索引用キーワード
889: @findex mt_gkz.kronecker_prd
890:
891: @table @t
892: @item mt_gkz.kronecker_prd(@var{A},@var{B})
893: :: It returns the Kronecker product of @var{A} and @var{B}.
894: @end table
895:
896: @comment --- 引数の簡単な説明 ---
897: @table @var
898: @item return
1.2 takayama 899: 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 900: @item A,B
901: list
902: @end table
903:
904:
905: @comment --- @example〜@end example は実行例の表示 ---
906:
907: @example
908: [2644] A=[[a,b],[c,d]];
909: [[a,b],[c,d]]
910: [2645] B=[[e,f],[g,h]];
911: [[e,f],[g,h]]
1.2 takayama 912: [2646] mt_gkz.kronecker_prd(A,B);
1.1 takayama 913: [ e*a f*a e*b f*b ]
914: [ g*a h*a g*b h*b ]
915: [ e*c f*c e*d f*d ]
916: [ g*c h*c g*d h*d ]
917: @end example
918:
919:
920:
921:
922:
923:
924:
925:
926: @node mt_gkz.secondary_eq,,, Secondary equation
927: @subsection @code{mt_gkz.secondary_eq}
928: @comment --- 索引用キーワード
929: @findex mt_gkz.secondary_eq
930:
931: @table @t
932: @item mt_gkz.secondary_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
933: :: It returns the secondary equation with respect to cocycles defined by Rvec.
934: @end table
935:
936:
937: @table @var
938: @item return
939: a list of coefficients of the Pfaffian system corresponding to the secondary equation (cf. equation (8) of [MT2020]).
940: @item A,Beta,Ap,Rvec,DirX
941: see @code{pfaff_eq}
942: @end table
943:
944:
945:
946: @comment --- ここで関数の詳しい説明 ---
947: @comment --- @itemize〜@end itemize は箇条書き ---
948: @comment --- @bullet は黒点付き ---
949: @itemize @bullet
950: @item
951: 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}.
952: @item
953: 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).
954: @item Options xrule, shift, b_ideal,cg.
955: Same as @code{pfaff_eq}.
956: @end itemize
957:
958: @comment --- @example〜@end example は実行例の表示 ---
959: Example:
960: @example
961: [2647] Beta=[b1,b2,b3]$
962: [2648] DirX=[dx1,dx4]$
963: [2649] Rvec=[[1,0,0,0],[0,0,1,0]]$
964: [2650] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
965: [2651] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
966: [2652] Xrule=[[x2,1],[x3,1]]$
1.2 takayama 967: [2653] P=mt_gkz.secondary_eq(A,Beta,Ap,Rvec,DirX|xrule=Xrule)$
1.1 takayama 968: --snip--
969: [2654] length(P);
970: 2
971: [2655] P[0];
972: [[(-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),
973: (-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
974: ((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,
975: (-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
976: ((-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),
977: (b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]
978: <--- Paffian matrix in x1 direction.
979: [2656] P[1];
980: [[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
981: ((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,
982: (-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
983: ((-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),
984: (b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]]
985: <--- Paffian matrix in x4 direction.
986: @end example
987:
988: @comment --- 参照(リンク)を書く ---
989: @table @t
990: @item Refer to
991: @ref{mt_gkz.pfaff_eq}
992: @end table
993: @comment おわり.
994:
995:
996:
997:
998: @node mt_gkz.generate_maple_file_IC,,, Secondary equation
999: @subsection @code{mt_gkz.generate_maple_file_IC}
1000: @comment --- 索引用キーワード
1001: @findex mt_gkz.generate_maple_file_IC
1002:
1003: @table @t
1004: @item mt_gkz.generate_maple_file_IC(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
1005: :: It returns the maple input for a solver of a Pfaffian system IntegrableConnections[RationalSolutions].
1006: @end table
1007:
1008: @comment --- 引数の簡単な説明 ---
1009: @table @var
1010: @item return
1011: 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}).
1012: @item A,Beta,Ap,Rvec,DirX
1013: see @code{pfaff_eq}.
1014: @end table
1015:
1016: @comment --- ここで関数の詳しい説明 ---
1017: @comment --- @itemize〜@end itemize は箇条書き ---
1018: @comment --- @bullet は黒点付き ---
1019: @itemize @bullet
1020: @item
1021: 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].
1022: @item
1023: 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,...].
1024: @item Options xrule, shift, b_ideal,cg.
1025: Same as @code{pfaff_eq}.
1026: @item Option filename.
1027: 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".
1028: @end itemize
1029:
1030: @comment --- @example〜@end example は実行例の表示 ---
1031: Example:
1032: @example
1033: [2681] Beta=[b1,b2,1/3]$
1034: [2682] DirX=[dx1,dx4]$
1035: [2683] Rvec=[[1,0,0,0],[0,0,1,0]]$
1036: [2684] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1037: [2685] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
1038: [2687] Xrule=[[x2,1],[x3,1]]$
1.2 takayama 1039: [2688] mt_gkz.generate_maple_file_IC(A,Beta,Ap,Rvec,DirX|xrule=Xrule,filename="Test.ml")$
1.1 takayama 1040:
1041:
1042: //A file named Test.ml is automatically generated as follows:
1043:
1044:
1045:
1046: with(OreModules);
1047: with(IntegrableConnections);
1048: with(linalg);
1049: C:=[Matrix([[(-2*x1^3*x4^2+4*x1^2*x4-2*x1)/(x1^4*x4^2-2*x1^3*x4+x1^2),
1050: (b2*x4)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1),0],[(b1)/(x1*x4-1),
1051: ((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,
1052: (-b2*x4)/(x1^2*x4-x1)],[(-b1)/(x1*x4-1),0,
1053: ((-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),
1054: (b2*x4)/(x1^2*x4-x1)],[0,(-b1)/(x1*x4-1),(b1)/(x1*x4-1),0]]),
1055: Matrix([[0,(b2)/(x1*x4-1),(-b2)/(x1*x4-1),0],[(b1*x1)/(x1*x4^2-x4),
1056: ((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,
1057: (-b2)/(x1*x4-1)],[(-b1*x1)/(x1*x4^2-x4),0,
1058: ((-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),
1059: (b2)/(x1*x4-1)],[0,(-b1*x1)/(x1*x4^2-x4),(b1*x1)/(x1*x4^2-x4),0]])];
1060: RatSols:=RationalSolutions(C,[x1,x4],['param',[b1,b2]]);
1061:
1062:
1063: /*
1064: If you run the output file on maple, you obtain a rational solution of
1065: the secondary equation.
1066: */
1067:
1068: [b2*(3*b1-1)/(b1*x1^2)]
1069: RatSols:=[3*b2/x1 ]
1070: [3*b2/x1 ]
1071: [3*b2-1 ]
1072:
1073: /*
1074: Note that the 4 entries of this vector correspond to entries of a 2 by 2 matrix.
1075: They are aligned as (1,1), (1,2), (2,1) (2,2) from the top.
1076: */
1077: @end example
1078:
1079: @*
1080:
1081: @comment --- 参照(リンク)を書く ---
1082: @table @t
1083: @item Refer to
1084: @ref{mt_gkz.pfaff_eq}
1085: @end table
1086: @comment おわり.
1087:
1088:
1089:
1090:
1091:
1092:
1093:
1094:
1095: @node mt_gkz.generate_maple_file_MR,,, Secondary equation
1096: @subsection @code{mt_gkz.generate_maple_file_MR}
1097: @comment --- 索引用キーワード
1098: @findex mt_gkz.generate_maple_file_MR
1099:
1100: @table @t
1101: @item mt_gkz.generate_maple_file_MR(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX},@var{D1},@var{D2})
1102: :: It returns the maple input for a solver of a Pfaffian system MorphismsRat[OreMorphisms].
1103: @end table
1104:
1105: @comment --- 引数の簡単な説明 ---
1106: @table @var
1107: @item return
1108: 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.
1109: @item A,Beta,Ap,Rvec,DirX
1110: see @code{pfaff_eq}.
1111: @item D1,D2
1112: Positive integers. D1 (resp. D2) is the upper bound of the degree of the numerator (resp. denominator) of the solution.
1113: @end table
1114:
1115: @comment --- ここで関数の詳しい説明 ---
1116: @comment --- @itemize〜@end itemize は箇条書き ---
1117: @comment --- @bullet は黒点付き ---
1118: @itemize @bullet
1119: @item
1120: 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]).
1121: @item
1122: 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.
1123: @item Options xrule, shift, b_ideal,cg.
1124: Same as @code{pfaff_eq}.
1125: @item Option filename.
1126: You can specify the file name as in @code{generate_maple_file_IC}.
1127: @item
1128: 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].
1129: @end itemize
1130:
1131: @comment --- @example〜@end example は実行例の表示 ---
1132: Example:
1133: @example
1134: [2668] Beta=[b1,b2,1/3]$
1135: [2669] DirX=[dx1,dx4]$
1136: [2670] Rvec=[[1,0,0,0],[0,0,1,0]]$
1137: [2671] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1138: [2672] Ap=[[1,1,0,0],[0,0,1,1],[0,0,0,0]]$
1139: [2673] Xvar=[x1,x4]$
1140: [2674] Xrule=[[x2,1],[x3,1]]$
1.2 takayama 1141: [2675] mt_gkz.generate_maple_file_MR(A,Beta,Ap,Rvec,DirX,2,2|xrule=Xrule)$
1.1 takayama 1142:
1143:
1144: //A file "auto-generated-MR.ml" is automatically generated as follows:
1145:
1146:
1147: with(OreModules);
1148: with(OreMorphisms);
1149: with(linalg);
1150: Alg:=DefineOreAlgebra(diff=[dx1,x1],diff=[dx4,x4],polynom=[x1,x4],comm=[b1,b2]);
1151: P:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
1152: -Matrix([[((b1+b2-4/3)*x1*x4-b1+4/3)/(x1^2*x4-x1),(-b2*x4)/(x1^2*x4-x1)],
1153: [(-b1)/(x1*x4-1),(b1*x4)/(x1*x4-1)],[(b2*x1)/(x1*x4-1),(-b2)/(x1*x4-1)],
1154: [(-b1*x1)/(x1*x4^2-x4),(1/3*x1*x4+b1-1/3)/(x1*x4^2-x4)]]);
1155: Q:=Matrix([[dx1,0],[0,dx1],[dx4,0],[0,dx4]])
1156: +Matrix([[((-b1-b2-2/3)*x1*x4+b1+2/3)/(x1^2*x4-x1),(b1)/(x1*x4-1)],
1157: [(b2*x4)/(x1^2*x4-x1),(-b1*x4)/(x1*x4-1)],[(-b2*x1)/(x1*x4-1),(b1*x1)/(x1*x4^2-x4)],
1158: [(b2)/(x1*x4-1),(-1/3*x1*x4-b1+1/3)/(x1*x4^2-x4)]]);
1159: RatSols:=MorphismsRat(P,Q,Alg,0,2,2);
1160:
1161: /*
1162: If you run the output file on maple, you obtain a vector RatSols.
1163: RatSols[1] is the rational solution of the secondary equation:
1164: */
1165:
1166: 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}})]
1167: [@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}})]
1168:
1169: /*
1170: 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}.
1171: */
1172: @end example
1173:
1174: @*
1175:
1176: @comment --- 参照(リンク)を書く ---
1177: @table @t
1178: @item Refer to
1179: @ref{mt_gkz.pfaff_eq}, @ref{mt_gkz.generate_maple_file_IC}.
1180: @end table
1181: @comment おわり.
1182:
1183:
1184:
1185:
1186:
1187:
1188:
1189:
1190:
1191:
1192:
1193:
1194: @node Normalizing constant,,, Cohomology intersection numbers
1195: @section Normalizing the cohomology intersection matrix
1196:
1197:
1198: @node mt_gkz.principal_normalizing_constant,,, Normalizing constant
1199: @subsection @code{mt_gkz.principal_normalizing_constant}
1200: @comment --- 索引用キーワード
1201: @findex mt_gkz.principal_normalizing_constant
1202:
1203: @table @t
1204: @item mt_gkz.principal_normalizing_constant(@var{A},@var{T},@var{Beta},@var{K})
1205: :: It returns the normalizing constant of the cohomology intersection matrix in terms of a regular triangulation T.
1206: @end table
1207:
1208: @comment --- 引数の簡単な説明 ---
1209: @table @var
1210: @item return
1.3 takayama 1211: 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 1212: @item A,Beta
1213: see @code{pfaff_eq}.
1214: @item T
1215: a regular triangulation of A.
1216: @item K
1217: The number of polynomial factors in the integrand. see [MT2020].
1218: @end table
1219:
1220: @comment --- ここで関数の詳しい説明 ---
1221: @comment --- @itemize〜@end itemize は箇条書き ---
1222: @comment --- @bullet は黒点付き ---
1223: @itemize @bullet
1224: @item
1.3 takayama 1225: 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 1226: @item
1.3 takayama 1227: One can find a regular triangulation by using a function @code{mt_gkz.regular_triangulation}.
1228: @item
1229: @code{mt_gkz.leading_terms} can be used more generally.
1.1 takayama 1230: @end itemize
1231:
1232: @comment --- @example〜@end example は実行例の表示 ---
1233: Example:
1234: @example
1235: [2676] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]]$
1236: [2677] Beta=[b1,b2,b3]$
1237: [2678] K=2$
1238: [2679] T=[[1,2,3],[2,3,4]]$
1.2 takayama 1239: [2680] mt_gkz.principal_normalizing_constant(A,T,Beta,K);
1.1 takayama 1240: (-b1-b2)/(b3*b1+b3*b2-b3^2)
1241: @end example
1242:
1243: @comment --- 参照(リンク)を書く ---
1244: @table @t
1245: @item Refer to
1.3 takayama 1246: @ref{mt_gkz.leading_terms}.
1.1 takayama 1247: @end table
1248: @comment おわり.
1249:
1250:
1251:
1252:
1253:
1254:
1255:
1256:
1.3 takayama 1257: @node mt_gkz.leading_terms,,, Normalizing constant
1258: @subsection @code{mt_gkz.leading_terms}
1.1 takayama 1259: @comment --- 索引用キーワード
1260: @findex mt_gkz.leading_terms
1261:
1262: @table @t
1.3 takayama 1263: @item mt_gkz.leading_terms(@var{A},@var{Beta},@var{W},@var{Q1},@var{Q2},@var{K},@var{N})
1.1 takayama 1264: :: It returns the W-leading terms of a cohomology intersection number specified by Q1 and Q2 up to W-degree=(minimum W-degree)+N.
1265: @end table
1266:
1267: @comment --- 引数の簡単な説明 ---
1268: @table @var
1269: @item return
1.3 takayama 1270: 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 1271: @item A,Beta
1272: see @code{pfaff_eq}.
1273: @item W
1274: a positive and integral weight vector.
1275: @item Q1,Q2
1276: @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}.
1277: @item K
1278: The number of polynomial factors in the integrand. see [MT2020].
1279: @item N
1280: A positive integer.
1281: @end table
1282:
1283: @comment --- ここで関数の詳しい説明 ---
1284: @comment --- @itemize〜@end itemize は箇条書き ---
1285: @comment --- @bullet は黒点付き ---
1286: @itemize @bullet
1287: @item
1288: 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}.
1289: @item
1.3 takayama 1290: 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 1291: @item
1292: 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.
1293: @item Option xrule.
1294: Same as @code{pfaff_eq}.
1295: @end itemize
1296:
1297: @comment --- @example〜@end example は実行例の表示 ---
1298: Example:
1299: @example
1300: [2922] Beta=[b1,b2,1/3];
1301: [b1,b2,1/3]
1302: [2923] Q=[[1,0,0],[0,1,0]];
1303: [[1,0,0],[0,1,0]]
1304: [2924] A=[[1,1,0,0],[0,0,1,1],[0,1,0,1]];
1305: [[1,1,0,0],[0,0,1,1],[0,1,0,1]]
1306: [2925] W=[1,0,0,0];
1307: [1,0,0,0]
1308: [2926] K=2;
1309: 2
1310: [2927] N=2;
1311: 2
1.3 takayama 1312: [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 1313: --snip--
1314: [2929] NC;
1315: [[(-3)/(x1),-5],[0,-4],[0,-3]]
1316:
1317:
1318: /*
1319: This output means that the W-leading term of the (1,2) entry of the cohomology
1320: 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
1321: intersection matrix is given by
1322: */
1323:
1324: [-(3*b1-1)/(b1*x1^2) -3/x1 ]
1325: [-3/x1 -(3*b2-1)/b2]]
1326:
1327:
1328: //divided by 2@math{\pi\sqrt{-1}}.
1329: @end example
1330:
1331:
1332: @comment --- 参照(リンク)を書く ---
1333: @table @t
1334: @item Refer to
1.3 takayama 1335: @ref{mt_gkz.leading_terms}, @ref{mt_gkz.generate_maple_file_IC}, @ref{mt_gkz.generate_maple_file_MR}.
1.1 takayama 1336: @end table
1337: @comment おわり.
1338:
1339:
1340:
1341:
1342:
1343: @node mt_gkz.leading_term_rat,,, Normalizing constant
1344: @subsection @code{mt_gkz.leading_term_rat}
1345: @comment --- 索引用キーワード
1346: @findex mt_gkz.leading_term_rat
1347:
1348: @table @t
1349: @item mt_gkz.leading_term_rat(@var{P},@var{W},@var{V})
1350: :: It returns the W-leading term of a rational function P depending on variables V.
1351: @end table
1352:
1353: @comment --- 引数の簡単な説明 ---
1354: @table @var
1355: @item return
1356: It returns the W-leading term of a rational function P.
1357: @item P
1358: a rational function.
1359: @item W
1360: a weight vector.
1361: @item V
1362: a list of variables of P.
1363: @end table
1364:
1365: @comment --- ここで関数の詳しい説明 ---
1366: @comment --- @itemize〜@end itemize は箇条書き ---
1367: @comment --- @bullet は黒点付き ---
1368: @itemize @bullet
1369: @item
1.3 takayama 1370: This function is supposed to be combined with @code{leading_terms} to compute the leading term of a cohomology intersection number.
1.1 takayama 1371: @item
1372: 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."
1373: @end itemize
1374:
1375:
1376: @comment --- 参照(リンク)を書く ---
1377: @table @t
1378: @item Refer to
1.3 takayama 1379: @ref{mt_gkz.leading_terms}.
1.1 takayama 1380: @end table
1381: @comment おわり.
1382:
1383:
1384:
1385:
1386:
1387:
1388: @node Regular triangulations,,, Cohomology intersection numbers
1389: @section Regular triangulations
1390:
1391: @comment --- 個々の関数の説明 ---
1392: @comment --- section 名を正確に ---
1393: @node mt_gkz.toric_gen_initial,,, Regular triangulations
1394: @node mt_gkz.regular_triangulation,,, Regular triangulations
1395: @node mt_gkz.top_standard_pairs,,, Regular triangulations
1396: @subsection @code{mt_gkz.toric_gen_initial}, @code{mt_gkz.regular_triangulation}, @code{mt_gkz.top_standard_pairs}
1397: @comment --- 索引用キーワード
1398: @findex mt_gkz.toric_gen_initial
1399: @findex mt_gkz.regular_triangulation
1400: @findex mt_gkz.top_standard_pairs
1401:
1402: @table @t
1403: @item mt_gkz.toric_gen_initial(@var{A},@var{W})
1404: @item mt_gkz.regular_triangulation(@var{A},@var{W})
1405: @item mt_gkz.top_standard_pairs(@var{A},@var{W})
1406: :: 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.
1407: @end table
1408:
1409: @comment --- 引数の簡単な説明 ---
1410: @table @var
1411: @item return
1412: @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}.
1413: @item return
1414: @code{regular_triangulation} returns a list of simplices of a regular triangulation @math{T_W} specified by the weight W.
1415: @item return
1416: @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.
1417: @item A
1418: a configuration matrix.
1419: @item W
1420: a positive weight vector.
1421: @end table
1422:
1423: @comment --- ここで関数の詳しい説明 ---
1424: @comment --- @itemize〜@end itemize は箇条書き ---
1425: @comment --- @bullet は黒点付き ---
1426: @itemize @bullet
1427: @item
1428: As for the definition of the standard pair, see Chapter 3 of [SST00].
1429: @item
1430: 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}}.
1431: @item
1.3 takayama 1432: 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 1433: @item
1.3 takayama 1434: These functions are utilized in @code{leading_terms}.
1.1 takayama 1435: @end itemize
1436:
1437: @comment --- @example〜@end example は実行例の表示 ---
1438: Example: An example of a non-unimodular triangulation and non-trivial standard pairs.
1439: @example
1440: [3256] A=[[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]];
1441: [[1,1,1,1,1],[0,1,0,2,0],[0,0,1,0,2]]
1442: [3257] W=[2,0,1,2,2];
1443: [2,0,1,2,2]
1.3 takayama 1444: [3258] mt_gkz.toric_gen_initial(A,W);
1.1 takayama 1445: --snip--
1446: [[x1*x5,x1*x4,x3^2*x4],[x1,x2,x3,x4,x5]]
1.3 takayama 1447: [3259] mt_gkz.regular_triangulation(A,W);
1.1 takayama 1448: --snip--
1449: [[2,4,5],[2,3,5],[1,2,3]]
1.3 takayama 1450: [3260] mt_gkz.top_standard_pairs(A,W);
1.1 takayama 1451: --snip--
1452: [[[[0,0],[0,1]],[2,4,5]],[[[0,0]],[2,3,5]],[[[0,0]],[1,2,3]]]
1453:
1454: /*
1455: This means that the regular triangulation of the configuration matrix A is
1456: given by @math{T=\{\{2,4,5\},\{2,3,5\},\{1,2,3\}\}}. The normalized volumes of these simplices
1457: are 2,1 and 1. Moreover, the top-dimensional standard pairs are
1458: @math{(1,\{2,4,5\}),(\partial_3,\{2,4,5\})}, @math{(1,\{2,3,5\})},@math{(1,\{1,2,3\})}.
1459: */
1460: @end example
1461:
1462:
1463:
1464: @comment --- 参照(リンク)を書く ---
1465: @table @t
1466: @item Refer to
1.3 takayama 1467: @ref{mt_gkz.leading_terms}.
1.1 takayama 1468: @end table
1469: @comment おわり.
1470:
1471:
1472:
1473:
1474:
1475:
1476:
1477: @comment --- おまじない ---
1478: @node Index,,, Top
1479: @unnumbered Index
1480: @printindex fn
1481: @printindex cp
1482: @iftex
1483: @vfill @eject
1484: @end iftex
1485: @summarycontents
1486: @contents
1487: @bye
1488: @comment --- おまじない終り ---
1489:
1490: @comment *********************************************************
1491: @comment ********* template
1492: @comment **********************************************************
1493: @comment --- 個々の関数の説明 ---
1494: @comment --- section 名を正確に ---
1495: @node mt_gkz.pfaff_eq,,, Pfaff equation for given cocycles
1496: @subsection @code{mt_gkz.pfaff_eq}
1497: @comment --- 索引用キーワード
1498: @findex mt_gkz.pfaff_eq
1499:
1500: @table @t
1501: @item mt_gkz.pfaff_eq(@var{A},@var{Beta},@var{Ap},@var{Rvec},@var{DirX})
1502: :: It returns the Pfaff equation for the GKZ system defined by @var{A} and @var{Beta} with respect to cocycles defined by @var{Rvec}.
1503: @end table
1504:
1505: @comment --- 引数の簡単な説明 ---
1506: @table @var
1507: @item return
1508: a list of coefficients of the Pfaff equation with respect to the direction @var{DirX}
1509: @item A
1510: the matrix A of the GKZ system.
1511: @item Beta
1512: ...
1513: @end table
1514:
1515: @comment --- ここで関数の詳しい説明 ---
1516: @comment --- @itemize〜@end itemize は箇条書き ---
1517: @comment --- @bullet は黒点付き ---
1518: @itemize @bullet
1519: @item
1520: The independent variables are @code{x1,x2,x3,...} (@math{x_1, x_2, x_3, \ldots}).
1521: @end itemize
1522:
1523: @comment --- @example〜@end example は実行例の表示 ---
1524: Example: Gauss hypergeometric system, see [GM2020] example ??.
1525: @example
1526: [1883] import("mt_gkz.rr");
1527: @end example
1528:
1529: @comment --- 参照(リンク)を書く ---
1530: @table @t
1531: @item Refer to
1532: @ref{mt_gkz.pfaff_eq}
1533: @end table
1534: @comment おわり.
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